Problem 18 - University Physics with Modern Physics

Question

Pendulum A has a bob of mass \(m\) hung from a string of length \(I_{i}\) pendulum \(B\) is identical to \(A\) except its bob has mass \(2 m\). Compare the frequencies of small oscillations of the two pendulums.

Step-by-Step Solution

Answer

Question: Compare the frequencies of small oscillations of two pendulums, A and B, where pendulum A has a bob of mass m and string length L, and pendulum B has a bob of mass 2m and the same string length L as pendulum A. Answer: The frequencies of small oscillations of pendulums A and B are the same.

Step 1: Recall the formula for the frequency of a simple pendulum

The formula for the frequency of a simple pendulum is given by: $$ f=\frac{1}{2\pi}\sqrt{\frac{g}{L}} $$ where \(f\) is the frequency, \(g\) is the acceleration due to gravity, and \(L\) is the length of the pendulum.

Step 2: Find the frequency of pendulum A

Since pendulum A has a length of \(L_i\) and a bob of mass \(m\), we can substitute these values into the frequency formula to find its frequency: $$ f_A=\frac{1}{2\pi}\sqrt{\frac{g}{L_i}} $$

Step 3: Find the frequency of pendulum B

Pendulum B has the same length as pendulum A, \(L_i\), but with a bob of mass \(2m\). Since the mass of the bob does not affect the frequency of a simple pendulum, the formula for the frequency of pendulum B is the same as for pendulum A. Hence, the frequency of pendulum B is: $$ f_B=\frac{1}{2\pi}\sqrt{\frac{g}{L_i}} $$

Step 4: Compare the frequencies of small oscillations of the two pendulums

We can now compare the frequencies \(f_A\) and \(f_B\): Since both pendulums have the same length \(L_i\) and the mass of the bob does not affect their frequencies, their frequencies will be the same. $$ f_A=f_B $$ In conclusion, the frequencies of small oscillations of pendulums A and B are the same, regardless of the mass of their bobs.