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Chapter 10: Problem 68

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

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

Step by step solution

01

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.
02

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 \(A_f = A_i/20\).
03

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: $$ \frac{E_i}{E_f} = \frac{A_i^2}{A_f^2} $$
04

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: $$ \frac{E_i}{E_f} = \frac{A_i^2}{\left(\frac{A_i}{20}\right)^2} $$
05

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.

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Most popular questions from this chapter

Martin caught a fish and wanted to know how much it weighed, but he didn't have a scale. He did, however, have a stopwatch, a spring, and a \(4.90-\mathrm{N}\) weight. He attached the weight to the spring and found that the spring would oscillate 20 times in 65 s. Next he hung the fish on the spring and found that it took 220 s for the spring to oscillate 20 times. (a) Before answering part (b), determine if the fish weighs more or less than \(4.90 \mathrm{N}\). (b) What is the weight of the fish?
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