Question

In the experiment, you will study an oscillator called a "torsion pendulum." In this case, the restoring "force" is the torsion constant of the wire that suspends the weight X and the inertial term is the rotational inertia of the suspended mass. You will compare the periods of a suspended sphere and of a suspended cube. The rotational inertia of a sphere is Is = 110msD2, where ms is the mass of the sphere and D is its diameter. The rotational inertia of a cube is Ic = 16mcS2, where mc is the mass of the cube and S is the length of its side. If the cube and the sphere are suspended from the same wire, what is the expected ratio of their periods, Tc/Ts? Assume that D = S,

Answer 1

The expected ratio of the periods of the suspended sphere and cube, Tc/Ts, can be found by taking the square root of the ratio of their rotational inertias. The rotational inertia of the sphere is 110msD^2 and the rotational inertia of the cube is 16mcS^2, where ms is the mass of the sphere, D is its diameter, mc is the mass of the cube, and S is the length of its side. Since D = S, the formula for the ratio of the periods simplifies to sqrt(16mc/(110ms)).

Step-by-step explanation:

The expected ratio of the periods, Tc/Ts, can be found by using the formulas for the rotational inertias of the sphere and the cube. For the sphere, the rotational inertia Is = 110msD^2, where ms is the mass of the sphere and D is its diameter. For the cube, the rotational inertia Ic = 16mcS^2, where mc is the mass of the cube and S is the length of its side. Since D = S, we can substitute S for D in the formulas.

The period of the pendulum is proportional to the square root of the rotational inertia. So, the ratio of the periods is given by:

Tc/Ts = sqrt(Ic/Is)

Substituting the formulas for rotational inertia:

Tc/Ts = sqrt((16mcS^2)/(110msD^2))

Since D = S:

Tc/Ts = sqrt((16mcS^2)/(110msS^2))

Canceling out the S^2 terms:

Tc/Ts = sqrt(16mc/(110ms))

Therefore, the expected ratio of the periods, Tc/Ts, is sqrt(16mc/(110ms)).

Answer 2

`(T_c)/(T_s)= 0.381`

Step-by-step explanation:

Given that

For sphere

Is = 110msD²

For cube

Ic = 16mcS²

D= S

We know that time period of torsion pendulum is directly proportional to the square root of the rotational inertia.

`T=2\pi \sqrt{(I)/(K)}`

`(T_s)/(T_c)= \sqrt{(I_s)/(I_c)}`

Now by putting the values

`(T_s)/(T_c)= \sqrt{(110)/(16)}`

`(T_s)/(T_c)= 2.62`

`(T_c)/(T_s)= 0.381`