AP Physics C Mechanics 5.4 Rotational Inertia Overview

Describe rotational inertia relative to an axis and apply mass distribution and the parallel-axis theorem to rigid systems.

Covered syllabus points

5.4.B—Describe the rotational inertia of a rigid system rotating about an axis that does not pass through the system’s…5.4.A—Describe the rotational inertia of a rigid system relative to a given axis of rotation

[Maximum number: 3]

A uniform rod of length L and mass m is attached to a pivot on a vertical pole, as shown in Figure 1. There is negligible friction between the rod and the pivot. A horizontal string connects Point Q on the rod to the pole. The rod makes an angle $\theta$ with the pole. A block of mass 3 m hangs from the rod at Point P. The center of mass of the rod is located at Point C.

A nonuniform rod is now attached to the pivot, as shown in Figure 3. There is negligible friction between the nonuniform rod and the pivot. The rod has a length of 1.2 m and a linear mass density $\lambda(x)=A+B x$ , where x is the distance from the pivot, $A=6.0 \mathrm{~kg} / \mathrm{m}$ , and $B=10.0 \mathrm{~kg} / \mathrm{m}^{2}$ .

Calculate the rotational inertia of the rod about the pivot.

(ii) For a correct substitution of $\lambda$ into an integral expression of rotational inertia 1 point
Example Response

I=\int_{0}^{1.2 \mathrm{~m}}(A+B x) x^{2} d x

For a correct integration 1 point
Example Response

I=\left.\left(\frac{A x^{3}}{3}+\frac{B x^{4}}{4}\right)\right|_{0} ^{1.2 \mathrm{~m}}

For a correct numeric answer with correct units 1 point
Example Response

I=8.64 \mathrm{~kg} \cdot \mathrm{~m}^{2}

Example Solution

\begin{aligned} & I=\int r^{2} d m \quad d m=\lambda d r \text { and } r=x \\ & I=\int_{0}^{1.2 \mathrm{~m}} \lambda x^{2} d x \\ & I=\int_{0}^{1.2 \mathrm{~m}}(A+B x) x^{2} d x \\ & I=\left.\left(\frac{A x^{3}}{3}+\frac{B x^{4}}{4}\right)\right|_{0} ^{1.2 \mathrm{~m}} \\ & I=\frac{(6.0 \mathrm{~kg} / \mathrm{m})(1.2 \mathrm{~m})^{3}}{3}+\frac{\left(10.0 \mathrm{~kg} / \mathrm{m}^{2}\right)(1.2 \mathrm{~m})^{4}}{4} \\ & I=8.64 \mathrm{~kg} \cdot \mathrm{~m}^{2} \end{aligned}

Total for part (d) for question 315 points

[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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Covered syllabus points

5.4 Rotational Inertia question 1

5.4 Rotational Inertia question 2