Rotational Motion And Torque

To determine the power delivered by the applied torque, we need to use the relationship between torque, angular velocity, and power. The power delivered by the torque is given by:

where:

• is the torque,
• is the angular velocity.

First, we need to find the angular velocity , which is the derivative of the angular position function with respect to time .

Given:

Find :

Evaluating at s:

Next, we find the torque from the second derivative of , which gives the angular acceleration .

The angular acceleration is:

The moment of inertia of a circular disk rotating about an axis perpendicular to its plane is given by:

Torque is related to angular acceleration by:

Substituting the values:

Now we can calculate the power:

Thus, the power delivered by the applied torque at s is .

To solve the problem of finding the ratio of the squares of the speeds and , we need to analyze the dynamics of a rolling sphere on an inclined plane.

The speed at the bottom of the inclined plane results from the conversion of potential energy to kinetic energy. For a solid sphere rolling without slipping on an incline:

where is the acceleration due to gravity, and is the height of the incline.

The height of the incline can be calculated using the length and the angle of inclination :

So, the velocity at the bottom can be expressed as:

Let's consider the two scenarios:

• When the angle is , the speed is:
• When the angle is , the speed is:

Now, to find the ratio , calculate the squares of both expressions:

Therefore, the ratio is:

This matches the given correct answer: . Thus, the correct option is .

Step 1:Find velocity} At :
Step 2:Linear momentum} Thus statement A is correct.
Step 3:Force At : Thus B is correct.
Step 4:Angular momentum} At : Thus C is correct.
Step 5:Torque} Thus D is correct. Hence the correct statements are