Question

The amplitude of oscillation of a pendulum decreases by a factor of 20.0 in \(120 \mathrm{s}\). By what factor has its energy decreased in that time?

Short answer

Answer: The energy of the pendulum decreases by a factor of 400.

Step-by-step solution

Understanding the relationship between energy and amplitude of pendulum oscillation

The energy of a pendulum is proportional to the square of its amplitude. The equation for the energy of the oscillating pendulum is \(E \propto A^2\). If the amplitude decreases by a certain factor, we want to find the factor by which the energy decreases.

Calculate the decrease in amplitude

The amplitude of the pendulum decreases by a factor of 20.0 in 120 seconds. Let's assume the initial amplitude was \(A_i\) so the final amplitude after 120 seconds would be \(A_f = A_i/20\).

Find the ratio of initial and final energy

We want to find the factor by which the energy has decreased. To find this, we can calculate the ratio of the initial energy \(E_i\) to the final energy \(E_f\). Using the relation \(E \propto A^2\), we get: $$ \frac{E_i}{E_f} = \frac{A_i^2}{A_f^2} $$

Substitute the values of initial and final amplitude

Substitute the values of initial and final amplitude \(A_i\) and \(A_f = A_i/20\) into the ratio equation: $$ \frac{E_i}{E_f} = \frac{A_i^2}{\left(\frac{A_i}{20}\right)^2} $$

Simplify the ratio

Simplify the equation to find the factor by which the energy has decreased: $$ \frac{E_i}{E_f} = \frac{A_i^2}{\frac{A_i^2}{400}} = 400 $$ The energy of the pendulum has decreased by a factor of 400 in 120 seconds.