An engineer has an odd-shaped 11.4 kg object and needs to find its rotational inertia about an axis through its center of mass. The object is supported on a wire stretched along the desired axis. The wire has a torsion constant κ = 0.428 N·m. If this torsion pendulum oscillates through 23 cycles in 45.9 s, what is the rotational inertia of the object

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I thought you could get Icm of the object through T=2pi x sqrt(Icm/K). Solving for Icm with (T/2pi)^2 x k = Icm
That however does not work. I really don't know what to do about it. Thanks in advance.

I figured out what I was doing wrong. 23cycles in 45.9s give the frequency. Not the period. I was using the frequency as T.

To find the rotational inertia (Icm) of the object, you can use the formula T = 2π√(Icm / κ), where T is the period of oscillation (time for one complete cycle) and κ is the torsion constant.

In the given problem, we are provided with the following information:
- Period of oscillation (T) = 45.9 s
- Number of cycles (n) = 23
- Torsion constant (κ) = 0.428 N·m

First, we need to find the time for one cycle (t) by dividing the total time by the number of cycles:
t = T / n = 45.9 s / 23 = 1.995 s

Now, we can use the formula T = 2π√(Icm / κ) and solve for Icm:
T = 2π√(Icm / κ)

Rearranging the formula, we get:
(Icm / κ) = (T / 2π)^2

Substituting the values, we have:
(Icm / 0.428) = (1.995 / (2π))^2

Simplifying further:
Icm = 0.428 * (1.995 / (2π))^2

Calculating this expression, you will find the rotational inertia (Icm) of the object. Remember to use the appropriate units for the final answer.

I hope this explanation helps! Let me know if you have any further questions.