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Find the constant force in the **(a)** cgs system and **(b)** SI system needed to accelerate a mass of 10,000 g moving along a straight line from a speed of 54 km/h to 108 km/h in 5 minutes.\n\nAssume the motion to be in the direction of the positive x-axis. Then if \( v_1 \) and \( v_2 \) are the velocities, we have f... | **(a) In the cgs system**\n\n\[\nm = 10^4 \text{ g}, \quad v_1 = 54 \text{ km/h} = 1.5 \times 10^4 \text{ cm/s}, \quad v_2 = 3.0 \times 10^4 \text{ cm/s}, \quad t = 300 \text{ s}\n\]\n\nThen\n\n\[\nF = ma = m \left( \frac{v_2 - v_1}{t} \right) = (10^4 \text{ g}) \left( \frac{1.5 \times 10^4 \text{ cm/s}}{3 \times 10^2 ... | So I've got this physics problem here. I need to find the constant force required to accelerate a mass of 10,000 grams from 54 kilometers per hour to 108 kilometers per hour in 5 minutes. And I need to express this force in both the cgs system and the SI system. Alright, let's break this down step by step.
First, I ne... |
A heavy uniform rod of length l is hinged at its upper end to a horizontal axle O, as shown in Fig. 428. When the rod is vertical, it is given an angular velocity ω_0 = 3 \sqrt{\frac{g}{l}}. Because of this, the rod performs half of a turn and leaves the axle O. Determine the path of the centre of gravity and the angul... | (1) The parabola y_c = \frac{l}{2} - \frac{2}{3l} x_c^2;\n\n(2) ω = \sqrt{\frac{3g}{l}}. | So I've got this physics problem here. Let's see what it's asking. There's a heavy uniform rod of length L, hinged at its upper end to a horizontal axle O. When the rod is vertical, it's given an angular velocity ω₀ equals 3 times the square root of (g over L). Because of this, the rod performs half a turn and leaves t... |
A particle moves in the \( xy \) plane in a force field given by \( \mathbf{F} = - x \mathbf{i} - y \mathbf{j} \). Prove that in general it will move in an elliptical path. | If the particle has mass \( m \), its equation of motion is\n\n\[m \frac{d^2 \mathbf{r}}{dt^2} = \mathbf{F} = -x \mathbf{i} - y \mathbf{j} \tag{1}\]\n\nor, since \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} \),\n\n\[m \frac{d^2x}{dt^2} \mathbf{i} + m \frac{d^2y}{dt^2} \mathbf{j} = -x \mathbf{i} - y \mathbf{j}\]\n\nThen\... | So I've got this problem here: there's a particle moving in the xy-plane, and it's in a force field given by F = -x i - y j. I need to prove that in general, it will move in an elliptical path. Okay, let's see.
First, I know that force is related to acceleration through Newton's second law, F = m a, where m is the mas... |
8.17 If \( f(x) \) is even, show that (a) \( a_n = \frac{2}{l} \int_0^l f(x) \cos \frac{n \pi x}{l} \, dx \), (b) \( b_n = 0 \). | (a) \[ a_n = \frac{1}{l} \int_{-l}^l f(x) \cos \frac{n \pi x}{l} \, dx = \frac{1}{l} \int_{-l}^0 f(x) \cos \frac{n \pi x}{l} \, dx + \frac{1}{l} \int_0^l f(x) \cos \frac{n \pi x}{l} \, dx \] Letting \( x = -u \), \[ \frac{1}{l} \int_0^{-l} f(x) \cos \frac{n \pi x}{l} \, dx = \frac{1}{l} \int_l^0 f(-u) \cos \left( \frac... | To solve this problem, we need to understand the properties of even functions and how they relate to Fourier series coefficients. Let's break it down step by step.
### Step 1: Understanding Even Functions
A function \( f(x) \) is even if it satisfies the condition:
\[
f(-x) = f(x) \quad \text{for all } x \text{ in t... |
A prism accelerates along a horizontal plane. Its side edge makes an angle **α** with the horizontal. What should the acceleration of the prism be to ensure that a weight, resting on the side edge, does not move with respect to the prism? | w = g \tan \alpha. | So I've got this problem here about a prism accelerating on a horizontal plane. The prism has a side edge that makes an angle α with the horizontal. There's a weight resting on this side edge, and I need to find out what acceleration the prism should have so that the weight doesn't move relative to the prism. Alright, ... |
A mechanical system \(A\) consists of a mass \(m_1\) which is rigidly connected at \(B\) to the piston of a dashpot. The mechanical system is suspended by a spring of stiffness \(c_1\) to a platform whose motion is defined by the law \(\xi = \xi(t)\). The frame of the dashpot of mass \(m_2\) is placed on a spring of st... | \[\nm_1 \ddot{x}_1 + \beta \dot{x}_1 - \beta \dot{x}_2 + (c_1 + c_2)x_1 - c_2x_2 = c_1 \xi(t);\n\]\n\[\nm_2 \ddot{x}_2 - \beta \dot{x}_1 + \beta \dot{x}_2 - c_2x_1 + c_2x_2 = 0.\n\] | Let's tackle this problem step by step. I'm a bit rusty on mechanical systems, but I think I can figure this out.
First, I need to understand the setup. There's a mechanical system A with a mass \( m_1 \) attached rigidly to the piston of a dashpot. This system is suspended by a spring with stiffness \( c_1 \) to a pl... |
A particle of mass \( m \) is repelled from a fixed centre \( O \) by a force which changes in accordance with the law: \( F = k^2 mr \), where \( r \) is a radius vector of the particle. Initially, the particle was located at \( M_0(a, 0) \) and had a velocity \( v_0 \) directed parallel to the axis \( y \). Find the ... | \[ \left( \frac{x}{a} \right)^2 - \left(\frac{ky}{v_0} \right)^2 = 1 \quad \text{(hyperbola)}. \] | So I have this problem here about a particle being repelled from a fixed center by a force that depends on the distance from the center. The force is given by \( F = k^2 m r \), where \( r \) is the radius vector of the particle. The particle starts at position \( M_0(a, 0) \) with an initial velocity \( v_0 \) directe... |
Solve \( \frac{d^2y}{dx^2} + 10 \frac{dy}{dx} + 25y = 20 \cos 2x \). | The complementary solution [by Problem C.13] is y_c = c_1 e^{-5x} + c_2 xe^{-5x} Since the right side has the term \cos 2x, we are led to the trial solution y_p = A \cos 2x + B \sin 2x Substitution into the given equation yields, after simplifying, (21A + 20B) \cos 2x + (21B - 20A) \sin 2x = 20 \cos 2x Equating coeffic... | To solve the differential equation \(\frac{d^2y}{dx^2} + 10 \frac{dy}{dx} + 25y = 20 \cos 2x\), we need to find both the complementary function (solution to the homogeneous equation) and the particular integral (solution to the non-homogeneous equation). The general solution will be the sum of these two parts.
### Ste... |
A weight of \( 20 \, \text{kgf} \) is to be lifted up an inclined plane through a distance of 6 m. The plane makes an angle of \( 30^\circ \) to the horizontal and the coefficient of friction is 0.01. Compute the work done to lift the weight. | \( 61.04 \, \text{kgfm} = 598 \, \text{joules}. \) | So I've got this problem here about lifting a weight up an inclined plane. Let me see if I can figure this out step by step. The weight is 20 kgf, and it's being lifted a distance of 6 meters up a plane that's inclined at 30 degrees to the horizontal. Also, there's a coefficient of friction of 0.01. I need to compute t... |
If \( \mathbf{A} = A_1 \mathbf{i} + A_2 \mathbf{j} + A_3 \mathbf{k} \) and \( \mathbf{B} = B_1 \mathbf{i} + B_2 \mathbf{j} + B_3 \mathbf{k} \), prove that \( \mathbf{A} \cdot \mathbf{B} = A_1 B_1 + A_2 B_2 + A_3 B_3 \). | \[\begin{align*}\mathbf{A} \cdot \mathbf{B} &= (A_1 \mathbf{i} + A_2 \mathbf{j} + A_3 \mathbf{k}) \cdot (B_1 \mathbf{i} + B_2 \mathbf{j} + B_3 \mathbf{k})\\&= A_1 \cdot (B_1 \mathbf{i} + B_2 \mathbf{j} + B_3 \mathbf{k}) + A_2 \cdot (B_1 \mathbf{i} + B_2 \mathbf{j} + B_3 \mathbf{k}) + A_3 \cdot (B_1 \mathbf{i} + B_2 \ma... | The dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is defined as the product of their magnitudes and the cosine of the angle between them. However, in component form, it's often expressed as the sum of the products of their corresponding components. Let's prove that \(\mathbf{A} \cdot \mathbf{B} = A_1 B_1... |
Find the least coefficient of friction for an inclined plane of angle \( \alpha \) in order that a solid cylinder will roll down it without slipping. | \n\[\n\frac{1}{3} \tan \alpha\n\] | So I've got this problem here about finding the least coefficient of friction for an inclined plane with an angle alpha, so that a solid cylinder can roll down without slipping. I need to figure out what the minimum coefficient of friction should be to prevent slipping.
First, I need to understand the forces acting on... |
A disk suspended from the end of an elastic wire performs torsion oscillations in a liquid. The moment of inertia of the disk about the axis of the wire is \( J \). The moment of the couple required to twist the wire through one radian equals \( c \). The moment of resistance to motion is \( \alpha S \omega \), where \... | T = \frac{4\pi J}{\sqrt{4cJ - \alpha^2 S^2}}. | To determine the period of oscillation of the disk in the liquid, we need to analyze the forces and torques acting on the disk during its torsional oscillations. Let's break down the problem step by step.
