An engineer has an odd-shaped object of mass 16 kg and needs to find its rotational inertia about an axis through its center of mass. The object is supported on a wire along the desired axis. The wire has a torsion constant κ = 0.4 N· m. If this torsion pendulum oscillates through 40 complete cycles in 50 s, what is the rotational inertia of this object?
I assume you use frequency somehow but, not sure how t change it into inertia
To find the rotational inertia of the object, we can use the formula for the period of a torsional pendulum:
T = 2π * √(I / κ)
Where:
T is the period of oscillation
I is the rotational inertia of the object
κ is the torsion constant of the wire
First, let's find the period of oscillation using the given information. We are told that the torsional pendulum completes 40 complete cycles in 50 seconds.
One complete cycle consists of an oscillation from one extreme end to the other and back to the starting position. So, the total time for 40 complete cycles is 50 seconds.
The period of oscillation (T) is the time taken for one complete cycle, which is the total time divided by the number of cycles:
T = 50 s / 40 = 1.25 s
Now we can rearrange the formula to solve for the rotational inertia (I):
I = (T / (2π))^2 * κ
Substituting the values we have:
I = (1.25 s / (2π))^2 * 0.4 N·m
Calculating this expression will give us the rotational inertia of the object.