A simple pendulum has a mass of 0.202 kg and a length of 1.00 m.It is displaced through an angle of 31.4° and then released.After a time,the maximum angle of swing is only 10.0°.How much energy has been lost to friction?
A)0.260 J
B)0.290 J
C)0.0301 J
D)1.00 J
E)0.320 J

Question

Quizplus · Accepted Answer

The energy lost to friction can be calculated by finding the difference in potential energy at the two different angles. The potential energy in a pendulum's swing is given by $PE = mgh$, where $m$ is the mass, $g$ is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$), and $h$ is the height relative to the lowest point of the swing. The height $h$ can be found using the formula $h = L - L\cos(\theta)$, where $L$ is the length of the pendulum and $\theta$ is the angle of displacement from the vertical in radians.For the initial displacement of $31.4^\circ$:- Convert the angle to radians: $31.4^\circ = 31.4 \times \frac{\pi}{180} = 0.548 \, \text{radians}$.- Calculate the initial height: $h_i = L - L\cos(0.548) = 1 - 1\cos(0.548) \approx 0.15 \, \text{m}$.- Calculate the initial potential energy: $PE_i = mgh_i = 0.202 \times 9.8 \times 0.15 \approx 0.297 \, \text{J}$.For the final displacement of $10.0^\circ$:- Convert the angle to radians: $10.0^\circ = 10.0 \times \frac{\pi}{180} = 0.175 \, \text{radians}$.- Calculate the final height: $h_f = L - L\cos(0.175) = 1 - 1\cos(0.175) \approx 0.015 \, \text{m}$.- Calculate the final potential energy: $PE_f = mgh_f = 0.202 \times 9.8 \times 0.015 \approx 0.030 \, \text{J}$.The energy lost to friction is the difference between the initial and final potential energies: $E_{lost} = PE_i - PE_f \approx 0.297 \, \text{J} - 0.030 \, \text{J} \approx 0.267 \, \text{J}$.The closest answer to this calculation is $0.260 \, \text{J}$, acknowledging some rounding differences in the calculation process.

A Simple Pendulum Has a Mass of 0