Torsional pendulum, logarithmic decrement

[tone999]

Homework Statement

The amplitude of a torsional vibration decreases so that the amplitude on the 100th cycle is 13% of the the amplitude of the first cycle. Determine the level of damping in terms of the logarithmic decrement.

Homework Equations

The Attempt at a Solution

Is this simply ln(100/13)= 2.04

or ln(13/100)= -2.04?

 

[Redbelly98]

It will be a positive number (use 100/13, not 13/100). However, you still need to account for the fact that it takes 100 cycles to reach 13%.

 

[Mene]

I would first clarify with the person providing the content what they mean by "damping" in this context. Damping can refer to the resistance of a system to oscillate, or it can refer to the decrease in amplitude over time. Without this clarification, it is difficult to provide an accurate response.

Assuming they are referring to the decrease in amplitude over time, then the correct equation to use would be:

ln(A_n/A_n+1) = 2πδ

Where A_n is the amplitude of the nth cycle and δ is the logarithmic decrement.

Using the given information, we can set up the following equation:

ln(13/A_100) = 2πδ

Solving for δ, we get:

δ = ln(13/A_100)/(2π)

Without knowing the amplitude of the first cycle, we cannot accurately determine the level of damping in terms of the logarithmic decrement. However, we can say that the larger the value of δ, the greater the level of damping in the system.