### Step 1: Identify the forces and torques involved
1. **Restoring Torque due to the wire:**
- The wire provi... |
The data given are the same as in the previous problem, except that to determine the coefficient \( c \), another experiment is carried out: a homogeneous circular disk of weight \( P \) and radius \( r \) is mounted on the body, whose moment of inertia is to be determined. If the period of oscillations of the body wit... | J_z = \frac{Pr^2}{2g} \cdot \frac{\tau_1^2}{\tau_2^2 - \tau_1^2}. | So I have this problem here, and it's similar to the previous one but with some differences. Let me try to understand what's being asked.
We have a body, and we need to find its moment of inertia \( J_z \) around the z-axis. To do this, we're doing two experiments. First, we measure the period of oscillations of the b... |
**(a)** Show that the function \( A \cos \omega t + B \sin \omega t \) can be written as \( C \cos (\omega t - \phi) \) where \( C = \sqrt{A^2 + B^2} \) and \( \phi = \tan^{-1}(B/A) \). **(b)** Find the amplitude, period and frequency of the function in (a). | **(a)** \n\n\[\nA \cos \omega t + B \sin \omega t = \sqrt{A^2 + B^2} \left( \frac{A}{\sqrt{A^2 + B^2}} \cos \omega t + \frac{B}{\sqrt{A^2 + B^2}} \sin \omega t \right)\n\]\n\n\[\n= \sqrt{A^2 + B^2}(\cos \phi \cos \omega t + \sin \phi \sin \omega t)\n\]\n\n\[\n= \sqrt{A^2 + B^2} \cos (\omega t - \phi) = C \cos (\omega t... | So I've got this problem here, and I need to show that the function \( a \cos \omega t + b \sin \omega t \) can be written as \( c \cos (\omega t - \phi) \), where \( c = \sqrt{a^2 + b^2} \) and \( \phi = \tan^{-1}(b/a) \). Then, part b asks for the amplitude, period, and frequency of this function. Alright, let's tack... |
A particle moves with simple harmonic motion in a straight line. Its maximum speed is 6 m/s and its maximum acceleration is 24 m/s. Find the period and frequency of the motion. | \( \pi/2 \, s, \, 2/\pi \, \text{Hz} \) | So I've got this problem here about a particle moving with simple harmonic motion. It says that the particle moves in a straight line with a maximum speed of 6 meters per second and a maximum acceleration of 24 meters per second squared. I need to find the period and frequency of this motion.
First, I need to recall t... |
Two particles have position vectors given by \( \mathbf{r}_1 = 2t\mathbf{i} - t^2\mathbf{j} + (3t^2 - 4t)\mathbf{k} \) and \( \mathbf{r}_2 = (5t^2 - 12t + 4)\mathbf{i} + t^3\mathbf{j} - 3t\mathbf{k} \). Find (a) the relative velocity and (b) the relative acceleration of the second particle with respect to the first at ... | (a) The velocities of the particles at \( t = 2 \) are respectively \[ \begin{align*} \mathbf{v}_1 &= \dot{\mathbf{r}}_1 = 2\mathbf{i} - 2t\mathbf{j} + (6t - 4)\mathbf{k} \bigg|_{t=2} = 2\mathbf{i} - 4\mathbf{j} + 8\mathbf{k} \\ \mathbf{v}_2 &= \dot{\mathbf{r}}_2 = (10t - 12)\mathbf{i} + 3t^2\mathbf{j} - 3\mathbf{k} \b... | So I've got this problem here about two particles with position vectors that depend on time, and I need to find the relative velocity and relative acceleration of the second particle with respect to the first at a specific time, which is t equals 2. Alright, let's break this down step by step.
First, I need to underst... |
What force should be applied in order to give a start to a 50-kgf carriage along the horizontal plane of a lathe bed? Lubrication is poor, so the coefficient of static friction is only 0.15. | \[\text{Ans. } 7.5 \text{ kgf}.\] | To determine the force required to start moving a 50-kgf carriage on a lathe bed with a coefficient of static friction of 0.15, we need to calculate the force of static friction and then apply a force slightly greater than that to overcome it.
First, let's understand the units. The mass of the carriage is given in kil... |
(1) To find the radius of gyration of an elliptic area, of uniform thickness and density, about its principal axes. | Let \( \rho \) represent the uniform density of the area, and \( r \) its indefinitely small thickness; then, \( x, y \) denoting the coordinates of any point of the curve referred to \( a, b \), as axes of coordinates, we have for the moment of inertia, about the axis \( a \), of a quadrant of the ellipse,\n\n\[ MK^2 ... | So I have this problem here: I need to find the radius of gyration of an elliptic area with uniform thickness and density about its principal axes. Okay, let's break this down. First, what is the radius of gyration? I remember it's a measure that describes the distribution of mass around an axis. It's kind of like how ... |
A box of the weight \( P \) rests on a rough horizontal surface whose coefficient of friction is \( f \) (Fig. 31). Determine at what angle \( \beta \) the minimum force \( Q \) must be applied to move the box, and the value of this force. | **Ans.** \( \beta = \arctan f \); \( Q_{\text{min}} = \frac{fP}{\sqrt{1+f^{2}}} \). | So I've got this problem here. There's a box sitting on a rough horizontal surface, and the coefficient of friction between the box and the surface is \( f \). The weight of the box is \( P \). I need to find out at what angle \( \beta \) I should apply a force \( Q \) to move the box, and also what the minimum value o... |
A chain suspended at its extremities from two tacks in the same horizontal line, forms itself into a cycloid; to find the unit of mass at any point of the string and the weight of the arc between this and the lowest point. Let \( \omega \) denote the weight of the arc; then taking the ordinary equations to the cycloid ... | m = \frac{r \, \sec \frac{1}{2} \theta}{4a g}, \, \omega = r \tan \frac{1}{2} \theta. | I'm trying to understand this problem about a chain forming a cycloid and finding the unit of mass at any point and the weight of the arc between that point and the lowest point. The equations given are for a cycloid:
\[ x = a (\theta + \sin \theta), \quad y = a (1 - \cos \theta), \]
and the expressions for mass per ... |
A trolley weighing P runs down an inclined track AB which then forms a circular loop of radius a (Fig. 273). Determine the height h from which the trolley should start from rest in order to run round the circular loop without leaving the surface of the latter. Find also the value of the pressure N of the trolley on the... | h ≥ 2.5a; N = P (2h/a - 2 + 3 cos φ). | So I've got this physics problem here, and I'm going to try to work through it step by step. It's about a trolley running down an inclined track, then going around a circular loop, and I need to find the height from which it should start so that it doesn't leave the surface of the loop at any point. Also, I need to fin... |
Show that the force field given by \( \mathbf{F} = x^2 y z \hat{i} - x y z^2 \hat{k} \) is non-conservative. | We have:\n\n\\[\n\\nabla \\times \\mathbf{F} = \\begin{vmatrix}\n\\hat{i} & \\hat{j} & \\hat{k} \\\n\\frac{\\partial}{\\partial x} & \\frac{\\partial}{\\partial y} & \\frac{\\partial}{\\partial z} \\\nx^2 y z & 0 & -x y z^2\n\\end{vmatrix} = -xz \\hat{i} + (x^2 y + y z^2) \\hat{j} - x^2 z \\hat{k}\n\\]\n\nThen since \\... | To show that the force field \(\mathbf{F} = x^2 y z \hat{i} - x y z^2 \hat{k}\) is non-conservative, we need to demonstrate that the work done by the force along a closed path is not zero. Alternatively, we can show that the curl of the vector field is not zero, which is a necessary and sufficient condition for a vecto... |
Find \(\int_C \mathbf{F} \cdot d\mathbf{r}\) where \(\mathbf{F} = (x - 3y)\mathbf{i} + (y - 2x)\mathbf{j}\) and \(C\) is the closed curve in the \(xy\) plane, \(x = 2 \cos t, y = 3 \sin t\), \(z = 0\) from \(t = 0\) to \(t = 2\pi\). | 6π | To find the line integral \(\int_C \mathbf{F} \cdot d\mathbf{r}\), where \(\mathbf{F} = (x - 3y)\mathbf{i} + (y - 2x)\mathbf{j}\) and \(C\) is the closed curve defined by \(x = 2 \cos t\), \(y = 3 \sin t\), \(z = 0\) for \(t\) from \(0\) to \(2\pi\), we can proceed as follows:
First, parameterize the curve \(C\). Give... |
A ladder of length \( l \) and weight \( W_l \) has one end against a vertical wall which is frictionless and the other end on the ground assumed horizontal. The ladder makes an angle \( \alpha \) with the ground. Prove that a man of weight \( W_m \) will be able to climb the ladder without having it slip if the coeffi... | Let the ladder be represented by \( AB \) in Fig. 9-29 and choose an \( xy \) coordinate system as indicated.\n\nThe most dangerous situation in which the ladder would slip occurs when the man is at the top of the ladder. Hence we would require that the ladder be in equilibrium in such case.\n\nThe forces acting on the... | To solve this problem, we need to analyze the forces acting on the ladder and ensure that the ladder does not slip when the man climbs it. We will consider both translational and rotational equilibrium conditions.
