In operation an aluminum flywheel within an engine gearbox increases its temperature by 40 $\circ$ C.Assuming the flywheel to be a uniform circular disk,calculate the percentage change in the moment of inertia of the flywheel. (Coefficient of linear expansion for Al = 24 $\times$ 10^-6/K.) A)0.002% B)0.2% C)0.1% D)0.001% E)1.0%

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Quizplus · Accepted Answer

The moment of inertia (I) of a uniform circular disk is given by $I = \frac{1}{2}MR^2$, where $M$ is the mass and $R$ is the radius. When the temperature increases, the radius $R$ changes due to thermal expansion. The linear expansion formula is $\Delta L = L_0\alpha\Delta T$, where $\Delta L$ is the change in length (or radius in this case), $L_0$ is the original length (radius), $\alpha$ is the coefficient of linear expansion, and $\Delta T$ is the change in temperature. Since the moment of inertia depends on the square of the radius, a small change in radius due to temperature will result in the change in the moment of inertia.Given $\alpha = 24 \times 10^{-6} /K$, and $\Delta T = 40K$, the relative change in radius $\Delta R/R = \alpha\Delta T = 24 \times 10^{-6} \times 40 = 9.6 \times 10^{-4}$. Since the moment of inertia depends on $R^2$, the percentage change in the moment of inertia will be $2 \times 9.6 \times 10^{-4} \times 100\% = 0.192\%$, which is approximately $0.2\%$.

In Operation an Aluminum Flywheel Within an Engine Gearbox Increases ∘\circ∘

In Operation an Aluminum Flywheel Within an Engine Gearbox Increases $\circ$