Oscillations: แบบฝึกหัด

A simple pendulum undergoes undamped simple harmonic motion. Which of the following statements correctly describes the relationship between the acceleration \( a \) of the pendulum bob and its displacement \( x \) from the equilibrium position?

An object of mass \( m \) undergoes simple harmonic motion with an amplitude \( x_0 \).

At which displacement \( x \) from the equilibrium position is the kinetic energy of the object equal to three times its potential energy?

Two particles, X and Y, are oscillating in simple harmonic motion about the same equilibrium position along the same straight line. Both particles have an amplitude of oscillation equal to A and the same frequency. There is a constant phase difference of \(\frac{\pi}{3}\) rad between their motions.

What is the maximum distance between the two particles during the oscillation?

A particle is undergoing simple harmonic motion about a fixed equilibrium position. Which statement correctly describes the relationship between its acceleration \( a \) and its displacement \( x \) from the equilibrium position?

A particle of mass \( 0.50\text{ kg} \) is undergoing simple harmonic motion. The variation with the square of its displacement \( x^2 \) of the square of its velocity \( v^2 \) is shown in the graph as a straight line. The intercept on the \( v^2 \)-axis is \( 0.16\text{ m}^2\text{s}^{-2} \) and the intercept on the \( x^2 \)-axis is \( 4.0 \times 10^{-4}\text{ m}^2 \).

What is the total energy of the oscillating particle?

A simple pendulum has a period of \( 2.0\text{ s} \). Calculate its angular frequency \( \omega \) in \( \text{rad s}^{-1} \).

Explain how the amplitude of a forced oscillation at the resonance frequency changes when the degree of damping in the system is increased.

A particle in simple harmonic motion has displacement \( x = x_0 \sin(\omega t) \). Determine the displacement, in terms of the amplitude \( x_0 \), at which the kinetic energy of the particle is exactly three times its potential energy.

(a) Define simple harmonic motion. [2]

(b) A small sphere of mass \( 0.40\text{ kg} \) is undergoing simple harmonic motion with a period of \( 0.80\text{ s} \) and an amplitude of \( 3.0\text{ cm} \). Calculate the maximum speed of the sphere. [2]

An object is oscillating with simple harmonic motion of amplitude \( x_0 \).

(a) Describe the interchange between kinetic energy and potential energy during one complete oscillation, starting from the equilibrium position. [2]
(b) Show that the displacement \( x \) at which the kinetic energy of the object is equal to three times its potential energy is given by \( x = \pm 0.50x_0 \). [3]