Oscillating systems: แบบฝึกหัด

In simple harmonic motion (SHM), which statement correctly describes the variation of the kinetic energy (\(E_k\)) of an oscillating mass with respect to its displacement (\(x\)) from the equilibrium position?

An object is undergoing SHM. When its displacement from the equilibrium position is \(4.0 \text{ cm}\), its acceleration is \(1.6 \text{ m/s}^2\). What is the period of the oscillation?

A mass of \(0.50 \text{ kg}\) is attached to a spring of stiffness \(50 \text{ N/m}\) and set into simple harmonic oscillation. What is the period of oscillation?

A simple pendulum has a period \(T\) when oscillating with a small amplitude on Earth. If the length of the pendulum is increased by \(21\%\) and it is moved to a planet where the gravitational field strength is \(1.21\) times the Earth's gravitational field strength, what is the new period, \(T'\), in terms of \(T\)?

An object undergoes simple harmonic motion with an amplitude of \(15 \text{ cm}\) and a frequency of \(4.0 \text{ Hz}\). Calculate the maximum speed of the object.

Define the condition for Simple Harmonic Motion (SHM) using acceleration \(a\) and displacement \(x\), and explain the physical significance of the constant of proportionality.

A mass undergoes Simple Harmonic Motion (SHM). Given its maximum speed \(v_{max}\) and maximum acceleration \(a_{max}\), derive the amplitude \(A\) in terms of these two quantities.

A mass \(m\) attached to a spring with spring constant \(k\) oscillates with period \(T\). If the mass is replaced by \(2m\), calculate the new spring constant \(k'\) required to maintain the period at \(T\).

A mass of $m = 0.50 \text{ kg}$ is attached to a vertical spring with a spring constant $k = 200 \text{ N m}^{-1}$. The mass is pulled down a distance of $4.0 \text{ cm}$ from its equilibrium position and released from rest. Air resistance is negligible.

(a) Show that the motion of the mass is simple harmonic and calculate the period of oscillation, $T$.

(b) Determine the maximum speed of the mass during its oscillation.

(c) Calculate the maximum kinetic energy and the total energy of the system.

(d) Explain why the variation of gravitational potential energy throughout the motion does not appear explicitly in the calculation of the total mechanical energy of the Simple Harmonic Oscillator when analyzing motion relative to the equilibrium position.

A mass $m = 0.20 \text{ kg}$ oscillates horizontally on a frictionless surface attached to a spring with spring constant $k$. The motion is Simple Harmonic Motion, and the total energy of the oscillating system is $0.045 \text{ J}$.

(a) If the amplitude of oscillation is $A = 5.0 \text{ cm}$, calculate the spring constant $k$ of the spring.

(b) Determine the maximum acceleration of the mass during its oscillation.

(c) Calculate the magnitude of the force acting on the mass when its velocity is exactly half of its maximum velocity.

(d) Describe the phase relationship between the displacement $x$ and the acceleration $a$ of the mass, stating the phase difference in radians.