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/s².
(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. Step 1: Identify the given values for part (a). The initial angular velocity \( \omega_0 \) is 0.250 rev/s, the angular acceleration \( \alpha \) is 0.900 rev/s², and the time \( t \) is 0.200 s. Use the kinematic equation for angular velocity: \( \omega = \omega_0 + \alpha t \).
2. Step 2: Substitute the given values into the equation \( \omega = \omega_0 + \alpha t \). This will give the angular velocity \( \omega \) after 0.200 s. Ensure the units are consistent (revolutions per second).
3. Step 3: For part (b), use the kinematic equation for angular displacement: \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \), where \( \theta \) is the angular displacement in revolutions. Substitute the given values for \( \omega_0 \), \( \alpha \), and \( t \).
4. Step 4: Calculate \( \theta \) by performing the arithmetic operations in the equation \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \). This will give the total angular displacement in revolutions during the 0.200 s interval.
5. Step 5: Interpret the results. The angular velocity \( \omega \) from part (a) represents how fast the turntable is spinning at the end of the time interval, while the angular displacement \( \theta \) from part (b) represents the total number of revolutions the turntable has completed during the 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 and is crucial for understanding rotational motion. In this problem, the initial angular velocity is given, and we need to calculate the final angular velocity after a certain time interval.

Angular Acceleration

Angular acceleration refers to the rate of change of angular velocity over time, measured in 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 constant angular acceleration allows us to determine the change in angular velocity over the specified time period.

Rotational Kinematics

Rotational kinematics involves the equations that describe the motion of rotating objects, analogous to linear kinematics for straight-line motion. Key equations relate angular displacement, angular velocity, and angular acceleration. In this question, we will use these equations to calculate both the final angular velocity and the total number of revolutions made by the turntable during the time interval.