Question

A uniform disk rotating with constant angular acceleration covers 50 complete rotations in the first 5 seconds after the start. Calculate the angular acceleration and the angular velocity at the end of 5 seconds.

(a) Angular Acceleration: 10 rad/s², Angular Velocity: 50 rad/s
(b) Angular Acceleration: 5 rad/s², Angular Velocity: 25 rad/s
(c) Angular Acceleration: 2 rad/s², Angular Velocity: 10 rad/s
(d) Angular Acceleration: 1 rad/s², Angular Velocity: 5 rad/s

Answer 1

The angular acceleration of the disk is approximately 25.13 rad/s², and the angular velocity at the end of 5 seconds is 125.66 rad/s. None of the provided options match these calculations.

Step-by-step explanation:

The student is asking about the angular acceleration and angular velocity of a disk that completes 50 rotations in 5 seconds. To solve for the angular acceleration, α, we can use the equation for angular position under constant angular acceleration, θ = αt²/2 + ω0t, where ω0 is the initial angular velocity, which we assume to be 0 since the disk starts from rest. Given that 1 complete rotation is 2π radians, the disk has covered 100π radians in 5 seconds. Substituting the given values into the equation, we get 100π = α(5²)/2.

Solving for α gives us α = 8π rad/s², which is approximately 25.13 rad/s². To find the angular velocity at the end of 5 seconds, we use the relation ω = αt, which yields ω = 25.13 rad/s² * 5 s, resulting in ω = 125.66 rad/s. So, the correct answer is none of the provided options.