### Step 1: Identify the forces acting on the ladder
1. **Weight of the ladder (\( W_l \))**: Acts downw... |
A pendulum consists of a bob of mass \( m \) which is suspended by an unstretched thread of length \( l \). The point of suspension of the pendulum moves in accordance with the law \(\xi = \xi_0(t)\) down the straight line making an angle \(\alpha\) with the horizontal. Derive the equation of motion of the pendulum. | \(\varphi + \frac{g}{l} \sin \varphi + \frac{\ddot{\xi}}{l} \cos (\varphi - \alpha) = 0\). | So I've got this problem here about a pendulum. Let's see what it's asking. There's a bob of mass \( m \) suspended by a thread of length \( l \), and the point of suspension is moving according to some law \(\xi = \xi_0(t)\) along a straight line that makes an angle \(\alpha\) with the horizontal. I need to derive the... |
(a) Suppose that the new position and momentum coordinates are taken to be \( w_\alpha \) and \( J_\alpha \) respectively. Prove that if \( \mathcal{H} \) is the new Hamiltonian,\n\n\[\dot{J}_\alpha = -\frac{\partial \mathcal{H}}{\partial w_\alpha}, \quad \dot{w}_\alpha = \frac{\partial \mathcal{H}}{\partial J_\alpha}.... | (a) By Hamilton's equations for the canonical coordinates \( Q_\alpha, P_\alpha \),\n\n\[\dot{P}_\alpha = -\frac{\partial \mathcal{H}}{\partial Q_\alpha}, \quad \dot{Q}_\alpha = \frac{\partial \mathcal{H}}{\partial P_\alpha}.\]\n\nThen since the new position and momentum coordinates are taken as \( Q_\alpha = w_\alpha ... | (a) To prove the given equations for the time evolution of the new position and momentum coordinates \( w_\alpha \) and \( J_\alpha \), we need to use the Hamiltonian formalism. In Hamiltonian mechanics, the time evolution of any dynamical variable is given by its Poisson bracket with the Hamiltonian. Specifically, for... |
A particle of mass \( m \) moves along a straight line under the influence of a constant force of magnitude \( F \). If its initial speed is \( v_0 \), find (a) the speed, (b) the velocity, and (c) the distance traveled after time \( t \). | (a) Assume that the straight line along which the particle \( P \) moves is the \( x \) axis, as shown in Fig. 3-6 above. Suppose that at time \( t \) the particle is at a distance \( x \) from origin \( O \). If \( i \) is a unit vector in the direction \( OP \) and \( v \) is the speed at time \( t \), then the veloc... | Let's tackle this problem step by step. We have a particle of mass \( m \) moving along a straight line under the influence of a constant force of magnitude \( F \), and its initial speed is \( v_0 \). We need to find its speed, velocity, and the distance traveled after time \( t \).
First, I need to recall Newton's s... |
A mass of 5000 kg moves on a straight line from a speed of 540 km/h to 720 km/h in 2 minutes. What is the impulse developed in this time? | **Method 1.**\n\nAssume that the mass travels in the direction of the positive \( x \) axis. In the SI system,\n\n\[v_1 = 540 \frac{\text{km}}{\text{h}} = \frac{540 \times 1000 \, \text{m}}{3600 \, \text{s}} = 1.5 \times 10^2 \frac{\text{m}}{\text{s}}\]\n\n\[v_2 = 720 \frac{\text{km}}{\text{h}} = \frac{720 \times 1000 ... | To find the impulse developed on the mass, we need to calculate the change in momentum of the object. Impulse is defined as the change in momentum, which is given by the formula:
\[ \text{Impulse} = m \Delta v \]
where:
- \( m \) is the mass of the object,
- \( \Delta v \) is the change in velocity.
First, let's con... |
A weight of \( 2000 \, \text{kgf} \) is moved up a plane inclined at an angle of \( 30^\circ \) to the horizontal. The coefficient of friction is 0.5. Determine the minimum work which should be done to lift the weight to a height of 5 m. | \( 18,660 \, \text{kgfm} = 183 \, \text{kilojoules}. \) | So I've got this physics problem here. Let me read it carefully.
"A weight of 2000 kgf is moved up a plane inclined at an angle of 30 degrees to the horizontal. The coefficient of friction is 0.5. Determine the minimum work which should be done to lift the weight to a height of 5 meters."
Alright, first things first,... |
The motion of a load released by an aircraft is given by the equations: \[ x = v_0 t, \, y = h - \frac{gt^2}{2} \] Axis \(Ox\) is horizontal and \(Oy\) is directed vertically upwards. **Determine:** 1. The equation of the path. 2. The velocity of the load (magnitude and direction) at the instant when it crosses the axi... | 1. \( y = h - g \frac{x^2}{2v_0^2} \); 2. \( v = \sqrt{v_0^2 + 2gh} \); \(\cos(v, x) = \frac{v_0}{v}\); \(\cos(v, y) = -\frac{\sqrt{2gh}}{v}\); 3. \( x = v_0 \sqrt{\frac{2h}{g}} \); 4. The vertical straight line at a distance \(v_0\) from the origin of the coordinates; \(v_{1y} = -g\). | So I've got this problem about the motion of a load released by an aircraft. The equations given are:
\[ x = v_0 t \]
\[ y = h - \frac{gt^2}{2} \]
Where:
- \(x\) is the horizontal position,
- \(y\) is the vertical position,
- \(v_0\) is the initial horizontal velocity,
- \(h\) is the initial height,
- \(g\) is ... |
Determine the velocity \(v\) and the acceleration \(w\) of the particle on the surface of the earth in the city of Leningrad taking into consideration only the rotation of the earth about its axis, and assuming that the latitude of the city of Leningrad is \(60^\circ\) and the radius of the earth is 6370 km. | \(v = 0.232 \text{ km/sec}; \quad w = 0.0169 \text{ m/sec}^2\). | So I need to find the velocity and acceleration of a particle on the Earth's surface in Leningrad, considering only the Earth's rotation about its axis. The latitude is 60 degrees, and the Earth's radius is 6370 km. Okay, let's break this down.
First, I need to understand what velocity and acceleration are in this con... |
Expand \( f(x) = \begin{cases} \n2 - x & 0 < x < 4 \\ \nx - 6 & 4 < x < 8 \n\end{cases} \) in a Fourier series of period 8. | \[\frac{16}{\pi^2} \left( \cos \frac{n\pi x}{4} + \frac{1}{3^2} \cos \frac{3\pi x}{4} + \frac{1}{5^2} \cos \frac{5\pi x}{4} + \cdots \right)\] | To expand the function \(f(x)\) in a Fourier series of period 8, we need to express it as a sum of sines and cosines that repeat every 8 units. The general form of a Fourier series with period \(2L\) is:
\[
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\... |
A tram runs along a straight track. After starting, the distance travelled by the tram is proportional to the cube of time. During the first minute the tram travelled 90 metres. Find the velocity and acceleration for \( t = 0 \) and \( t = 5 \) sec. Sketch the distance, velocity and acceleration curves. | \( v_0 = 0; \quad w_0 = 0; \quad v_5 = \frac{15}{8} \) m/min; \( w_5 = 45 \) m/min\(^2\). | So I've got this problem here about a tram that's running along a straight track. It says that the distance traveled by the tram is proportional to the cube of time, and that in the first minute, it traveled 90 meters. I need to find the velocity and acceleration at t=0 and t=5 seconds, and also sketch the distance, ve... |
Solve \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 4x = 0.\) | Letting x = e^{\alpha t}, we find \alpha^2 + 4\alpha + 4 = 0 or \alpha = -2, -2. Then the general solution is x = c_1e^{-2t} + c_2te^{-2t} = e^{-2t}(c_1 + c_2t). | To solve the differential equation \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 4x = 0\), we'll follow these steps:
1. **Identify the type of differential equation:**
This is a second-order linear homogeneous differential equation with constant coefficients.
2. **Form the characteristic equation:**
For a differential... |
A particle moves along the x axis in a force field having potential V = 1/2κx^2, κ > 0. (a) Determine the points of equilibrium and (b) investigate the stability. | (a) Equilibrium points occur where ∇V = 0 or in this case dV/dx = κx = 0 or x = 0 Thus there is only one equilibrium point, at x = 0. (b) Method 1. Since d^2V/dx^2 = k > 0, it follows that at x = 0, V is a minimum. Thus by Theorem 2.10, page 38, x = 0 is a point of stability. This is also seen from Problem 2.36 where i... | So I've got this problem here about a particle moving along the x-axis in a force field with a potential given by V = 1/2 κ x², where κ is greater than zero. I need to find the points of equilibrium and then investigate their stability. Alright, let's break this down step by step.
First, I need to understand what equi... |
Assume that a continuous string, which is fixed at its endpoints and vibrates transversely, is replaced by \( N \) particles of mass \( m \) at equal distances from each other. Determine the equations of motion of the particles.\n\nWe assume that the particles are connected to each other by taut, elastic strings having... | The solution involves deriving the equations of motion for the particles using Newton's second law and considering the forces due to neighboring particles. The derived equation is:\n\n\[\n\ddot{Y}_v = \frac{T}{ma} (Y_{v-1} - 2Y_v + Y_{v+1})\n\]\n\nThis equation accounts for the transverse forces acting on each particle... | So I've been trying to understand this problem about modeling a vibrating string with particles. The setup is that instead of dealing with a continuous string, we're breaking it down into N particles of mass m, all equally spaced apart by a distance a. The string is fixed at both ends, and we're only considering transv... |
A bicycle transmission consists of a chain passing over a gear \( A \) with 26 teeth and a gear \( B \) with 9 teeth. The gear \( B \) is rigidly connected with the wheel \( C \) of diameter 70 cm (Fig. 196). Determine the speed of the bicycle when the gear \( A \) makes one revolution per second while the wheel \( C \... | 22.87 km/h. | So I've got this problem about a bicycle transmission, and I need to figure out the speed of the bicycle based on some given information about the gears and the wheel. Let me see if I can break this down step by step.
