Chapter 4, Problem 4.7P

Calculate the maximum values of the amplitudes of the response functions shown in Figures 3-22 and 3-24. Obtain numerical values for β = 0.2ω0 when a = 2 m/s2, ω0 = 1 rad/s, and t0 = 0.

The total energy of a simple harmonic oscillator with amplitude 3.00 cm is 0.500 J. a. What is the kinetic energy of the system when the position of the oscillator is 0.750 cm? b. What is the potential energy of the system at this position? c. What is the position for which the potential energy of the system is equal to its kinetic energy? d. For a simple harmonic oscillator, what, if any, are the positions for which the kinetic energy of the system exceeds the maximum potential energy of the system? Explain your answer. FIGURE P16.73

C, N A uniform plank of length L and mass M is balanced on a fixed, semicircular bowl of radius R (Fig. P16.19). If the plank is tilted slightly from its equilibrium position and released, will it execute simple harmonic motion? If so, obtain the period of its oscillation.

Plot a velocity resonance curve for a driven, damped oscillator with Q = 6, and show that the full width of the curve between the points corresponding to is approximately equal to ω0/6.

A particle of mass m moving in one dimension has potential energy U(x) = U0[2(x/a)2 (x/a)4], where U0 and a are positive constants. (a) Find the force F(x), which acts on the particle. (b) Sketch U(x). Find the positions of stable and unstable equilibrium. (c) What is the angular frequency of oscillations about the point of stable equilibrium? (d) What is the minimum speed the particle must have at the origin to escape to infinity? (e) At t = 0 the particle is at the origin and its velocity is positive and equal in magnitude to the escape speed of part (d). Find x(t) and sketch the result.

The angular frequency and amplitude of a simple pendulum are w and A respectively. At a displacement x from the mean position , if its KE is K and PE is U , find the ratio K/U.

A simple harmonic oscillator consists of a spring with a force constant of k = 59.5 N/m connected to a block with a mass of m = 1.05 kg. Assume that the surface supporting the block is flat and frictionless, and there is no air resistance. At time t = 0, the spring is stretched 16.5 cm from its natural length and the block is moving at a speed of 1.95 m/s in the −x direction, as shown below. (a) Find x(t), the displacement of the block from equilibrium as a function of time. Hint: you’ll need to find the constants ω (in rad/s), A (in cm), ? (in radians) for the function: x(t) = Acos(ωt + φ). (b) What is the velocity of the block (in m/s) at t = 2.00 s? (c) What is the acceleration of the block (in m/s2) at t = 2.00 s?