Question:

A disk of radius 0.5 m is rotating with an angular velocity of 4 rad/s. The disk accelerates uniformly at a rate of 2 rad/s² for a period of 3 seconds and then decelerates uniformly at a rate of 3 rad/s² for another 2 seconds.

(a) What is the final angular velocity of the disk?

(b) Calculate the total angle through which the disk has rotated during the entire 5-second period.

(c) Determine the average angular acceleration of the disk during the 5-second period.

(d) What is the total time taken by the disk to come to rest from its initial angular velocity?

Answer:

(a) To find the final angular velocity of the disk, we can use the formula:

𝑣 = 𝑣0 + 𝑎𝑡

where 𝑣 is the final angular velocity, 𝑣0 is the initial angular velocity, 𝑎 is the angular acceleration, and 𝑡 is the time.

Given 𝑣0 = 4 rad/s, 𝑎 = -3 rad/s² (negative sign indicates deceleration), and 𝑡 = 2 s, we have:

𝑣 = 4 + (-3 × 2) = 4 - 6 = -2 rad/s

Therefore, the final angular velocity is -2 rad/s (negative sign indicating the opposite direction of rotation).

(b) To find the total angle through which the disk has rotated during the entire 5-second period, we need to calculate the angular displacement during each phase of acceleration and deceleration.

For the phase of acceleration, we can use the formula:

𝜃 = 𝜃0 + 𝜔0𝑡 + 0.5𝑎𝑡²

where 𝜃 is the angular displacement, 𝜃0 is the initial angular displacement (assumed to be 0), 𝜔0 is the initial angular velocity, 𝑎 is the angular acceleration, and 𝑡 is the time.

Given 𝜔0 = 4 rad/s, 𝑎 = 2 rad/s² (during acceleration), and 𝑡 = 3 s, we have:

𝜃_acceleration = 0 + (4 × 3) + 0.5 × 2 × (3)² = 12 + 9 = 21 rad

For the phase of deceleration, we can use the same formula:

𝜃_deceleration = 0 + (-2 × 2) + 0.5 × (-3) × (2)² = -4 - 6 = -10 rad

Adding the angular displacements from both phases, we find:

Total angular displacement = 𝜃_acceleration + 𝜃_deceleration = 21 + (-10) = 11 rad

Therefore, the total angle through which the disk has rotated during the entire 5-second period is 11 radians.

(c) The average angular acceleration of an object can be calculated using the formula:

Average angular acceleration = (final angular velocity - initial angular velocity) / time

We already know the final angular velocity is -2 rad/s, the initial angular velocity is 4 rad/s, and the time is 5 s. Therefore:

Average angular acceleration = (-2 - 4) / 5 = -6 / 5 rad/s²

Therefore, the average angular acceleration of the disk during the 5-second period is -1.2 rad/s².

(d) To find the total time taken by the disk to come to rest from its initial angular velocity, we can use the formula:

𝑡 = (𝑣 - 𝑣0) / 𝑎

where 𝑣 is the final angular velocity (0 rad/s for rest) and 𝑣0 is the initial angular velocity (4 rad/s).

Substituting the values, we have:

𝑡 = (0 - 4) / (-3) = 4 / 3 s

Therefore, the total time taken by the disk to come to rest from its initial angular velocity is approximately 1.33 seconds.