First, there's a chain passing over two gears: gear A has 26 teeth, and gear B has 9 teeth. Gear B is... |
If \( \mathbf{A} = t \mathbf{i} - \sin t \mathbf{k} \) and \( \mathbf{B} = \cos t \mathbf{i} + \sin t \mathbf{j} + \mathbf{k} \), find \( \frac{d}{dt}(\mathbf{A} \cdot \mathbf{B}) \). | \(-t \sin t\) | So I have these two vector functions, A and B, which depend on t. A is given by t i minus sine of t k, and B is cosine of t i plus sine of t j plus k. I need to find the derivative with respect to t of their dot product. So, first things first, I need to find the dot product of A and B, and then take its derivative wit... |
A bead slides without friction on a frictionless wire in the shape of a cycloid [Fig. 11-10] with equations\n\n\[x = a(\theta - \sin \theta), \quad y = a(1 + \cos \theta)\]\n\nwhere \(0 \leq \theta \leq 2\pi\). Find (a) the Lagrangian function, (b) the equation of motion. | (a) Kinetic energy \(T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2)\)\n\n\[= \frac{1}{2}ma^2[(1 - \cos \theta)\dot{\theta}]^2 + [-\sin \theta \dot{\theta}]^2 = ma^2(1 - \cos \theta)\dot{\theta}^2\]\n\nPotential energy \(V = mgy = mga(1 + \cos \theta)\)\n\nThen\n\n\[\text{Lagrangian } L = T - V = ma^2(1 - \cos \theta)\dot{\the... | So I've got this problem here about a bead sliding on a cycloid wire. The equations for the cycloid are given as:
\[ x = a(\theta - \sin \theta) \]
\[ y = a(1 + \cos \theta) \]
And theta ranges from 0 to 2π. I need to find the Lagrangian function and the equation of motion. Okay, let's start by understanding what a L... |
Find the area of the triangle with vertices at \(P(2, 3, 5)\), \(Q(4, 2, -1)\), \(R(3, 6, 4)\). | \[\mathbf{PQ} = (4-2)\mathbf{i} + (2-3)\mathbf{j} + (-1-5)\mathbf{k} = 2\mathbf{i} - \mathbf{j} - 6\mathbf{k}\] \[\mathbf{PR} = (3-2)\mathbf{i} + (6-3)\mathbf{j} + (4-5)\mathbf{k} = \mathbf{i} + 3\mathbf{j} - \mathbf{k}\] Area of triangle \[= \frac{1}{2} |\mathbf{PQ} \times \mathbf{PR}| = \frac{1}{2} \begin{vmatrix} \m... | To find the area of a triangle in three-dimensional space given its vertices, we can use the concept of vectors and the cross product. The area of the triangle formed by points \(P\), \(Q\), and \(R\) can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \|\vec{PQ} \times \vec{PR}\|
\]
where \(\vec{PQ}\) ... |
Derive the equation of motion of a mathematical pendulum of mass \( m \) which is suspended by an elastic string. In equilibrium the string has a length \( l \) and its stiffness is \( c \). | If φ is the angle between the pendulum and the vertical and \( z \) is a relative elongation of the string, then the equations of motion are\n\n\[\n(1+z) \ddot{\phi} + 2\dot{z}\dot{\phi} + \frac{g}{l} \sin \phi = 0;\n\]\n\n\[\n\ddot{z} - (1+z)\dot{\phi}^2 + \frac{c}{m} z + \frac{g}{l} (1-\cos \phi) = 0.\n\] | To derive the equation of motion for a mathematical pendulum of mass \( m \) suspended by an elastic string with equilibrium length \( l \) and stiffness \( c \), we need to consider both the gravitational force and the elastic force acting on the mass. Let's proceed step by step.
### Step 1: Define the System
- **Ma... |
A plane starts diving without any initial vertical speed. The air resistance is proportional to the square of the speed. Find the dependence between the vertical speed at a given instant, the path travelled, and the maximum speed of diving. | \(\text{Ans.}\ v = v_{\max} \sqrt{1 - e^{-2gs/v_{\max}^2}}\) | Let's tackle this problem step by step. We have a plane starting to dive without any initial vertical speed, and there's air resistance proportional to the square of the speed. We need to find how the vertical speed depends on the path traveled and the maximum diving speed.
First, let's understand the forces acting on... |
The rotation of the shaft of an electric motor is given by the equation \(\varphi = \omega t (\omega = \text{const})\). A rod \(OA\) of length \(l\) is attached to the shaft at an angle of \(90^\circ\) (Fig. 175). As the electric motor is not fastened down, it makes horizontal harmonic oscillations on its base defined ... | w_a = \omega^2 \sqrt{a^2 + l^2}. | So I've got this problem here involving an electric motor and some moving parts. Let me try to understand what's going on.
First, there's an electric motor whose shaft is rotating with a constant angular velocity ω, and the rotation is given by φ = ωt. So, the shaft is spinning at a constant rate.
Then, there's a rod... |
Determine the equation of motion and the path of a point on a rim of a locomotive wheel of radius \( R = 1 \, \text{m} \), if the locomotive moves along a straight track with a constant speed of \( 20 \, \text{m/sec} \). It is assumed that the wheel rolls without sliding. Take the origin of the coordinate system to be ... | The cycloid \[ x = 20t - \sin 20t; \quad y = 1 - \cos 20t. \] | So I have this problem about a locomotive wheel rolling without sliding along a straight track. The wheel has a radius of 1 meter, and the locomotive is moving at a constant speed of 20 meters per second. I need to find the equation of motion and the path of a point on the rim of the wheel. The coordinate system is set... |
Two particles having masses \( m \) and \( M \) are traveling on the \( x \) axis (assumed frictionless) with velocities \( v_{i1} \) and \( V_{i1} \) respectively. Suppose that they collide and that after the collision (impact) their velocities are \( v_{i2} \) and \( V_{i2} \) respectively. Prove that the velocities ... | By the conservation of momentum,\n\n\[\n\text{Total momentum before impact} = \text{total momentum after impact}\n\]\n\n\[\nmv_{i1} + MV_{i1} = mv_{i2} + MV_{i2}\n\]\n\nor\n\n\[\nm v_1 + M V_1 = m v_2 + M V_2\n\]\n\nLet \( z \) and \( X \) be the respective coordinates of the particles. Then the center of mass is given... | So I've got this problem here about two particles colliding on the x-axis, and I need to prove that the velocity of their center of mass stays the same before and after the collision. Okay, let's break this down.
First, I need to recall what the center of mass is. The center of mass of a system is like the average pos... |
6.29 An \( xyz \) coordinate system is rotating with angular velocity \( \mathbf{\omega} = 5i - 4j - 10k \) relative to a fixed \( XYZ \) coordinate system having the same origin. Find the velocity of a particle fixed in the \( xyz \) system at the point \( (3, 1, -2) \) as seen by an observer fixed in the \( XYZ \) sy... | \( 18i - 20j + 17k \) | To find the velocity of a particle fixed in the rotating \( xyz \) coordinate system as seen by an observer in the fixed \( XYZ \) system, we need to use the concept of rotational velocity. The velocity of a particle in a rotating frame is given by the cross product of the angular velocity vector and the position vecto... |
A perfectly elastic sphere impinges with a given velocity and in a given direction against a smooth plane; to determine the velocity and direction of reflection. | Let u, v, denote the velocities of incidence and of reflection, and α, β, the angles which the directions of the motion before and after impact make with a normal to the plane. The resolved parts of the velocities parallel to the plane are u sin α and v sin β, and, at right angles to it, u cos α and v cos β. But the pl... | So I've got this problem here: there's a perfectly elastic sphere that's hitting a smooth plane with a certain velocity and direction, and I need to find out how it bounces off—what its velocity and direction will be after reflection. Okay, let's break this down.
First, since the sphere is perfectly elastic and the pl... |
Find the volume of a parallelepiped whose edges are given by \( \mathbf{A} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}, \mathbf{B} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}, \mathbf{C} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} \). | 31 | To find the volume of a parallelepiped defined by three vectors \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\), we can use the scalar triple product formula. The volume \(V\) is given by the absolute value of the scalar triple product of the three vectors:
\[ V = |\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})| \... |
A particle travels with uniform angular speed \(\omega\) around a circle of radius \(b\). Prove that its projection on a diameter oscillates with simple harmonic motion of period \(2\pi/\omega\) about the center. | Choose the circle in the \(xy\) plane with center at the origin O as in Fig. 4-10 below. Let \(Q\) be the projection of particle \(P\) on diameter \(AB\) chosen along the \(x\) axis.\n\nIf the particle is initially at \( B \), then in time \( t \) we will have \( \angle BOP = \theta = \omega t \). Then the position of ... | So I've got this problem here: a particle is moving with uniform angular speed \(\omega\) around a circle of radius \(b\), and I need to prove that its projection on a diameter oscillates with simple harmonic motion of period \(2\pi/\omega\) about the center. Alright, let's break this down.
First, I need to understand... |
8.19. Find the transverse displacement of a vibrating string of length \( l \) with fixed endpoints if the string is initially given a displacement \( f(x) \) from its equilibrium position and then released. | Let the transverse displacement of any point \( x \) of the string at time \( t \) be \( Y(x, t) \). Since the ends \( x = 0 \) and \( x = l \) of the string are fixed, we must have \( Y(0, t) = 0 \) and \( Y(l, t) = 0 \). Since the initial displacement is \( f(x) \), we have \( Y(x, 0) = f(x) \); and since the initial... | So I've got this problem here about finding the transverse displacement of a vibrating string. The string has a length \(l\), and both endpoints are fixed. Initially, the string is displaced from its equilibrium position by a function \(f(x)\), and then it's released. I need to find the displacement over time.
