Problem : A disk of mass 2 kg and radius .5 m is hung from a wire, then rotated a small angle such that it engages in torsional oscillation. The period of oscillation is measured at 2 seconds. Given that the moment of inertia of a disk is given by I = , find the torsional constant, κ , of the wire.
To solve this problem we use the equation for the period of a torsional oscillator:
Solving for κ ,
We are given T , and must simply compute I . Given the dimensions of the disk, we can simply plug into the formula we are given for the moment of inertia: I = = = .25 . Thus:
Problem : The disk from problem 1 is replaced with an object of unknown mass and shape, and rotated such that it engages in torsional oscillation. The period of oscillation is observed to be 4 seconds. Find the moment of inertia of the object.
To find the moment of inertia we use the same equation:
Solving for I,
>From last problem we know that κ = , and we are given the period (4 seconds). Thus:
In the last two problems we have established a method for determining the moment of inertia of any object.
Problem : A pendulum of length L is displaced an angle θ , and is observed to have a period of 4 seconds. The string is then cut in half, and displaced to the same angle θ . How does this effect the period of oscillation?
We turn to our equation for the period of the pendulum:
Clearly if we reduce the length of the pendulum by a factor of 2 we reduce the period of the oscillation by a factor of 4.
Problem : A pendulum is commonly used to calculate the acceleration due to gravity at various points around the earth. Often areas with low acceleration indicate a cavity in the earth in the area, many times filled with petroleum. An oil prospector uses a pendulum of length 1 meter, and observes it to oscillate with a period of 2 seconds. What is the acceleration due to gravity at this point?
We use the familiar equation:
Solving for g:
g | = | ||
= | = 9.87 m/s^{2} |
Problem : What is the angular velocity of a particle moving in uniform circular motion that has the same period as a mass of 2 kg on a spring with constant 8 N/m?
Recall from our comparison of circular and oscillatory motion that the angular velocity of a particle in circular motion corresponds with the angular frequency of a particle in oscillatory motion. We know the angular frequency of the mass- spring system:
Thus we can infer that the particle moves in a circle with constant angular velocity of 2 rad/s.