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Chapter 7: Problem 8

Angular displacement \((\theta)\) of a flywheel varies with time as \(\theta=a t+b t^{2}+c t^{3}\) then angular acceleration is given by (a) \(a+2 b t-3 c t^{2}\) (b) \(2 b-6 t\) (c) \(a+2 b-6 t\) (d) \(2 b+6 c t\)

Short Answer

Expert verified
The angular acceleration is given by \(2 b + 6 c t\) which corresponds to option (d).

Step by step solution

01

Understand the provided equation for angular displacement

The provided equation for angular displacement is \(\theta = a t + b t^2 + c t^3\). This equation indicates that the angular displacement \(\theta\) of the flywheel at a given time \(t\) depends on some constants \(a\), \(b\), and \(c\). The term \(a t\) represents the linear component of displacement, \(b t^2\) is the acceleration component, and \(c t^3\) is jerk (rate of change of acceleration).
02

Differentiate the angular displacement to get angular velocity

Since angular velocity is the first derivative of angular displacement with respect to time, we'll take the derivative of our displacement equation to get angular velocity. Using the power rule for differentiation, the derivative of \(a t\) is \(a\), the derivative of \(b t^2\) is \(2 b t\), and the derivative of \(c t^3\) is \(3 c t^2\). So, our equation for angular velocity is \(\omega = a + 2 b t + 3 c t^2\).
03

Differentiate the angular velocity to get angular acceleration

Angular acceleration is the first derivative of angular velocity with respect to time. So we'll now take the derivative of our velocity equation to get angular acceleration. The derivative of \(a\) is 0 (since it's a constant), the derivative of \(2 b t\) is \(2 b\), and the derivative of \(3 c t^2\) is \(6 c t\). Therefore, our equation for angular acceleration is \(\alpha = 2 b + 6 c t\).

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Most popular questions from this chapter

The moment of inertia of a solid sphere of density \(\rho\) and radius \(R\) about its diameter is (a) \(\frac{105}{176} R^{5} \rho\) (b) \(\frac{105}{176} R^{2} \rho\) (c) \(\frac{176}{105} R^{5} \rho\) (d) \(\frac{176}{105} R^{2} \rho\)

Angular displacement \((\theta)\) of a flywheel varies with time as \(\theta=a t+b t^{2}+c t^{3}\) then angular acceleration is given by (a) \(a+2 b t-3 c t^{2}\) (b) \(2 b-6 t\) (c) \(a+2 b-6 t\) (d) \(2 b+6 c t\)

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