First, ... |
Given a space curve \( C \) with position vector \[ \mathbf{r} = 3 \cos 2t \, \mathbf{i} + 3 \sin 2t \, \mathbf{j} + (8t - 4)\mathbf{k} \] (a) Find a unit tangent vector \( \mathbf{T} \) to the curve. (b) If \( \mathbf{r} \) is the position vector of a particle moving on \( C \) at time \( t \), verify in this case tha... | (a) A tangent vector to \( C \) is \[ \frac{d\mathbf{r}}{dt} = -6 \sin 2t \, \mathbf{i} + 6 \cos 2t \, \mathbf{j} + 8\mathbf{k} \] The magnitude of this vector is \[ |\frac{d\mathbf{r}}{dt}| = \frac{ds}{dt} = \sqrt{(-6 \sin 2t)^2 + (6 \cos 2t)^2 + (8)^2} = 10 \] Then a unit tangent vector to \( C \) is \[ \mathbf{T} = ... | So I've got this problem here about a space curve given by the position vector \(\mathbf{r} = 3 \cos 2t \, \mathbf{i} + 3 \sin 2t \, \mathbf{j} + (8t - 4)\mathbf{k}\). There are two parts: first, to find a unit tangent vector \(\mathbf{T}\) to the curve, and second, to verify that if \(\mathbf{r}\) is the position vect... |
The maximum range of a projectile when fired down an inclined plane is twice the maximum range when fired up the inclined plane. Find the angle which the incline makes with the horizontal. | \( \sin^{-1} 1/3 \) | So I've got this problem here about projectiles fired up and down an inclined plane, and it says that the maximum range when fired down the plane is twice the maximum range when fired up the plane. I need to find the angle that the incline makes with the horizontal. Hmm, I'm a bit rusty on projectile motion on inclined... |
A particle moves along the space curve defined by \(x = e^{-t} \cos t\), \(y = e^{-t} \sin t\), \(z = e^{-t}\). Find the magnitude of the (a) velocity and (b) acceleration at any time \(t\). | (a) \(\sqrt{3} e^{-t}\), (b) \(\sqrt{5} e^{-t}\) | Given the parametric equations for the particle's position:
\[
x = e^{-t} \cos t, \quad y = e^{-t} \sin t, \quad z = e^{-t}
\]
We are to find the magnitude of the velocity and acceleration at any time \( t \).
### Part (a): Magnitude of the Velocity
First, we need to find the velocity vector \(\mathbf{v}(t)\), whic... |
To find the form of the catenary when the central force is attractive and varies inversely as the square of the distance; the unit of mass being invariable. Let \( AOB \) (fig. 65) be the catenary; \( S \) the centre of force; \( SO \) the radius vector which meets the curve at right angles. \( r = \) the tension at \(... | R = \int Fmdr = \int k \frac{c^2}{r^2} mdr = C' - \frac{mkc^2}{r}, \quad r = C' - mkc, \quad t = R = r + mkc - \frac{mkc^2}{r}, \quad d\theta = \frac{Cdr}{r \left( \left( r + mkc - \frac{mkc^2}{r} \right)^2 - C^2 \right)^{\frac{1}{2}}}, \quad d\theta = \frac{crdr}{r \left( \left( r + mkc - \frac{mkc^2}{r} \right)^2 - c... | I'm having trouble understanding this problem about finding the form of a catenary under an attractive central force that varies inversely as the square of the distance. The unit of mass is constant, and there are some given equations and substitutions. I need to follow through the steps to understand how the different... |
A 3-kgf particle moves to the left on a horizontal line with the velocity of 5 m/sec. A constant force, directed to the right, is applied to the particle. In 30 sec the force value is reduced to 0 and then the velocity of the particle becomes 55 m/sec towards the right. Determine the value of the force and the work it ... | 0.612 kgf; 459 kgfm = 4.5 kilojoules. | I have this problem here. Let's see what it's asking.
We have a 3-kg particle moving to the left at 5 m/s on a horizontal line. Then, a constant force is applied to the right. This force is reduced to zero in 30 seconds, and after that, the particle's velocity becomes 55 m/s to the right. I need to find the value of t... |
An aircraft flies horizontally. The air resistance is proportional to the square of the speed. At a speed of 1 m/sec the air resistance equals 0.05 kgf. The tractive force is constant and equals 3080 kgf and it makes an angle of \( 10^\circ \) with the direction of the movement of the aircraft. Determine the maximum sp... | **Ans.** \(v_{\text{max}} = 246 \, \text{m/sec}.\) | I have this problem about an aircraft flying horizontally, and I need to find its maximum speed. Let's see what's given and how to approach this.
First, the aircraft is flying horizontally, and there's air resistance that's proportional to the square of the speed. So, the drag force \( D \) can be expressed as:
\[ D ... |
A ball is thrown horizontally in the northern hemisphere. (a) Would the path of the ball, if the Coriolis force is taken into account, be to the right or to the left of the path when it is not taken into account as viewed by the person throwing the ball? (b) What would be your answer to (a) if the ball were thrown in t... | (a) to the right, (b) to the left | So I've got this question about a ball being thrown horizontally, and it's considering the Coriolis force. It's divided into two parts: first, for the northern hemisphere, and then for the southern hemisphere. I need to figure out whether the ball's path would be deflected to the right or to the left compared to its pa... |
A small ball of weight \( p \) knotted to an inelastic string slides along a smooth horizontal surface. A free end of the string is pulled through a hole in the plane with a constant velocity \( a \). At the origin, the string is straight, the distance between the hole and the ball is \( R \), and the projection of the... | Considering the hole as the origin of polar coordinates and an angle \( \varphi_0 = 0 \), the motion of the ball is given by\n\n\[\nr = R-at; \quad \varphi = \frac{v_0 t}{R-at} \quad T = \frac{p v_0^2 R^2}{g \, (R-at)^3}\n\] | So I've got this physics problem here, and I need to figure out how a ball moves when it's attached to a string that's being pulled through a hole with a constant velocity. Let's break this down step by step.
First, let's understand the setup. There's a smooth horizontal surface, and on it, there's a small ball of wei... |
A particle moves in a circle of radius 20 cm. If its tangential speed is 40 cm/s, find (a) its angular speed, (b) its angular acceleration, (c) its normal acceleration. | (a) 2 rad/s, (b) 0 rad/s\(^2\), (c) 80 cm/s\(^2\) | Let's tackle this problem step by step. We have a particle moving in a circle with a given radius and tangential speed. We need to find its angular speed, angular acceleration, and normal acceleration.
First, let's understand what each term means:
1. **Angular speed (ω)**: This is how fast the particle is moving in t... |
Two rigid bodies rotate independently about a common fixed axis with constant angular velocities ω_1 and ω_2. Their moments of inertia with respect to this axis are J_1 and J_2, respectively. What must the angular velocity of both bodies be, if they are connected while rotating? | \(\omega = \frac{J_1 \omega_1 + J_2 \omega_2}{J_1 + J_2}\). | So I've got this problem here about two rigid bodies rotating independently about a common fixed axis with constant angular velocities ω₁ and ω₂. Their moments of inertia with respect to this axis are J₁ and J₂, respectively. The question is, what must the angular velocity of both bodies be if they are connected while ... |
An object is projected vertically upward from the earth's surface with initial speed \(v_0\). Neglecting air resistance, (a) find the speed at a distance \(H\) above the earth's surface and (b) the smallest velocity of projection needed in order that the object never return. | (a) Let \(r\) denote the radial distance of the object at time \(t\) from the center of the earth, which we assume is fixed [see Fig. 5-14]. If \(M\) is the mass of the earth and \(R\) is its radius, then by Newton's universal law of gravitation and Problem 5.29, the force between \(m\) and \(M\) is\n\[ \mathbf{F} = -\... | Let's tackle this problem step by step. We have an object being projected vertically upward from the Earth's surface with an initial speed \(v_0\). We need to find its speed at a distance \(H\) above the Earth's surface and also determine the smallest velocity required for the object to never return to Earth.
First, l... |
The weight on a vertical spring undergoes forced vibrations according to the equation \( d^2x/dt^2 + 4x = 8 \sin \omega t \) where \( x \) is the displacement from the equilibrium position and \( \omega > 0 \) is a constant. If at \( t = 0 \), \( x = 0 \) and \( dx/dt = 0 \), find \( (a) \) \( x \) as a function of \( ... | (a) \( x = (8 \sin \omega t - 4\omega \sin 2t)/(4 - \omega^2) \) if \( \omega \neq 2; \; x = \sin 2t - 2t \cos 2t \) if \( \omega = 2 \) (b) \( \omega = 2 \) or period = \(\pi\) | Let's tackle this problem step by step. We have a differential equation that describes the motion of a weight on a vertical spring undergoing forced vibrations:
\[ \frac{d^2x}{dt^2} + 4x = 8 \sin \omega t \]
with initial conditions:
\[ x(0) = 0, \quad \frac{dx}{dt}(0) = 0 \]
Our tasks are:
(a) Find \( x \) as a fu... |
Prove that \( \frac{d}{du}(\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \cdot \frac{d\mathbf{B}}{du} + \frac{d\mathbf{A}}{du} \cdot \mathbf{B} \), where \(\mathbf{A}\) and \(\mathbf{B}\) are differentiable functions of \( u \). | ### Method 1. \[ \frac{d}{du} ( \mathbf{A} \cdot \mathbf{B} ) = \lim_{\Delta u \to 0} \frac{(\mathbf{A} + \Delta \mathbf{A}) \cdot (\mathbf{B} + \Delta \mathbf{B}) - \mathbf{A} \cdot \mathbf{B}}{\Delta u} \] \[ = \lim_{\Delta u \to 0} \frac{\mathbf{A} \cdot \Delta \mathbf{B} + \Delta \mathbf{A} \cdot \mathbf{B} + \Delt... | To prove that the derivative of the dot product of two vector functions \(\mathbf{A}(u)\) and \(\mathbf{B}(u)\) with respect to \(u\) is given by
\[
\frac{d}{du}(\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \cdot \frac{d\mathbf{B}}{du} + \frac{d\mathbf{A}}{du} \cdot \mathbf{B},
\]
we will proceed step by step.
