An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev/s^2. (a) Compute the angular velocity of the turntable after 0.200 s. (b) Through how many revolutions has the turntable spun in this time interval?

Verified step by step guidance

1. Identify the given values: initial angular velocity (\(\omega_0\)) = 0.250 rev/s, angular acceleration (\(\alpha\)) = 0.900 rev/s^2, and time (\(t\)) = 0.200 s.
2. Use the formula for final angular velocity with constant acceleration: \(\omega = \omega_0 + \alpha t\). Substitute the given values to find \(\omega\), the angular velocity after 0.200 s.
3. To find the total number of revolutions, use the formula for angular displacement (\(\theta\)) with constant acceleration: \(\theta = \omega_0 t + \frac{1}{2} \alpha t^2\).
4. Substitute the given values into the angular displacement formula to calculate \(\theta\), which represents the total angular displacement in revolutions.
5. The result from step 4 gives the total number of revolutions the turntable has spun in the given time interval.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second or revolutions per second. It indicates the rate of change of angular displacement over time. In this question, the initial angular velocity is given, and we need to calculate the final angular velocity after a certain time interval using the formula: final angular velocity = initial angular velocity + (angular acceleration × time).

Angular Acceleration

Angular acceleration refers to the rate of change of angular velocity over time, measured in units such as revolutions per second squared or radians per second squared. It indicates how quickly an object is speeding up or slowing down in its rotation. In this scenario, the turntable has a constant angular acceleration, which allows us to apply the kinematic equations for rotational motion to find the final angular velocity and the total number of revolutions.

Revolutions and Angular Displacement

Revolutions are a way to quantify the number of complete rotations an object makes around an axis. Angular displacement, measured in radians or revolutions, represents the angle through which an object has rotated. To find the total number of revolutions during the time interval, we can use the formula for angular displacement, which incorporates both the initial angular velocity and the angular acceleration, allowing us to calculate how far the turntable has spun in the given time.