Problem 16 - University Physics with Modern Physics

Question

At \(t =\) 0 a grinding wheel has an angular velocity of 24.0 rad/s. It has a constant angular acceleration of 30.0 rad/s\(^2\) until a circuit breaker trips at \(t =\) 2.00 s. From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. (a) Through what total angle did the wheel turn between \(t =\) 0 and the time it stopped? (b) At what time did it stop? (c) What was its acceleration as it slowed down?

Step-by-Step Solution

Answer

(a) 540 rad, (b) 12.3 s, (c) -8.17 rad/s².

Step 1: Calculate the Angle Turned Until the Circuit Breaker Trips

To find the angle turned before the circuit breaker trips, we use the equation for rotational kinematics:\[\theta_1 = \omega_0 t + \frac{1}{2} \alpha t^2\]where \(\omega_0 = 24.0\, \text{rad/s}\), \(\alpha = 30.0\, \text{rad/s}^2\), and \(t = 2.00\, \text{s}\). Substituting these values, we get:\[\theta_1 = 24.0 \times 2.00 + \frac{1}{2} \times 30.0 \times (2.00)^2 = 48.0 + 60.0 = 108.0\, \text{rad}\]

Step 2: Calculate the Total Angle Turned

The total angle turned by the wheel is the sum of the angle turned before the circuit breaker trip and the angle turned while coasting:\[\theta_\text{total} = \theta_1 + 432.0 = 108.0 + 432.0 = 540.0\, \text{rad}\]

Step 3: Find the Angular Velocity When the Circuit Breaker Tripped

We use the angular velocity formula:\[\omega = \omega_0 + \alpha t\]Substituting the known values:\[\omega = 24.0 + 30.0 \times 2.00 = 84.0\, \text{rad/s}\]

Step 4: Calculate the Time Taken to Stop After the Circuit Breaker Trip

As the wheel coasts to stop turning, we use the kinematic equation:\[\theta_2 = \frac{\omega^2}{2 \alpha_2}\]We know \(\theta_2 = 432.0\, \text{rad}\) and we had just calculated \(\omega = 84.0\, \text{rad/s}\). Solving for \(\alpha_2\):\[432.0 = \frac{(84.0)^2}{2 \alpha_2}\rightarrow \alpha_2 = \frac{(84.0)^2}{2 \times 432.0} = -8.17\, \text{rad/s}^2\]

Step 5: Find the Total Time to Stop

The total time to stop is the sum of time before the circuit breaker tripped (2 seconds) and the time taken to stop after that, which we calculate by using:\[\omega = \alpha_2 \times t_2\]\[0 = 84.0 + (-8.17) t_2 \,\Rightarrow \, t_2 = \frac{84.0}{8.17} = 10.3 \text{s}\]The total time is then:\[t_\text{total} = 2 + 10.3 = 12.3\, \text{s}\]

Step 6: Conclusion Based on Calculations

The total angle turned by the wheel from start until it stopped is 540 rad, it stopped after a total of 12.3 s, and the acceleration as it slowed down was -8.17 rad/s².

Key Concepts

Rotational Motion

Rotational motion refers to the movement of a body around a fixed axis. Unlike linear motion, which deals with straight paths, rotational motion involves angular displacement, angular velocity, and angular acceleration.
Angular displacement is how much an object has rotated, measured in radians. It tells us about the "spin" of an object.
Angular velocity is the rate of change of an object's angular position. It's similar to how speed describes motion along a straight path, but it applies to objects spinning around an axis.
Angular acceleration refers to the rate at which angular velocity changes over time. This happens when a force acts upon a rotating object, like when you accelerate a car around a corner.

• Angular displacement: \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \)
• Angular velocity: \( \omega = \omega_0 + \alpha t \)
• Angular acceleration: \( \alpha = \frac{\Delta \omega}{\Delta t} \)

Understanding these terms provides the foundation for analyzing systems where rotation is a key aspect, such as the spinning of wheels or gears.

Angular Acceleration

When the speed of rotation of an object changes, we say it has angular acceleration. Angular acceleration tells us how quickly the speed of rotation is changing.
In the original problem, we had a grinding wheel with a starting angular velocity of 24.0 rad/s and an angular acceleration of 30.0 rad/s² for the first 2 seconds.
This means the wheel's speed was increasing at that rate while the circuit breaker was active. Once the circuit breaker tripped, a different acceleration rate came into effect, slowing the wheel down.

• Positive angular acceleration implies that the speed of rotation is increasing.
• Negative angular acceleration, or deceleration, means the object is slowing down, as seen in the coasting phase of the wheel.

The change from positive to negative angular acceleration is a common scenario in real-world applications. For example, when a car brakes, its wheels experience negative angular acceleration.

Kinematic Equations

Kinematic equations in rotational motion are similar to those for linear motion. They help us predict future behavior by relating angular displacement, velocity, time, and acceleration.
The core kinematic equation used in the exercise was:

• \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \)

This equation helped determine how far the wheel turned before the circuit breaker tripped, based on initial conditions.
Additionally, to find the time it took to stop after the circuit breaker tripped, we used:

• \( \omega = \alpha t \)

This allowed us to determine that the wheel took 10.3 seconds to coast to a stop after the circuit breaker tripped.

• It's crucial to remember that for these equations to work correctly, angular acceleration must remain constant.
• Understanding these equations makes it easier to solve complex problems involving rotating objects.

These kinematic equations are fundamental tools in physics, not only describing rotation but also allowing the prediction and analysis of rotational systems. They are key to solving rotational dynamics issues in engineering and other applied sciences.