Principles of Engineering Mechanics: Volume 2 Dynamics -- The Analysis of Motion by Millard F. Beatty | 9780387237046 | 0387237046 | Holooly

Mechanics Dynamics, Mechanical Engineering

Principles of Engineering Mechanics: Volume 2 Dynamics -- The Analysis of Motion

85 SOLVED PROBLEMS

Question: 10.10

A homogeneous thin rod of mass m and length l is connected to a vertical shaft S by a smooth hinge bearing at Q. The shaft rotates with a constant angular velocity Ω, as shown in Fig. 10.11. (i) Derive the equation of motion of the rod. (ii) Determine as a function of θ the hinge bearing reaction ...

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(i). The central point

\mathcal{Q}

...

Question: 8.5

Find the kinetic energy of the antenna system in Example 8.3, page 314. ...

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The center of mass C of the two coil system has ve...

Question: 11.6

(i) Derive the equations for the uniaxial motion of the spring-mass system shown in Fig. 11.1. The supporting surface is smooth and all springs are linearly elastic and unstretched initially. (ii) Determine the motion of the system for the special symmetric case when m1 = m2 = m and k1 = k2 = k. ...

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Solution of (i). The holonomic constraints for the...

Question: 11.5

(a) Derive the Lagrange equations of motion for a heavy bead of mass m that slides freely in a smooth circular tube of radius a, as the tube spins with constant angular speed Φ = ω about its fixed vertical axis, as shown in the diagram for Problem 6.66. Obtain the first integral of the equation ...

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Solution of (a). Introduce the spherical coordinat...

Question: 11.10

A rigid body shown in Fig. 11.4 is driven by a torque μ(t) about a fixed, principal body axis k in a smooth bearing at H . (i) Apply (11.73) to derive the equationof motionfor the body.(ii) Repeatthe derivation from (11.38). Show that the result has the familiar form of the equation of motion of a ...

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Solution of (i). The system is holonomic with one ...

Question: 11.8

Two particle s of equal mass m are attached to the ends of a massless rigid rod of length ℓ initially oriented parallel to the y-axis of a frame ψ = {O;ik} and at rest on a smooth horizontal surface. An instantaneous impulsive normal force P = Pi acts on the particle closer to O. Determine the ...

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Following the impulse, the center of mass of the s...

Question: 11.7

A nonconservative holonomic system having two degrees of freedom with generalized coordinates (q1, q2) and corresponding generalized forces QN1 = -mb²vq·1, QN2 = 0, has a Lagrangian function L = 1/2 ma² sin² q1 + mb² (q·2 + a/b cos q1)² + 1/2 mb² (q·1 + c)², (11.48a)2 b 2 in which a, b, c, and m ...

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Notice that

q_2

is ignorable and ...

Question: 11.4

Derive the equation of motion for a particle P that falls from rest in a Stokes medium . ...

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The kinetic energy of P is

T=\frac{1}{2} ...

Question: 11.1

Apply Lagrange’s equations (11.15) to derive the equations of unconstrained motion of a particle in cylindrical coordinates. ...

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The three independent generalized coordinates and ...

Question: 10.13

A physical pendulum shown in Fig. 10.15 swings in its vertical plane of symmetry about a smooth, fixed axle at Q. The pendulum is released from rest at the placement θo with angular speed ωo. (i) Apply the work—energy principle to derive the equation of motion of the pendulum. Confirm the solution ...

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Solution of (i). The idea here is to obtain the wo...

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