### Step... |
Let \( \delta \) be a complete solution of the Hamilton-Jacobi equation containing the \( n \) constants \( \beta_1, \ldots, \beta_n \). Let \n\n\[J_\alpha = \oint p_\alpha dq_\alpha.\]\n\nProve that the \( J_\alpha \) are functions of the \( \beta_\alpha \) only.\n\nWe have \n\n\[\delta = S_1(q_1, \beta_1, \ldots, \be... | We have \n\n\[\delta = S_1(q_1, \beta_1, \ldots, \beta_n) + \cdots + S_n(q_n, \beta_1, \ldots, \beta_n) - \beta_1 t\]\n\nwhere the constant \( \beta_1 = E \), the total energy. Now\n\n\[p_\alpha = \frac{\partial \delta}{\partial q_\alpha} = \frac{dS_\alpha}{dq_\alpha}.\]\n\nThus\n\n\[J_\alpha = \oint p_\alpha dq_\alpha... | I'm trying to understand this proof about the action variables \( J_\alpha \) being functions of the constants \( \beta_\alpha \) only. Let's break it down step by step.
First, we have the complete solution \( \delta \) of the Hamilton-Jacobi equation, which is expressed as:
\[
\delta = S_1(q_1, \beta_1, \ldots, \bet... |
An axle of the driving wheel of a car moves along a horizontal surface in a straight line. It is acted upon by a horizontal motive force \( F \). The radius of gyration of the wheel about an axis, passing through the centre of gravity perpendicular to its plane, equals \(\rho\). The coefficient of sliding friction betw... | F \leq fP \frac{r^2 + \rho^2}{\rho^2}. | So I've got this problem here about a car's driving wheel moving along a horizontal surface. There's a horizontal force F acting on it, and I need to figure out what conditions F has to satisfy so that the wheel runs without sliding. Okay, let's break this down.
First, I need to understand all the components involved.... |
A uniform rectangular board \(KLMN\), (fig. 40), is placed upon a rough inclined plane \(AB\); supposing the inclination of the plane \(AB\) to the horizon to be gradually increased, to find whether the equilibrium of the board will be disturbed by the commencement of a rolling or of a sliding motion. | First suppose that the board begins to slide; let \(R\) be the whole of the reaction of the plane at right angles to itself on the board, \(\mu\) the coefficient of friction, and \(\phi\) the inclination of the plane at the commencement of sliding. Then, resolving forces parallel to the inclined plane,\n\n\[ \mu R = W ... | I've got this problem here about a rectangular board on an inclined plane, and I need to figure out whether it will start rolling or sliding as the angle of the incline increases. Let's see... The board is uniform, which probably means it has uniform density, so its center of mass should be at its geometric center. The... |
Prove that if \( a \) and \( b \) are non-collinear, then \( xa + yb = 0 \) implies \( x = y = 0 \). | Suppose \( x \neq 0 \). Then \( xa + yb = 0 \) implies \( xa = -yb \) or \( a = -(x/y)b \), i.e., \( a \) and \( b \) must be parallel to the same line (collinear), contrary to hypothesis. Thus \( x = 0 \); then \( yb = 0 \), from which \( y = 0 \). | To prove that if \( a \) and \( b \) are non-collinear vectors, then the equation \( xa + yb = 0 \) implies that \( x = 0 \) and \( y = 0 \), we can proceed as follows:
1. **Understanding Non-Collinear Vectors**: Non-collinear vectors are vectors that do not lie on the same line or parallel lines. In other words, one ... |
(1) A particle is projected with a given velocity and in a given direction, and is acted upon by a constant force in parallel lines; to determine the path of the particle. | Let the axis of \( x \) be taken so as to pass through the initial place of the particle, and let the axis of \( y \) be taken parallel to the constant force which acts towards the axis of \( x \). Let \( f \) denote the constant force. Then the tangential resolved part being \( -f \frac{dy}{ds} \), and the normal one ... | So, I've got this problem here: a particle is projected with a given velocity and direction, and it's acted upon by a constant force that's acting along parallel lines. I need to determine the path of the particle. Okay, let's break this down.
First, I need to understand what's being asked. The particle has an initial... |
A weight \(P\) suspended from a spring is acted on by a disturbing force defined by the conditions: \(F = 0\) for \(t < 0; F = \frac{t}{\tau} F_0\) for \(0 < t < \tau; F = F_0\) for \(t < \tau\). The spring has a stiffness \(c\). Determine the motion of the weight and find the amplitude of oscillations for \(t > \tau\)... | \[ x = \frac{F_0}{c} \left[ 1 - \frac{2}{k \tau} \cos k \left( t - \frac{\tau}{2} \right) \sin \frac{k \tau}{2} \right]; \quad k = \sqrt{\frac{cg}{P}}; \] \n\[ A = \frac{2F_0}{kc \tau} \sin \frac{k \tau}{2}. \] | So I've got this problem here about a weight suspended from a spring that's being acted upon by a disturbing force. Let me try to understand what's going on.
First, there's a weight \( p \) hanging from a spring with stiffness \( c \). So, without any external forces, the weight would just hang there at equilibrium. B... |
A particle acted on by gravity is projected in a uniform medium, of which the resistance varies as the velocity, with a given velocity and at a given angle of inclination to the horizon; to find after what interval of time the particle will arrive at its greatest altitude. | Let \(k\) be the resistance for a unit of velocity, \(u\) the velocity and \(\alpha\) the angle of projection, and let \(y\) be the height through which the particle has ascended at the end of the time \( t \). Then \(\frac{d^2 y}{d t^2} = -g - k \frac{dy}{dt}, \log \left( g + k \frac{dy}{dt} \right) = C - kt ;\) but \... | So I've got this problem here about a particle being projected in a uniform medium where the resistance varies as the velocity. It's acted on by gravity, and it's launched with a given velocity at a given angle to the horizon. I need to find out after what interval of time the particle will arrive at its greatest altit... |
Masses \( m_1 \) and \( m_2 \) are located on smooth inclined planes of angles \( \alpha_1 \) and \( \alpha_2 \) respectively and are connected by an inextensible string of negligible mass which passes over a smooth peg at \( A \) [Fig. 3-26]. Find the accelerations of the masses. | The accelerations are in magnitude equal to \[ \frac{m_1 \sin \alpha_1 - m_2 \sin \alpha_2}{m_1 + m_2} g \] | So I'm trying to figure out this problem with two masses on inclines connected by a string over a peg. Let's see, masses \( m_1 \) and \( m_2 \) are on smooth inclined planes with angles \( \alpha_1 \) and \( \alpha_2 \), respectively, and they're connected by an inextensible string that goes over a smooth peg at point... |
The landing speed of an aircraft is 100 km/h. Determine its deceleration at landing on the track of \( l = 100 \) m length if the deceleration is constant. | \( w = 3.86 \) m/sec\(^2\). | So I've got this problem here: an aircraft lands with a speed of 100 kilometers per hour, and it comes to a stop on a track that's 100 meters long. The deceleration is constant, and I need to find out what that deceleration is. Alright, let's break this down.
First, I need to make sure all my units are consistent. The... |
One end of a heavy chain is attached to a fixed point \( A \), and the other to a weight which is placed on a rough horizontal plane passing through \( A \), and the chain hangs through a slit in the horizontal plane; to find the greatest distance of the weight from \( A \), at which equilibrium is possible. If \( a \)... | \frac{x}{a} = \left\{ \frac{1 + \{1 + \mu^2(1 + n)^2\}^{\frac{1}{2}}}{\mu(1+n)} \right\}^{\frac{1}{\mu \log{n}}}. | I'm trying to solve this problem about a chain attached to a fixed point and a weight on a rough horizontal plane. The chain hangs through a slit in the plane, and I need to find the greatest distance of the weight from the fixed point A where equilibrium is possible. The variables given are:
- \( a \): length of the ... |
If \( \mathbf{T} \) is a unit tangent vector to a space curve \( C \), show that \( \frac{d\mathbf{T}}{ds} \) is normal to \( \mathbf{T} \). | Since \( \mathbf{T} \) is a unit vector, we have \( \mathbf{T} \cdot \mathbf{T} = 1 \). Then differentiating with respect to \( s \), we obtain \[ \mathbf{T} \cdot \frac{d\mathbf{T}}{ds} + \frac{d\mathbf{T}}{ds} \cdot \mathbf{T} = 2\mathbf{T} \cdot \frac{d\mathbf{T}}{ds} = 0 \quad \text{or} \quad \mathbf{T} \cdot \frac... | So I've got this problem here: If T is a unit tangent vector to a space curve C, I need to show that the derivative of T with respect to arc length s, which is dT/ds, is normal to T. Alright, let's see how to approach this.
