Question:

A disk with a radius of 0.2 meters is rotating with an angular velocity of 5 radians per second. The disk starts from rest and completes 20 revolutions before coming to a stop. What is the time it takes for the disk to come to a stop?

(A) 2.5 seconds (B) 4 seconds (C) 5 seconds (D) 10 seconds

Answer:

To solve this problem, we need to use the rotational kinematic equation that relates angular acceleration, initial angular velocity, final angular velocity, and time:

ωf = ωi + αt

Where: ωf = final angular velocity in radians per second ωi = initial angular velocity in radians per second α = angular acceleration in radians per second squared t = time in seconds

In this problem, the disk starts from rest, so the initial angular velocity (ωi) is zero. The final angular velocity (ωf) is also zero because the disk comes to a stop. We need to determine the time it takes for the disk to stop, so we rearrange the equation and solve for time:

ωf = ωi + αt

0 = 5 + αt

Since the disk starts from rest, its initial angular velocity (ωi) is zero. Therefore, the equation becomes:

0 = αt

Since the final angular velocity is also zero, the angular acceleration (α) must be negative, indicating that the disk is decelerating. The negative sign indicates a change in the direction of rotation.

We know that the disk completes 20 revolutions before coming to a stop. One revolution is equal to 2π radians. Therefore, the total angular displacement (θ) can be calculated as follows:

θ = (number of revolutions) * (2π radians/revolution) θ = 20 * 2π

Now, we can calculate the angular acceleration (α) using the kinematic equation for rotational motion:

θ = ωi * t + (½) α * t^2

Since the initial angular velocity (ωi) is zero, the equation becomes:

θ = (½) α * t^2

Substituting the value of angular displacement (θ) and the number of revolutions, we get:

20 * 2π = (½) α * t^2

Rearranging the equation, we get:

α * t^2 = 20 * 2π * 2

Simplifying,

α * t^2 = 80π

We need to determine the time (t), so we divide both sides of the equation by α:

t^2 = (80π) / α

Since α is negative in this case, we need to use the absolute value of α when calculating t. Taking the square root of both sides, we get:

t = √[(80π) / |α|]

The angular acceleration (α) can be calculated using:

α = (ωf - ωi) / t

As ωi = 0 and ωf = 0, we get:

α = 0 / t α = 0

Substituting this value back into the equation for t:

t = √[(80π) / |0|] t = √[(80π) / 0] t = √(∞)

Since division by zero is undefined, we can conclude that the time it takes for the disk to stop is infinite. Therefore, the correct answer is (E) None of the above.

Note: This question is designed to test conceptual understanding and problem-solving skills related to rotational kinematics and dynamics. Students are expected to understand and apply the equations of rotational motion, as well as the relationships between angular velocity, angular acceleration, time, and angular displacement.