Problem 18 - University Physics with Modern Physics

Question

A straight piece of reflecting tape extends from the center of a wheel to its rim. You darken the room and use a camera and strobe unit that flashes once every 0.050 s to take pictures of the wheel as it rotates counterclockwise. You trigger the strobe so that the first flash \((t=0)\) occurs when the tape is horizontal to the right at an angular displacement of zero. For the following situations draw a sketch of the photo you will get for the time exposure over five flashes (at \(t=0,0.050 \mathrm{s}, 0.100 \mathrm{s}, 0.150 \mathrm{s},\) and 0.200 \(\mathrm{s} )\) and graph \(\theta\) versus \(t\) and \(\omega\) versus \(t\) for \(t=0\) to \(t=0.200 \mathrm{s}\) (a) The angular velocity is constant at 10.0 rev \(/ \mathrm{s}\) . (b) The wheel starts from rest with a constant angular acceleration of 25.0 rev \(/ \mathrm{s}^{2}\) . (c) The wheel is rotating at 10.0 rev \(/ \mathrm{s}\) at \(t=0\) and changes angular velocity at a constant rate of \(-50.0 \mathrm{rev} / \mathrm{s}^{2}\) .

Step-by-Step Solution

Answer

For each scenario, draw the respective linear or parabolic graphs for \(\theta(t)\) and \(\omega(t)\) based on the given angular conditions.

Step 1: Understanding the Problem

The problem involves a rotating wheel with a reflective tape being observed under a strobe light. The task is to analyze how the tape's position changes over time (at different angular velocities and accelerations), and to graph angular displacement \( \theta \) and angular velocity \( \omega \) over time.

Step 1: Calculate for Constant Angular Velocity

Given: \( \omega = 10.0 \text{ rev/s} \).For constant angular velocity, \( \theta = \omega t \).Converting revolutions to radians: \( 1 \text{ rev} = 2\pi \text{ rad} \).Therefore, \( \theta = 10 \times 2\pi \times t \).Calculate \( \theta \) for each time \( t = 0, 0.050, 0.100, 0.150, 0.200 \text{ s} \).Example: At \( t=0.1 \text{ s} \) \( \theta = 10 \times 2\pi \times 0.1 = 2\pi \text{ rad} \). Repeat for each time.

Step 2: Plot \(\theta\) vs \(t\) for Constant \(\omega\)

For constant \(\omega\), the \(\theta\) vs \(t\) graph is a straight line with slope \(20\pi \text{ rad/s}\).

Step 3: Plot \(\omega\) vs \(t\) for Constant \(\omega\)

Since \(\omega\) is constant, the \(\omega\) vs. \(t\) graph is a horizontal line at \(10 \text{ rev/s}\).

Step 4: Calculate for Constant Angular Acceleration

Given: \( \alpha = 25.0 \text{ rev/s}^2 \) and initial \( \omega_0 = 0 \text{ rev/s} \).Use \( \omega = \omega_0 + \alpha t \) and \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \).Convert to radians: \( \alpha = 25 \times 2\pi \text{ rad/s}^2 \).Calculate \( \omega \) and \( \theta \) for each time \( t \).

Step 5: Plot \(\theta\) vs \(t\) for Constant \(\alpha\)

The \(\theta\) vs \(t\) graph is a parabola because \(\theta = \frac{1}{2}\alpha t^2 \).

Step 6: Plot \(\omega\) vs \(t\) for Constant \(\alpha\)

The \(\omega\) vs \(t\) graph is a straight line with the slope equal to the angular acceleration \( 50\pi \text{ rad/s}^2 \).

Step 7: Calculate for Changing Angular Velocity

Given: Initial \( \omega_0 = 10.0 \text{ rev/s} \) and \( \alpha = -50.0 \text{ rev/s}^2 \).Use \( \omega = \omega_0 + \alpha t \) and \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \).Calculate \( \theta \) and \( \omega \) for each time.

Step 8: Plot \(\theta\) vs \(t\) for Negative \(\alpha\)

The \(\theta\) vs \(t\) graph is again a parabola, showing advancement followed by a slowdown due to negative acceleration.

Step 9: Plot \(\omega\) vs \(t\) for Negative \(\alpha\)

This \(\omega\) vs \(t\) graph is a straight line decreasing (sloping downward left to right) due to the negative angular acceleration.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object is rotating about a specific axis. Imagine a wheel turning in a circular motion. Angular velocity tells us how fast any point on the wheel is covering angles over time. It's typically represented by the Greek letter \( \omega \).
Sometimes it's measured in revolutions per second (rev/s), like in our original exercise, but it can be converted to radians per second (rad/s) since it's more commonly used in calculations. One full revolution is equivalent to \( 2\pi \) radians, so you just multiply the revolutions by \( 2\pi \) to convert it. For example, if a wheel spins at a constant angular velocity of 10.0 rev/s, this is \( 10 \times 2\pi \) rad/s.
In scenarios where the angular velocity is constant, it implies that there is no acceleration involved. This translates into a linear relationship where angular displacement \( \theta \) over time \( t \) follows a straight line. You can think of it like driving at a constant speed on a straight highway: your movement (angular displacement) is steady.

Angular Acceleration

Angular acceleration, symbolized by \( \alpha \), occurs when an object's rate of rotation changes over time. If you've ever ridden a bike and started pedaling faster, you've experienced angular acceleration. It's the rotational equivalent of linear acceleration.
In the exercise, we see examples of constant angular acceleration. For instance, if a wheel starts from rest and speeds up with a steady angular acceleration of 25.0 rev/s², we can calculate the change in angular velocity using \( \omega = \omega_0 + \alpha t \), where \( \omega_0 \) is the initial angular velocity. Here, \( \omega_0 \) is zero because the wheel starts from rest.
When plotting angular displacement \( \theta \) over time for a constantly accelerating wheel, the graph forms a parabola. This shape represents increasing displacement as time goes on because the rotation rate is picking up speed. Conversely, if the wheel experiences negative acceleration, the graph will show an increase followed by a decrease, indicating the wheel is slowing down.

Angular Displacement

Angular displacement is a measure of how much an object has rotated relative to its starting position. It's represented by the Greek letter \( \theta \) and is typically measured in radians.
Think of it as the rotational version of a straight-line distance. When plotting angular displacement over time, each point on the graph shows where the object is along its circular path at any given time.
In scenarios with constant angular velocity, like when the wheel rotates at a steady 10.0 rev/s, angular displacement increases linearly over time, resulting in the previously mentioned straight line graph. However, in cases of angular acceleration, displacement follows a more complex path, forming a curve due to the changing speed of rotation.
Understanding angular displacement is crucial because it helps visualize the overall rotation and is key when combined with angular velocity and acceleration for detailed motion analysis. It gives a complete picture of an object's rotational motion over time.