Final answer:
The fraction of the pendulum's remaining energy after 3 minutes is found by taking the ratio of the squares of the final to initial amplitudes. Given the reduction in amplitude from 5m to 2m, the pendulum retains 16% of its original energy.
Step-by-step explanation:
The question involves finding out what fraction of the pendulum's original energy remains after 3 minutes, given that its amplitude has decreased. To determine this, we need to understand that the potential energy (PE) at the height of the pendulum swing directly correlates to the square of the amplitude. If the amplitude decreases, the potential energy decreases proportionally to the square of the reduction in amplitude.
To find the fraction of the remaining energy, you would use the ratio of the squares of the final amplitude to the initial amplitude. Originally, the pendulum was released from an arc length of 5 meters, and after 3 minutes, it reached an arc length of just 2 meters.
Here's the calculation:
Energyfinal/Energyinitial = (Amplitudefinal/Amplitudeinitial)^2
= (2m/5m)^2
= 0.16
Therefore, the pendulum retains 16% of its original energy after 3 minutes, which is the answer in percentage form.