Chapter 14, Problem 1P

(a)

To determine

Whether the statement is true or false.

Explanation of Solution

Introduction:

Simple harmonic oscillator is any system which when displaced from its equilibrium position, exhibits a restoring force where restoring force is directly proportional to the displacement.

The time period in the simple harmonic oscillator is the time taken to complete full oscillation.

Write the expression for the time period of the simple harmonic oscillator.

  $T=2\pi \sqrt{\frac{m}{k}}$

Here, $T$ is the time period of the simple harmonic oscillator, $k$ is the force constant and $m$ is the mass of the particle.

Conclusion:

The time period of a simple harmonic oscillator does not depend on the amplitude. Thus, the statement is not correct.

(b)

To determine

Whether the statement is true or false.

Explanation of Solution

Write the expression for the time period of the simple harmonic oscillator.

  $T=2\pi \sqrt{\frac{m}{k}}$

The frequency in the simple harmonic oscillator is inversely proportional to the time period.

  $f\propto \frac{1}{T}$   ............. (1)

Here, $f$ is the frequency of the simple harmonic oscillator

Substitute $2\pi \sqrt{\frac{m}{k}}$ for $T$ in equation (1).

  $\begin{array}{l}f=\frac{1}{2\pi \sqrt{\frac{m}{k}}}\\ f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}\end{array}$

Thus, the frequency of the simple harmonic oscillator does not depend on amplitude, it depends on two factors one is mass and the other is force constant.

Conclusion:

The frequency of the simple harmonic oscillator does not depend on the amplitude. Thus, the statement is correct.

(c)

To determine

Whether the statement is true or false.

Explanation of Solution

The condition for the simple harmonic oscillator is that force should be directly proportional to the displacement and the displaced object tends to move in its equilibrium position. The restoring force always acts in the direction opposite to the particle displacement. For any system to be simple harmonic the acceleration of the particle is in the opposite direction from its equilibrium position and acceleration should be proportional to the displacement from the equilibrium position.

Conclusion:

The net force in the particle is one-dimensional motion which is directly proportional to the displacement made by the particle from its equilibrium position. Thus, the statement is correct.