A compact disc (CD), initially at rest, reaches an angular speed of 40 rad/s in 5.0 s.
a. What is the magnitude of its angular acceleration?
b. How many revolutions does the CD make in 5.0 s?
a. acceleration = speed / time ... 40 / 5.0
b. (average speed) * time
...[(0 + 40)/2] * 5.0
b. 40rad/s * 1rev/6.28rad * 5s = 31.85 rev.
To answer these questions, we can use the equations of rotational motion. Let's break down the problem step by step.
a. To find the magnitude of the angular acceleration (α), we can use the formula:
α = (ωf - ωi) / t,
where α is the angular acceleration, ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time.
In this case, the initial angular velocity (ωi) is 0 rad/s (since the CD is initially at rest), the final angular velocity (ωf) is 40 rad/s, and the time (t) is 5.0 s.
Substituting these values into the formula, we get:
α = (40 rad/s - 0 rad/s) / 5.0 s = 8.0 rad/s^2.
Therefore, the magnitude of the angular acceleration is 8.0 rad/s^2.
b. To find the number of revolutions the CD makes in 5.0 s, we can use the formula:
θ = ωi * t + (1/2) * α * t^2,
where θ is the angle traversed in radians, ωi is the initial angular velocity, t is the time, and α is the angular acceleration.
In this case, the initial angular velocity (ωi) is 0 rad/s (since the CD is initially at rest), the time (t) is 5.0 s, and the angular acceleration (α) is 8.0 rad/s^2.
Substituting these values into the formula, we get:
θ = 0 rad/s * 5.0 s + (1/2) * 8.0 rad/s^2 * (5.0 s)^2
= 0 rad + (1/2) * 8.0 rad/s^2 * 25 s^2
= 0 rad + 100 rad
Therefore, the CD makes 100 revolutions in 5.0 s.