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Chapter 11: Problem 15

A wheel is spinning at 45 rpm with its axis vertical. After 15 s, it's spinning at 60 rpm with its axis horizontal. Find (a) the magnitude of its average angular acceleration and (b) the angle the average angular acceleration vector makes with the horizontal.

Short Answer

Expert verified
The magnitude of its average angular acceleration is \(0.1\pi \: \text{rad/s}^2\) and the vector makes an angle of 45 degrees with the horizontal.

Step by step solution

01

Calculate the initial and final angular velocities

Convert the rotational speed from rotations per minute (rpm) to radians per second (rad/s) since angular acceleration is usually measured in rad/s^2. Use the conversion factor \(1 \: \text{rotation} = 2\pi \: \text{radians}\) and \(1 \: \text{minute} = 60 \: \text{seconds}\). The initial angular speed, \(\omega_i = 45 \: \text{rpm} = 45 * 2\pi / 60 = 4.5\pi \: \text{rad/s}\). The final angular speed, \(\omega_f = 60 \: \text{rpm} = 60 * 2\pi / 60 = 6\pi \: \text{rad/s}\)
02

Calculate the average angular acceleration

Use the formula of average angular acceleration, \(a = \frac{\Delta\omega}{\Delta t}\) where \(\Delta\omega = \omega_f - \omega_i\). Substitute the values, we get \(\Delta\omega = 6\pi - 4.5\pi = 1.5\pi \: \text{rad/s}\). Given that \(\Delta t = 15s\), substitute these values into the formula to find the average angular acceleration, \(a = 1.5\pi/15 = 0.1\pi \: \text{rad/s}^2\).
03

Calculate the angle of the angular acceleration vector

The axis has changed from vertical to horizontal, meaning there are now components in the horizontal and vertical directions. From symmetry, we can see that the angle the resultant vector makes with the positive horizontal x-axis is 45 degrees. This is because the initial direction of angular acceleration is vertical (towards negative y-axis) and the final direction is horizontal (towards positive x-axis). So the angle is 45 degrees to the horizontal.

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