C.1 簡諧運動：練習題

An object is undergoing simple harmonic motion (SHM). Which of the following conditions must be satisfied for the motion to be classified as SHM?

A mass \( m \) is attached to a vertical spring with spring constant \( k \) and oscillates with a period \( T \). If the mass is replaced by a mass of \( 2m \) and the spring is replaced by one with a spring constant of \( \frac{1}{2}k \), what is the new period of oscillation?

A particle executes SHM with an amplitude \(A\). What is the ratio of its kinetic energy to its potential energy when the displacement is \(x = \frac{1}{3} A\)?

A particle undergoes simple harmonic motion with amplitude \(A\). Which graph correctly represents the variation of the particle's kinetic energy \(E_k\) with its displacement \(x\)?

A particle oscillates with simple harmonic motion. The displacement \( x \) of the particle varies with time \( t \) according to the equation \( x = A \cos(\omega t) \). What is the magnitude of the maximum acceleration of the particle?

An object undergoing simple harmonic motion has a maximum displacement of \(0.05 \text{ m}\) and an angular frequency of \(10 \text{ rad s}^{-1}\). Calculate the magnitude of its maximum acceleration.

The displacement-time graph for a particle undergoing simple harmonic motion is shown. The graph follows the function \(x = A \sin(\omega t)\). If the time interval between two consecutive points where the displacement is zero is \(0.40 \text{ s}\), determine the frequency of the oscillation.

A mass-spring system oscillates with an amplitude \(A\). Determine the displacement \(x\) from the equilibrium position, in terms of \(A\), where the kinetic energy of the mass is exactly three times its potential energy.

A simple pendulum consists of a small mass suspended from a fixed point by a string of length \(1.60 \text{ m}\). The mass is displaced by a small angle and released so that it performs simple harmonic motion.
(a) State the relationship between the acceleration and the displacement for an object undergoing simple harmonic motion.
(b) Calculate the period of the oscillation. Take \(g = 9.81 \text{ m s}^{-2}\).
(c) If the length of the string is reduced to \(0.40 \text{ m}\), calculate the new frequency of the oscillation.
(d) State the position of the mass during the oscillation at which the kinetic energy is at its maximum.

A mass of 0.50 kg is attached to a horizontal spring on a frictionless surface. The mass is pulled 0.12 m from its equilibrium position and released from rest, performing simple harmonic motion (SHM). The spring constant is \(200 \text{ N m}^{-1}\).
(a) Calculate the period of the oscillation.
(b) Determine the maximum speed of the mass.
(c) Calculate the magnitude of the acceleration when the mass is at a displacement of 0.06 m.