Problem 8 - University Physics with Modern Physics

Question

A wheel is rotating about an axis that is in the \(z\) -direction. The angular velocity \(\omega_{z}\) is \(-6.00 \mathrm{rad} / \mathrm{s}\) at \(t=0,\) increases linearly with time, and is \(+8.00 \mathrm{m} / \mathrm{s}\) at \(t=7.00 \mathrm{s}\) . We have taken counterclockwise rotation to be positive. (a) Is the angular acceleration during this time interval positive or negative? (b) During what time interval is the speed of the wheel increasing? Decreasing? (c) What is the angular displacement of the wheel at \(t=7.00 \mathrm{s} ?\)

Step-by-Step Solution

Answer

(a) Positive angular acceleration; (b) Decreases 0-3s, increases 3-7s; (c) 7.00 rad displacement.

Step 1: Determine Angular Acceleration

Angular acceleration \(\alpha\) is defined as the change in angular velocity \(\omega\) over the change in time \(t\). We start by calculating the angular acceleration:\[\alpha = \frac{\omega_{\text{final}} - \omega_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} = \frac{8.00 \, \text{rad/s} - (-6.00 \, \text{rad/s})}{7.00 \, \text{s} - 0 \, \text{s}} = \frac{14.00 \, \text{rad/s}}{7.00 \, \text{s}} = 2.00 \, \text{rad/s}^2.\]The angular acceleration is positive because \(2.00 \, \text{rad/s}^2\) is a positive value.

Step 2: Analyze Speed Increase and Decrease

To determine when the speed is increasing or decreasing, consider the relationship between the direction of angular velocity and acceleration:- If both have the same sign, the speed increases.- If they have opposite signs, the speed decreases.Initially, \(\omega_z(0) = -6.00 \, \text{rad/s}\), so the wheel is rotating in the negative direction. As \(\alpha = +2.00 \, \text{rad/s}^2\), initially, \(\omega\) and \(\alpha\) have opposite signs, meaning the speed is decreasing.The speed will increase once \(\omega\) becomes positive because \(\alpha\) remains positive. Solving \(\omega_z(t) = 0\) gives when the sign changes:\[-6.00 + 2.00t = 0 \2.00t = 6.00 \t = 3.00 \, \text{s}.\]Therefore, the speed decreases from \(t=0 \) to \( t=3.00\, \text{s} \) and increases from \(t=3.00\, \text{s}\) to \(t=7.00\, \text{s}\).

Step 3: Calculate Angular Displacement

Angular displacement \(\theta\) is calculated using the formula:\[\theta = \omega_{\text{initial}}t + \frac{1}{2}\alpha t^2.\]Substitute the known values:\[\theta = (-6.00 \, \text{rad/s})(7.00 \, \text{s}) + \frac{1}{2}(2.00 \, \text{rad/s}^2)(7.00 \, \text{s})^2 \= -42.00 \, \text{rad} + \frac{1}{2}(2)(49) \= -42.00 \, \text{rad} + 49.00 \, \text{rad} \= 7.00 \, \text{rad}.\]So, the angular displacement \(\theta\) of the wheel at \(t=7.00 \, \text{s}\) is \(7.00 \, \text{rad}\).

Key Concepts

Angular Acceleration

Angular acceleration is a key concept in rotational motion that refers to how the angular velocity of an object changes over time. To clearly understand this, think of it as the rotational counterpart of linear acceleration. It helps us determine whether an object is speeding up or slowing down its rotational movement.In our example, we calculate the angular acceleration \( \alpha \) using the formula:

• \( \alpha = \frac{\omega_{\text{final}} - \omega_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} \)

Here, we subtract the initial angular velocity from the final angular velocity, and divide by the time interval over which the change occurred.
For the wheel, the angular velocity changes from \(-6.00 \, \text{rad/s} \) to \(+8.00 \, \text{rad/s} \) over \(7.00 \, \text{s} \). Plug these values into the formula:

• \( \alpha = \frac{8.00 \, \text{rad/s} - (-6.00 \, \text{rad/s})}{7.00 \, \text{s}} = 2.00 \, \text{rad/s}^2 \)

The positive angular acceleration indicates that the wheel's rotational speed is increasing in the positive direction.

Angular Velocity

Angular velocity describes how fast something is spinning or rotating. It gives us a way to quantify the speed of rotation and its direction around a specific axis, which in this case is the z-direction.Consider angular velocity \( \omega \) as the rate at which angular displacement changes with time:

• \( \omega(t) = \omega_{\text{initial}} + \alpha t \)

Initially, the wheel has an angular velocity of \(-6.00 \, \text{rad/s} \). As time progresses and the angular acceleration takes effect, this velocity changes.
Between \(t=0\) and \(t=3.00\, \text{s} \), the angular velocity is negative, meaning the wheel rotates in the direction opposite to the positive axis. After \(3.00\, \text{s}\), since both angular acceleration and velocity become positive, the wheel spins faster in the positive (counterclockwise) direction.
Angular velocity is crucial in understanding how rotational speed and direction evolve over time.

Angular Displacement

Angular displacement is the measure of the angle through which an object has rotated or moved along a circular path around a particular axis. It tells us how far the object has turned and in which direction.To calculate the angular displacement \( \theta \), you can use the formula:

• \( \theta = \omega_{\text{initial}} t + \frac{1}{2} \alpha t^2 \)

This formula accounts for both the initial angular velocity and the effect of angular acceleration over a time period.
In our scenario, substitute the given values of \( \omega_{\text{initial}} = -6.00 \, \text{rad/s} \), \( \alpha = 2.00 \, \text{rad/s}^2 \), and \( t = 7.00 \, \text{s} \):

• \( \theta = (-6.00 \, \text{rad/s})(7.00 \, \text{s}) + \frac{1}{2}(2.00 \, \text{rad/s}^2)(7.00 \, \text{s})^2 \)

The calculation results in \(7.00 \, \text{rad} \), indicating the total angular displacement by the end of\( t=7.00 \, \text{s} \).
This concept is essential for comprehending how much and in which way an object has rotated around its axis over time.