AP Physics C Mechanics 5.4: Rotational Inertia

Describe a rigid system’s rotational inertia relative to a specified axis and connect it to the distribution of mass around that axis.

[Maximum number: 3]

A uniform disk of radius R and mass $m_{\mathrm{d}}$ is attached to a vertical pole by a horizontal axle that passes through the center of the disk. Friction between the axle and the disk is negligible. A lump of clay of mass $m_{\mathrm{c}}$ is attached to the edge of the disk at Point A. The size of the lump of clay is small compared with the radius of the disk. A horizontal string is connected from the pole to the edge of the disk at Point A. The string makes an angle $\theta$ with the line between Point A and the axle, as shown in Figure 1.

A nonuniform disk is now attached to the axle. The lump of clay is attached to the disk at Point B, as shown in Figure 3. The clay has mass $m_{\mathrm{c}}=0.60 \mathrm{~kg}$ and the disk has a radius $R=0.30 \mathrm{~m}$ . The mass density of the disk varies radially and can be modeled by $\rho(r)=\beta r$ , where r is the radial distance from the axle and $\beta=4.0 \mathrm{~kg} / \mathrm{m}^{3}$ .

Calculate the rotational inertia of the disk about the axle.

(i) For using integration to calculate rotational inertia 1 point
For one of the following: 1 point

- Substituting $\rho(2 \pi r) d r$ for d m

- Indicating the correct limits of integration

Example Response

I=\int_{0}^{0.3 \mathrm{~m}} r^{2} \rho(2 \pi r) d r

For a correct answer of $I=0.012 \mathrm{~kg} \cdot \mathrm{~m}^{2}$ , including units 1 point

Example Response

I=0.012 \mathrm{~kg} \cdot \mathrm{~m}^{2}

Example Solution

\begin{aligned} & I=\int r^{2} d m \quad d m=\rho d A \text { and } d A=2 \pi r d r \\ & I=\int_{0}^{R} r^{2} \rho d A \\ & I=\int_{0}^{0.3 \mathrm{~m}} 2 \pi \beta r^{4} d r \\ & I=\left.\frac{2 \pi \beta R^{5}}{5}\right|_{0} ^{0.3 \mathrm{~m}} \\ & I=\frac{2 \pi\left(4.0 \mathrm{~kg} / \mathrm{m}^{3}\right)(0.3 \mathrm{~m})^{5}}{5} \\ & I=0.012 \mathrm{~kg} \cdot \mathrm{~m}^{2} \end{aligned}

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5.4.A—Describe the rotational inertia of a rigid system relative to a given axis of rotation question 1