First off, I know that T is a unit tangent vector, which means that its magnitude is always 1, ... |
A particle of mass \( 20 \, \text{g} \) is attracted by a force to a fixed centre in accordance with Newton's law of gravitation. During 50 sec the particle completely traces out an ellipse with semi-axes 10 and 8 cm long, respectively. Determine the maximum and minimum magnitudes of the force of gravitation \( F \). | \( F_{\text{max}} = 19.7 \, \text{dynes}; \, F_{\text{min}} = 1.2 \, \text{dynes}. \) | So I've got this problem here involving a particle moving in an elliptical orbit under the influence of gravity, following Newton's law of gravitation. The particle has a mass of 20 grams, and it traces out an ellipse with semi-major axis 10 cm and semi-minor axis 8 cm in 50 seconds. I need to find the maximum and mini... |
A train moves at constant speed along a horizontal straight track. The weight of the train, excluding the weight of the locomotive, is 180,000 kgf. What is the tractive force exerted by the locomotive, if the resistance to motion is 0.005 of the train pressure on the rails? | 900 kgf. | Let's tackle this problem step by step. So, we have a train moving at a constant speed along a horizontal straight track. The weight of the train, excluding the weight of the locomotive, is 180,000 kgf. We need to find the tractive force exerted by the locomotive, given that the resistance to motion is 0.005 times the ... |
A body of weight \( Q = 12 \, \text{kgf} \) is attached to the end of a spring. It undergoes harmonic vibrations. The reading on the stop-watch shows that the body performed 100 complete vibrations per 45 sec. After this an additional weight \( Q_1 = 6 \, \text{kgf} \) is hung on the spring. Determine the period of vib... | T_1 = T \sqrt{\frac{Q + Q_1}{Q}} = 0.55 \, \text{sec}. | So I've got this problem here about harmonic vibrations and springs. Let me try to understand what's going on.
We have a body with weight \( Q = 12 \, \text{kgf} \) attached to a spring, and it's undergoing harmonic vibrations. The stop-watch reading shows that it performs 100 complete vibrations in 45 seconds. Then, ... |
A particle of mass \( m \) moves in a force field of potential \( V \). Write (a) the Hamiltonian and (b) Hamilton's equations in spherical coordinates \((r, \theta, \phi)\). | **(a)** The kinetic energy in spherical coordinates is\n\n\[ \nT = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2 + r^2 \sin^2 \theta \, \dot{\phi}^2) \n\tag{1}\n\]\n\nThen the Lagrangian is\n\n\[ \nL = T - V = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2 + r^2 \sin^2 \theta \, \dot{\phi}^2) - V(r, \theta, \phi) \n\tag{2}\n\... | So I have this problem here: a particle of mass \( m \) is moving in a force field with a potential \( V \). I need to write down the Hamiltonian and Hamilton's equations in spherical coordinates \((r, \theta, \phi)\). Okay, let's break this down step by step.
First, I need to recall what the Hamiltonian is. In classi... |
Give a set of generalized coordinates needed to completely specify the motion of each of the following: (a) a particle constrained to move on an ellipse, (b) a circular cylinder rolling down an inclined plane, (c) the two masses in a double pendulum [Fig. 11-3] constrained to move in a plane. | (a) Let the ellipse be chosen in the xy plane of Fig. 11-1. The particle of mass m moving on the ellipse has coordinates (x, y). However, since we have the transformation equations x = a cos θ, y = b sin θ, we can specify the motion completely by use of the generalized coordinate θ.\n\n(b) The position of the cylinder ... | So I've got this problem here that asks for the generalized coordinates needed to completely specify the motion of three different systems. I need to find these for:
(a) A particle constrained to move on an ellipse,
(b) A circular cylinder rolling down an inclined plane,
(c) The two masses in a double pendulum, cons... |
A street lamp hangs at point \( B \) in the middle of the wire cable \( ABC \). The ends of this wire cable are fastened to the hooks \( A \) and \( C \) located on the same level (Fig. 5). Determine the tensions \( T_1 \) and \( T_2 \) in the cable parts \( AB \) and \( BC \), if the weight of the street lamp is 15 kg... | \( T_1 = T_2 = 750 \, \text{kgf} \). | So I've got this problem here about a street lamp hanging from a wire cable. The lamp is at point B, and the cable is attached to points A and C, which are at the same level. The lamp weighs 15 kgf, and the total length of the cable is 20 meters. There's also this sag of 0.1 meters at point B. I need to find the tensio... |
A particle travels so that its acceleration is given by \[ \mathbf{a} = 2e^{-t} \mathbf{i} + 5 \cos t \mathbf{j} - 3 \sin t \mathbf{k} \] If the particle is located at \( (1, -3, 2) \) at time \( t = 0 \) and is moving with a velocity given by \( 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k} \), find (a) the velocity and (b)... | (a) \[ \mathbf{a} = \frac{d\mathbf{v}}{dt} = 2e^{-t} \mathbf{i} + 5 \cos t \mathbf{j} - 3 \sin t \mathbf{k} \] Integrating, \[ \mathbf{v} = \int (2e^{-t} \mathbf{i} + 5 \cos t \mathbf{j} - 3 \sin t \mathbf{k}) dt \] \[ = -2e^{-t} \mathbf{i} + 5 \sin t \mathbf{j} + 3 \cos t \mathbf{k} + \mathbf{c}_1 \] Since \( \mathbf{... | Let's tackle this problem step by step. We have the acceleration of a particle as a function of time, and we're given its initial position and velocity at time \( t = 0 \). We need to find the velocity and displacement at any time \( t > 0 \).
First, recall that acceleration is the derivative of velocity with respect ... |
Find the velocity of escape for an object on the surface of the moon. Use the fact that the acceleration due to gravity on the moon’s surface is approximately 1/6 that on the earth and that the radius of the moon is approximately 1/4 of the earth’s radius. | 2.29 km/s | So I need to find the escape velocity for an object on the surface of the moon. I know that the escape velocity from a celestial body can be found using the formula:
\[ v_{\text{esc}} = \sqrt{\frac{2GM}{r}} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the celestial body,
- \( r \) is... |
A 10-kgf homogeneous ball is held in equilibrium by two strings \(AB\) and \(CD\), lying in the same vertical plane, and forming an angle of 150° (Fig. 12). The string \(AB\) is inclined at 45° to the horizontal. Determine the tensions in the strings. | \(T_B = 19.3\) kgf; \(T_C = 14.1\) kgf. | So I've got this problem here about a 10-kgf ball being held in equilibrium by two strings, AB and CD, in the same vertical plane, with an angle of 150° between them. String AB is inclined at 45° to the horizontal. I need to find the tensions in both strings.
First, I need to understand the situation. There's a ball b... |
Prove that c = ae where c is the distance from the center to the focus of the ellipse. a is the length of the semi-major axis and e is the eccentricity. | From Fig. 5-4, page 119, we have c = CO = CV - OV = a - a(1 - e) = ae. An analogous result holds for the hyperbola [see Problem 5.73(c), page 139]. | I'd be happy to help you with that proof. So, we're dealing with an ellipse here, and we want to show that the distance from the center to a focus, which is denoted by \( c \), is equal to the product of the length of the semi-major axis, \( a \), and the eccentricity, \( e \). In other words, we want to prove that \( ... |
A body of weight \( P \) is given a push and moves along a rough horizontal plane. It travels the distance \( s = 24.5 \, \text{m} \) during 5 sec and then comes to rest. Determine the coefficient of friction \( f \). | **Ans.** \( f = 0.2 \). | So I've got this problem here. There's a body being pushed on a rough horizontal plane. It travels 24.5 meters in 5 seconds and then comes to rest. I need to find the coefficient of friction, denoted by \( f \). Alright, let's see how to approach this.
First, I need to recall some basic principles of motion and fricti... |
A uniform rod of length 0.6 m and mass 3 kg rotates with angular speed 10 radians per second about an axis perpendicular to it and passing through its center. Find the kinetic energy of rotation. | 4.5 J | To find the kinetic energy of rotation of the rod, we can use the formula for rotational kinetic energy:
\[ K = \frac{1}{2} I \omega^2 \]
where \( K \) is the kinetic energy, \( I \) is the moment of inertia of the rod about the axis of rotation, and \( \omega \) is the angular speed.
First, we need to determine the... |
Let \( (r, \theta) \) represent the polar coordinates describing the position of a particle. If \( \mathbf{r_1} \) is a unit vector in the direction of the position vector \( \mathbf{r} \) and \( \mathbf{\theta_1} \) is a unit vector perpendicular to \( \mathbf{r} \) and in the direction of increasing \( \theta \) [see... | (a) If \( r \) is the position vector of the particle at any time \( t \), then \( \partial \mathbf{r}/\partial r \) is a vector tangent to the curve \( \theta = \text{constant}, \) i.e. a vector in the direction of \( r \) (increasing \( r \)). A unit vector in this direction is thus given by\n\n\[\n\mathbf{r_1} = \fr... | So I've got this problem here about polar coordinates and unit vectors. I need to show two things: first, that the unit vector in the direction of the position vector \(\mathbf{r}\) is \(\mathbf{r_1} = \cos \theta \, \mathbf{i} + \sin \theta \, \mathbf{j}\), and the unit vector perpendicular to \(\mathbf{r}\) in the di... |
Find a unit normal to the surface \(2x^2 + 4yz - 5z^2 = -10\) at the point \(P(3, -1, 2)\). | By Problem 1.44, a vector normal to the surface is \(\nabla (2x^2 + 4yz - 5z^2) = 4x\mathbf{i} + 4z\mathbf{j} + (4y - 10z)\mathbf{k} = 12\mathbf{i} + 8\mathbf{j} - 24\mathbf{k}\) at \( (3, -1, 2) \).\n\nThen a unit normal to the surface at \(P\) is\n\n\[\n\frac{12\mathbf{i} + 8\mathbf{j} - 24\mathbf{k}}{\sqrt{(12)^2 + ... | To find a unit normal vector to the surface defined by the equation \(2x^2 + 4yz - 5z^2 = -10\) at the point \(P(3, -1, 2)\), we can follow these steps:
1. **Define the Surface as a Level Set:**
The given equation \(2x^2 + 4yz - 5z^2 = -10\) can be thought of as a level set of the function \(f(x, y, z) = 2x^2 + 4yz... |
Prove that the projection of **A** on **B** is equal to \( \mathbf{A} \cdot \mathbf{b} \), where \( \mathbf{b} \) is a unit vector in the direction of **B**. | Through the initial and terminal points of **A**, pass planes perpendicular to **B** at **G** and **H** respectively as in the adjacent Fig. 1-19; then Projection of **A** on **B** = \( \frac{GH}{EF} = A \cos \theta = \mathbf{A} \cdot \mathbf{b} \). | I'm going to try to prove that the projection of vector A onto vector B is equal to A · b, where b is a unit vector in the direction of B. Let's see how to approach this.
