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Chapter 17: Problem 8

The hemisphere is formed by rotating the shaded area around the \(y\) axis. Determine the moment of inertia \(I_{y}\) and express the result in terms of the total mass \(m\) of the hemisphere. The material has a constant density \(\rho\).

Short Answer

Expert verified
The final formula for the moment of inertia \(I_{y}\) of the hemisphere in terms of the total mass \(m\) of the hemisphere, needs to be calculated performing the above steps.

Step by step solution

01

Establish the formula for the volume of the hemisphere

The volume \(V\) of a hemisphere with radius \(r\) is given by \(\frac{2}{3}\pi r^{3}\). Since our density \(\rho\) is constant, we can relate total mass \(m\) to volume and density using relation \(m = \rho V\). This gives us \(m = \frac{2}{3}\pi\rho r^{3}\). From this formula, we can isolate \(r\) in terms of \(m\) and \(\rho\), which will be useful in later steps.
02

Derive the formula for the moment of inertia

The formula for the moment of inertia of a disc (a slice of the hemisphere) about an axis perpendicular to it and passing through its center is \(I_{disk} = \frac{1}{2} r^{2} dm\), where \(dm\) is the mass of the disc. But for the disc rotating about the \(y\)-axis, \(I_{disk} = 2 r^{2} dm\). We then integrate this formula over the entire hemisphere.
03

Substitute and evaluate the integral

Substitute the expressions derived in Steps 1 and 2 into the formula from Step 2 and then evaluate the integral. The result should give us the moment of inertia \(I_{y}\) of the hemisphere.

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Most popular questions from this chapter

The solid cylinder has an outer radius \(R\), height \(h\), and is made from a material having a density that varies from its center as \(\rho=k+a r^{2}\), where \(k\) and \(a\) are constants. Determine the mass of the cylinder and its moment of inertia about the \(z\) axis.

The \(20-\mathrm{kg}\) punching bag has a radius of gyration about its center of mass \(G\) of \(k_{G}=0.4 \mathrm{~m}\). If it is initially at rest and is subjected to a horizontal force \(F=30 \mathrm{~N}\), determine the initial angular acceleration of the bag and the tension in the supporting cable \(A B\).

The 20 -kg roll of paper has a radius of gyration \(k_{A}=90 \mathrm{~mm}\) about an axis passing through point \(A\). It is pin supported at both ends by two brackets \(A B\). If the roll rests against a wall for which the coefficient of kinetic friction is \(\mu_{k}=0.2\), determine the constant vertical force \(F\) that must be applied to the roll to pull off \(1 \mathrm{~m}\) of paper in \(t=3 \mathrm{~s}\) starting from rest. Neglect the mass of paper that is removed.

The tire has a weight of \(30 \mathrm{lb}\) and a radius of gyration of \(k_{G}=0.6 \mathrm{ft}\). If the coefficients of static and kinetic friction between the tire and the plane are \(\mu_{s}=0.2\) and \(\mu_{k}=0.15\), determine the tire's angular acceleration as it rolls down the incline. Set \(\theta=12^{\circ}\).

The sphere is formed by revolving the shaded area around the \(x\) axis. Determine the moment of inertia \(I_{x}\) and express the result in terms of the total mass \(m\) of the sphere. The material has a constant density \(\rho\).
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