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Chapter 8: Problem 15

The flywheel of an engine is rotating at \(25.2 \mathrm{rad} / \mathrm{s}\). When the engine is turned off, the flywheel decelerates at a constant rate and comes to rest after \(19.7 \mathrm{~s}\). Calculate \((a)\) the angular acceleration (in \(\mathrm{rad} / \mathrm{s}^{2}\) ) of the flywheel, (b) the angle (in rad) through which the flywheel rotates in coming to rest, and \((c)\) the number of revolutions made by the flywheel in coming to rest.

Short Answer

Expert verified
The angular acceleration of the flywheel is \(-1.28\) rad/s\(^2\), the angle through which the flywheel rotates in coming to rest is \(497.04\) rad, and the number of revolutions made by the flywheel in coming to rest is \(79.1\) revolutions.

Step by step solution

01

Calculate the Angular Acceleration

The angular acceleration can be found by using the formula for acceleration which is \(\alpha = (ω_f - ω_i)/ t\). Here, \(ω_f\) (final angular velocity) is \(0\) rad/s, \(ω_i\) (initial angular velocity) is \(25.2\) rad/s and \(t\) (time) is \(19.7\) s. Hence, \(\alpha = (0 - 25.2) / 19.7 = -1.28 \) rad/s\(^2\). The negative sign indicates that it is a deceleration.
02

Calculate the Angle

The angle or displacement can be found by using the kinematic equation \(\theta = ω_i t + 0.5*α*t^2\). Here, \(\theta\) is the angle, \(ω_i\) is \(25.2\) rad/s, \(α\) is \(-1.28\) rad/s\(^2\) and \(t\) is \(19.7\) s. Hence, \(\theta = 25.2 * 19.7 + 0.5*(-1.28)*(19.7)^2 = 497.04\) rad.
03

Calculate the Number of Revolutions

The number of revolutions can be calculated by understanding that \(1\) revolution equals to \(2π\) radians. To convert, divide the total angle of \(497.04\) rad by \(2π\). Hence, the number of revolutions \(N\) is \(497.04 / 2π = 79.1\) revolutions.

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