First, I need to recall what a projection of one vector onto another is. The projection of A onto B is the shadow that A casts onto B, so to speak, ... |
A skier moves down an inclined slope of angle \( 45^\circ \) without using his stick. The coefficient of friction between the skis and snow is \( f = 0.1 \). The air resistance to the skier’s motion is \( r = \alpha v^2 \), where \( \alpha = \text{const} \) and \( v \) is the skier’s speed. When the skier attains the s... | **Ans.** \(v_{1 \text{ max}} = 108 \, \text{km/h}; \quad v_{2 \text{ max}} = 111 \, \text{km/h}.\) | To solve this problem, we need to determine the maximum speed a skier can attain on an inclined slope under two different conditions regarding friction. Let's break down the problem step by step.
### Given Data:
- Incline angle: \( \theta = 45^\circ \)
- Coefficient of friction (initial): \( f = 0.1 \)
- Air resistanc... |
A thread with a load \(P\) at one end runs over a wheel mounted on a horizontal axle. At a particular instant the load is allowed to fall from rest with a constant acceleration \(w_0\) thus setting the wheel in motion. Find the total acceleration of the points on the rim of the wheel as a function of the height \(h\) t... | \(w = \frac{w_0}{R} \sqrt{R^2 + 4h^2}\). | So I have this problem here. There's a wheel mounted on a horizontal axle, and a thread is running over it with a load \(P\) at one end. This load is allowed to fall from rest with a constant acceleration \(w_0\), which makes the wheel spin. I need to find the total acceleration of the points on the rim of the wheel as... |
If\n\n\[ f(x) = A + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n \pi x}{l} + b_n \sin \frac{n \pi x}{l} \right) \]\n\nprove that by making suitable assumptions concerning term by term integration of infinite series, that for \( n = 1, 2, 3, \ldots \),\n\n(a) \( a_n = \frac{1}{l} \int_{-l}^{l} f(x) \cos \frac{n \pi x}{l}... | (a) Multiplying\n\n\[ f(x) = A + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n \pi x}{l} + b_n \sin \frac{n \pi x}{l} \right) \tag{1} \]\n\nby \( \cos \frac{m \pi x}{l} \) and integrating from \(-l\) to \(l\), using Problem 8.14, we have\n\n\[ \int_{-l}^{l} f(x) \cos \frac{m \pi x}{l} \, dx = A \int_{-l}^{l} \cos \frac{m... | So I've got this problem here about Fourier series. It gives me this function:
\[ f(x) = A + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n \pi x}{l} + b_n \sin \frac{n \pi x}{l} \right) \]
And I need to prove these formulas for the coefficients \( a_n \), \( b_n \), and \( A \) using term-by-term integration of the inf... |
Calculate the angular speed of the earth about its axis. | Since the earth makes one revolution \([2\pi \, \text{radians}]\) about its axis in approximately 24 hours = 86,460 s, the angular speed is\n\n\[ \omega = \frac{2\pi}{86,460} = 7.27 \times 10^{-5} \, \text{rad/s} \]\n\nThe actual time for one revolution is closer to 86,164 s and the angular speed 7.29 \times 10^{-5} \,... | To calculate the angular speed of the Earth about its axis, we need to determine how fast the Earth is rotating. Angular speed (ω) is defined as the rate of change of angular displacement and is given by the formula:
\[ \omega = \frac{\theta}{t} \]
where θ is the angular displacement in radians and t is the time take... |
Two perfectly elastic spheres meet directly with equal velocities; to find the relation between their magnitudes, that after collision one of them may remain at rest. | If m, m', denote their masses, m' corresponding to the one which remains at rest, m' : m :: 3 : 1. | So I've got this problem here: two perfectly elastic spheres meet directly with equal velocities, and I need to find the relation between their masses such that after the collision, one of them remains at rest. Okay, let's break this down.
First off, since the spheres are perfectly elastic, that means both kinetic ene... |
A particle is acted upon by the forces \( \mathbf{F_1} = 5\mathbf{i} - 10\mathbf{j} + 15\mathbf{k} \), \( \mathbf{F_2} = 10\mathbf{i} + 25\mathbf{j} - 20\mathbf{k} \), and \( \mathbf{F_3} = 15\mathbf{i} - 20\mathbf{j} + 10\mathbf{k} \). Find the force needed to keep the particle in equilibrium. | The resultant of the forces is\n\n\\[\n\\mathbf{R} = \\mathbf{F_1} + \\mathbf{F_2} + \\mathbf{F_3} = (5\\mathbf{i} - 10\\mathbf{j} + 15\\mathbf{k}) + (10\\mathbf{i} + 25\\mathbf{j} - 20\\mathbf{k}) + (15\\mathbf{i} - 20\\mathbf{j} + 10\\mathbf{k}) = 30\\mathbf{i} - 5\\mathbf{j} + 5\\mathbf{k}\n\\]\n\nThen the force nee... | So I've got this problem here about forces acting on a particle, and I need to find out what additional force is required to keep the particle in equilibrium. Let me see if I can figure this out.
First, I need to understand what equilibrium means in this context. If a particle is in equilibrium, that means the net for... |
Each spring of a railway car carries the load of \( P \, \text{kgf}. \) When in equilibrium, the spring is deflected 5 cm due to the load applied. Determine the period \( T \) of natural vibrations of the car on the springs. The stiffness of the spring is proportional to the deflection sag of the spring. | T = 0.45 \, \text{sec}. | So I have this problem here about a railway car sitting on springs, and I need to find the period of its natural vibrations. Let's see what I've got.
First, each spring carries a load of P kgf, and when equilibrium is reached, the spring deflects by 5 cm due to this load. I need to find the period T of the natural vib... |
A pendulum consists of a slider of mass \( M \) and a ball of mass \( m \) connected by a rod of length \( l \), as shown in Fig. 504. The slider slides without friction along the horizontal plane. The rod is free to oscillate about an axle which is connected to the slider. A spring of stiffness \( c \) is tied with it... | These frequencies are the roots of the equation\n\[ k^4 - \left[ \frac{c}{M} + \frac{g}{l} \frac{M + m}{M} \right] k^2 + \frac{c}{M} \frac{g}{l} = 0. \] | So I've got this physics problem here, and I need to find the frequencies of small oscillations for this pendulum system. Let's see what I've got:
- There's a slider of mass \( M \) moving without friction on a horizontal plane.
- Attached to this slider is a ball of mass \( m \), connected by a rod of length \( l \)... |
A particle of mass \( m \) moves in a straight line acted upon by a constant resisting force of magnitude \( F \). If it starts with a speed of \( v_0 \),\n- (a) how long will it take before coming to rest and\n- (b) what distance will it travel in this time? | (a) \( \frac{mv_0}{F} \), (b) \( \frac{mv_0^2}{2F} \) | So I've got this problem here. There's a particle with mass \( m \) moving in a straight line, and there's a constant resisting force of magnitude \( F \) acting against it. It starts with a speed \( v_0 \), and I need to figure out two things: first, how long it will take before it comes to rest, and second, what dist... |
Referring to Problem 3.1, show that the speed of the particle at any position \( x \) is given by \( v = \sqrt{v_0^2 + (2F/m)x} \). | **Method 1.**\n\nFrom (3) of Problem 3.1, we have \( t = m(v - v_0)/F \). Substituting into (4) and simplifying, we find \( x = (m/2F)(v^2 - v_0^2) \). Solving for \( v \) we obtain the required result.\n\n**Method 2.**\n\nFrom (1) of Problem 3.1, we have\n\n\[\n\frac{dv}{dt} = \frac{F}{m}, \quad \text{i.e.} \quad \fra... | So I'm trying to solve this problem, and I need to find the speed of a particle at any position \( x \), given that the force acting on it is constant. The problem refers to Problem 3.1, but I don't have the specific details here, so I'll assume that in Problem 3.1, we have a particle experiencing a constant force \( F... |
A horizontal crane beam of the length \( l \) is hinged at one end, and at the other end \( B \) it is suspended from the wall by a brace rod \( BC \) forming an angle of inclination \( \alpha \) with the horizontal (Fig. 50). A load \( P \) moves along the beam, and its position is defined by the distance \( x \) to t... | T = \frac{Px}{l \sin \alpha}. | So I've got this problem here about a horizontal crane beam that's hinged at one end and supported by a brace rod at the other end. The beam is horizontal, and the brace rod is at an angle alpha with the horizontal. There's a load P that can move along the beam, and I need to find the tension T in the brace rod as a fu... |
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