A horizontal disc rotating freely about a vertical axis makes 90 revolutions per minute reduces to 60,then find the moment of inertia of the disc?

1. mr2
2. 3/2mr2
3. 2mr2
4. 3mr2

(the last ‘2’in each option should be considered as square, please)

More information is required on the torque and how long it takes to slow down. If it is "rotating freely" then why is it slowing down at all?

Your question is incomplete. What effort have you made to solve it yourself?

To find the moment of inertia of the disc, we can use the concept of conservation of angular momentum.

The angular momentum (L) of a rotating object is given by the equation L = Iω, where I is the moment of inertia and ω is the angular velocity.

Given that the disc initially makes 90 revolutions per minute, we first need to convert this to the angular velocity in radians per second.

We know that one revolution is equivalent to 2π radians, so the initial angular velocity (ω1) can be calculated as follows:

ω1 = (90 revolutions/minute) * (2π radians/revolution) * (1 minute/60 seconds)
= 3π radians/second

Next, we are given that the disc reduces to 60 revolutions per minute. We can follow the same steps to calculate the new angular velocity (ω2):

ω2 = (60 revolutions/minute) * (2π radians/revolution) * (1 minute/60 seconds)
= 2π radians/second

Now, using the conservation of angular momentum, we can set up the following equation:

I1 * ω1 = I2 * ω2

Rearranging the equation, we can solve for I1 (the initial moment of inertia):

I1 = (I2 * ω2) / ω1

Plugging in the values we calculated for ω1 and ω2, we get:

I1 = (I2 * 2π) / (3π)
= 2I2 / 3

So, the moment of inertia of the disc is 2/3 times the moment of inertia when it rotates with a reduced speed.

Now let's consider the options provided:

1. mr^2: This is not the correct option because we found that the moment of inertia is equal to 2I2 / 3.

2. 3/2mr^2: This is not the correct option as it does not match our calculated result.

3. 2mr^2: This is not the correct option either because we found that the moment of inertia is equal to 2I2 / 3.

4. 3mr^2: This is the correct option because when we substitute I2 with 3mr^2 (where m is the mass of the disc and r is the radius), we get 2(3mr^2) / 3 = 2mr^2, which matches our calculated result.

Therefore, the correct answer is option 4), 3mr^2.