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Chapter 8: Problem 5

A wheel rotates with an angular acceleration \(\alpha_{z}\) given by $$ \alpha_{z}=4 a t^{3}-3 b t^{2} $$ where \(t\) is the time and \(a\) and \(b\) are constants. If the wheel has an initial angular velocity \(\omega_{0}\), write the equations for \((a)\) the angular velocity and \((b)\) the angle turned through as functions of time.

Short Answer

Expert verified
The angular velocity as a function of time is given by \(\omega = at^{4} - bt^{3} + \omega_0\) and the angle turned as a function of time is given by \(\theta = \frac{a}{5}t^{5} - \frac{b}{4}t^{4} + \omega_0t\).

Step by step solution

01

Identifying the quantities and interpreting the problem

We are given that the wheel has an initial angular velocity denoted by \(\omega_{0}\), with an angular acceleration \(\alpha_{z}\) denoted by the equation \(\alpha_{z}=4 a t^{3}-3 b t^{2}\). The task is to calculate the angular velocity and the angle turned.
02

Calculate Angular Velocity

To find the angular velocity (\(\omega\)), we need to integrate the angular acceleration (\(\alpha_z\)) with time. This is because velocity is the time integral of acceleration. Integrating \(\alpha_{z}=4 a t^{3}-3 b t^{2}\) with respect to time (t) gives \(\int (4 a t^{3}-3 b t^{2})dt = at^{4} - bt^{3} + C\), where C is the constant of integration. Since at t = 0, \(\omega = \omega_0\), the integration constant C will be equal to \(\omega_0\). Hence, the equation of angular velocity as a function of time is given by \(\omega = at^{4} - bt^{3} + \omega_0\)
03

Calculate Angle Turned

To find the angle turned (\(\theta\)), we integrate the angular velocity (\(\omega\)) with time. This is because displacement (or in this case angle turned) is the time integral of velocity. Integrating \(\omega = at^{4} - bt^{3} + \omega_0\) with respect to time (t) gives \(\int (at^{4} - bt^{3} + \omega_0)dt = \frac{a}{5}t^{5} - \frac{b}{4}t^{4} + \omega_0t + D\), where D is the constant of integration. Since at t = 0, \(\theta = 0\), the integration constant D will be equal to zero. Hence, the equation for the angle turned as a function of time is given by \(\theta = \frac{a}{5}t^{5} - \frac{b}{4}t^{4} + \omega_0t\)

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