Spinning down. You start a spinning wheel by rotating it with an angular acceleration of 10 rad/s2 for 3 s. (a) What is the angular velocity of the wheel as a function of time for the first 3 s. (b) Find the angle of the wheel as a function of time for the first 3 s. After releasing the spinning wheel, it slows down at a constant rate of 0.1 rad/s2. (c) What is the angular velocity of the wheel as a function of time after the first 3 s. (d) Find the angle of the wheel as a function of time after the first 3s. (e) How long time does the wheel take to stop? (f) How much longer would the wheel take to stop if you spun it for 6 seconds instead?

Angular acceleration is the rate at which the angular velocity of a spinning object changes over time. It's like how speed changes when you press or release the gas pedal in a car. In mathematical terms, angular acceleration \( \alpha \) is given by the formula: \[ \alpha = \frac{d\omega}{dt} \] where \( \omega \) is angular velocity and \( t \) is time. For example, if you give a spinning wheel an angular acceleration of 10 rad/s² for 3 seconds, it means every second, its speed increases by 10 rad/s.

This concept is crucial in solving problems involving rotational motion because it helps determine how quickly or slowly an object will spin up or slow down.

Angular velocity is the rate at which an object spins around an axis. It's similar to how fast something is moving in a straight line but instead applied to rotation. The formula for angular velocity \( \omega(t) \) with constant angular acceleration is: \[ \omega(t) = \omega_0 + \alpha t \] where \( \omega_0 \) is the initial angular velocity, \( \alpha \) is the angular acceleration, and \( t \) is the time.

For example, if you start a wheel from rest ( \( \omega_0 = 0 \) ) and accelerate it at 10 rad/s² for 3 seconds, its angular velocity at the end of 3 seconds will be:
\( \omega(3) = 0 + 10 \times 3 = 30 \ rad/s \). Angular velocity is key in understanding how quickly an object is spinning at any given time during its motion.

Deceleration, or negative angular acceleration, is the rate at which the angular velocity of a spinning object decreases. It's like hitting the brakes while driving; the car slows down over time. When you stop applying a force to the wheel, it will naturally start to slow down due to deceleration. For instance, the wheel slows down at a constant rate of 0.1 rad/s² after you release it.

The formula for angular velocity during deceleration is the same, just with a negative sign: \[ \omega(t) = \omega_0 - \alpha t \] For example, if the wheel had an angular velocity of 30 rad/s when released, it would slow down as: \[ \omega(t) = 30 - 0.1 \times t \] Deceleration helps determine how long it will take for the wheel to come to a stop.

Rotational kinematics involves the study of rotational motion without considering the forces causing it. It's similar to linear kinematics but applied to objects rotating about an axis. The basic equations are similar but involve angular quantities.

For example, to find the angular displacement \( \theta \) for a spinning wheel with constant angular acceleration: \[ \theta(t) = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2 \] where \( \theta_0 \) is the initial angular position, \( \omega_0 \) is the initial angular velocity, and \( \alpha \) is the angular acceleration.

For instance, if the wheel starts from rest and has an angular acceleration of 10 rad/s² for 3 seconds, its angular displacement is: \[ \theta(3) = 0 + 0 + 0.5 \times 10 \times 3^2 = 45 \ rad \] Rotational kinematics allows us to calculate angles, speeds, and times for rotating objects.

The time it takes for a spinning wheel to stop depends on its initial angular velocity and the rate of deceleration. Using the angular deceleration formula: \[ \omega = \omega_0 - \alpha t \] we rearrange it to solve for time \( t \): \[ t = \frac{\omega_0}{\alpha} \] For example, if the wheel has an initial angular velocity of 30 rad/s and a deceleration rate of 0.1 rad/s²: \[ t = \frac{30}{0.1} = 300 \ seconds \] This means it will take 300 seconds to come to a complete stop. Understanding this helps predict when a rotating object will stop moving, which is crucial for planning and safety.