Chapter 14, Problem 85A

To determine

The energy required to make a pulse with twice the amplitude in a stretched spring.

Explanation of Solution

Introduction:

The energy of sound wave or pulse is related to its amplitude. The energy required to make a pulse with twice the amplitude is four times of the energy.

Energy of waves is directly proportional to the square of the amplitude of the wave. Wave is actually a displacement acted upon by a restoring force. In a spring the restoring force is given as,

  $F=kx$    .......(1)

Here $F$ is the force, $k$ is the spring constant and $x$ is the amplitude or displacement.

Energy is given by the integral of force,

  $E={\displaystyle \underset{0}{\overset{x}{\int }}Fdx}={\displaystyle \underset{0}{\overset{x}{\int }}kxdx}=\frac{k{x}^2}{2}$    .......(2)

Here, $E$ is the energy.

From equation (2) it is clear that, the energy is equal to the square of the displacement.

In case of a pulse, this displacement is the amplitude. If amplitude is twice then the energy required is four times.

Conclusion:

Hence, the energy required to make a pulse with twice the amplitude is four times of the energy.