Problem 15 - College Physics

Question

\(\cdot\) A wheel turns with a constant angular acceleration of 0.640 \(\mathrm{rad} / \mathrm{s}^{2} .\) (a) How much time does it take to reach an angular velocity of \(8.00 \mathrm{rad} / \mathrm{s},\) starting from rest? (b) Through how many revolutions does the wheel turn in this interval?

Step-by-Step Solution

Answer

(a) 12.5 s; (b) 7.96 revolutions.

Step 1: Understanding the Problem

We know the angular acceleration \( \alpha = 0.640 \, \mathrm{rad/s}^2 \) and the final angular velocity \( \omega = 8.00 \, \mathrm{rad/s} \). The initial angular velocity \( \omega_0 = 0 \, \mathrm{rad/s} \) since it starts from rest. We need to find the time \( t \) it takes to reach this angular velocity.

Step 2: Applying the Angular Kinematic Equation

Use the equation for angular velocity with constant angular acceleration: \( \omega = \omega_0 + \alpha t \). Here, \( \omega_0 = 0 \) and \( \omega = 8.00 \, \mathrm{rad/s} \). So, \( 8.00 = 0 + 0.640 t \).

Step 3: Solving for Time

Rearrange the equation to solve for \( t \): \( t = \frac{8.00}{0.640} \).

Step 4: Calculating the Time

Compute \( t \): \( t = 12.5 \, \mathrm{s} \). The time taken to reach the angular velocity is 12.5 seconds.

Step 5: Finding the Angular Displacement

To find the number of revolutions, first find the angular displacement \( \theta \) using \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \). Here \( \omega_0 = 0 \), \( \theta = \frac{1}{2} \times 0.640 \times (12.5)^2 \).

Step 6: Calculating Angular Displacement

Compute \( \theta \): \( \theta = 0.320 \times 156.25 = 50.0 \, \mathrm{rad} \).

Step 7: Converting Radians to Revolutions

Each revolution is \( 2\pi \) radians, so divide the angular displacement by \( 2\pi \) to convert to revolutions: \( \text{revolutions} = \frac{50.0}{2\pi} \approx 7.96 \).

Step 8: Conclusion

The wheel turns approximately 7.96 revolutions in this interval.

Key Concepts

Angular Acceleration

Angular acceleration tells us how quickly the angular velocity of an object is changing over time. It is similar to how linear acceleration works, but instead of change in linear velocity, it is the change in angular velocity. In this case, we have a wheel that starts rotating from rest and gradually speeds up. This happens because it is experiencing a constant angular acceleration, which is given as 0.640 rad/s². Angular acceleration can be represented by the symbol \( \alpha \). When thinking about angular acceleration:

• The higher the angular acceleration, the quicker the object speeds up its rotation.
• If angular acceleration is constant, the rate at which angular velocity increases is steady.
• In practical terms, a constant angular acceleration means the wheel will rotate faster and faster, evenly, over time.

Understanding this concept is crucial as it relates directly to how we solve problems involving rotating objects, analyzing the changes in their speed as they turn.

Angular Velocity

Angular velocity is the measure of how fast something is rotating. While linear velocity tells us how fast something is moving in a straight line, angular velocity refers to the speed of rotation around an axis. In this exercise, we want to find out how long it takes for a wheel to reach an angular velocity of 8.00 rad/s starting from rest. The formula we use here is \( \omega = \omega_0 + \alpha t \), where \( \omega \) is the angular velocity, \( \omega_0 \) is the initial angular velocity, \( \alpha \) is the angular acceleration, and \( t \) is time.Key points to remember about angular velocity:

• Measured in radians per second (rad/s).
• In this exercise, the initial angular velocity \( \omega_0 \) is zero because the wheel starts from rest.
• We solve for time \( t \) using the known values, substituting into the angular velocity formula.

The solution shows how we rearrange the formula to calculate the time it takes for the wheel to reach 8.00 rad/s, revealing an important step in analyzing rotational motion.

Angular Displacement

Angular displacement refers to the angle through which an object rotates during a given time interval. It is the rotational equivalent of linear displacement. In the problem, we calculate how many radians the wheel rotates in a given time until it hits the desired angular velocity.The formula used is: \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \), where \( \theta \) represents angular displacement.Important aspects of angular displacement:

• Measured in radians, as it reflects the rotation angle.
• In scenarios where the initial angular velocity \( \omega_0 \) is zero, the formula simplifies a bit.
• The inclusion of \( \alpha t^2/2 \) shows how angular displacement grows as a function of time squared, demonstrating non-linear growth as time progresses.

By calculating \( \theta \), we determine how far the wheel turns in radians, understanding how the cumulative effect of constant acceleration translates into angular rotation.

Radians to Revolutions Conversion

Converting between radians and revolutions is crucial in problems involving rotational motion. Radians describe circular motion directly related to the radius of a circle, while revolutions count complete turns of the circle.Key points about the conversion process:

• One full revolution equals \( 2\pi \) radians.
• To convert radians to revolutions, divide the angular displacement in radians by \( 2\pi \).
• Helps in understanding the rotational distance in a more intuitive way for practical applications, such as how many times a wheel turns.

In this exercise, the angular displacement calculated was 50.0 radians. Once we convert this to revolutions—\( \frac{50.0}{2\pi} \)—we get approximately 7.96 revolutions. This conversion not only solidifies understanding of the rotation but helps visualize how many full spins the wheel has completed.