---
title: "A solid disk-shaped pulley of total mass \\(M = 3.0 \\text{ kg}\\) and radius \\(R = 0.100 \\text{ m}\\) is mounted on a horizontal axle. The pulley is manufactured such that its area mass density \\(\\sigma\\) (in \\(\\text{kg/m}^2\\)) increases with the radial distance \\(r\\) from the center according to the equation \\(\\sigma(r) = cr\\), where \\(c\\) is a positive constant. Given that the total mass of the pulley is \\(M\\), the constant \\(c\\) can be expressed as \\(c = \\dfrac{3M}{2\\pi R^3}\\)."
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url: "https://nerd-notes.com/ubq/117790/"
date_modified: "2026-08-04T07:57:02+00:00"
---

# A solid disk-shaped pulley of total mass \(M = 3.0 \text{ kg}\) and radius \(R = 0.100 \text{ m}\) is mounted on a horizontal axle. The pulley is manufactured such that its area mass density \(\sigma\) (in \(\text{kg/m}^2\)) increases with the radial distance \(r\) from the center according to the equation \(\sigma(r) = cr\), where \(c\) is a positive constant. Given that the total mass of the pulley is \(M\), the constant \(c\) can be expressed as \(c = \dfrac{3M}{2\pi R^3}\).

A solid disk-shaped pulley of total mass \(M = 3.0 \text{ kg}\) and radius \(R = 0.100 \text{ m}\) is mounted on a horizontal axle. The pulley is manufactured such that its area mass density \(\sigma\) (in \(\text{kg/m}^2\)) increases with the radial distance \(r\) from the center according to the equation \(\sigma(r) = cr\), where \(c\) is a positive constant. Given that the total mass of the pulley is \(M\), the constant \(c\) can be expressed as \(c = \dfrac{3M}{2\pi R^3}\).

![A cross-sectional view of a solid disk-shaped pulley of radius R mounted on a central horizontal axle. A string is wrapped around the outer edge of the pulley on the right side and extends straight downward. A rectangular block labeled with mass m_0 hangs from the bottom end of the string. The pulley is lightly shaded with a gradient that gets darker toward the outer edge to represent increasing density. A solid line extends from the center of the pulley to the outer edge, labeled R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830221-lpqrEP.jpg)

**Part a)** Using integral calculus, **derive** an expression for the theoretical rotational inertia \(I_{theo}\) of the pulley in terms of \(M\) and \(R\). *(3 points)*

**Part b)** A light string is wrapped around the outer edge of the pulley and attached to a hanging block of mass \(m_0\). The system is released from rest. Assuming friction in the axle is negligible, **derive** an expression for the magnitude of the downward acceleration \(a\) of the block in terms of \(m_0\), \(M\), \(R\), the experimental rotational inertia \(I_{exp}\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** Students perform an experiment to determine the actual rotational inertia of the pulley. They attach blocks of varying mass \(m_0\) and measure the resulting acceleration \(a\) of the block using a motion sensor. Their data is recorded in the table below. | Trial | \(m_0 \text{ (kg)}\) | \(a \text{ (m/s}^2\text{)}\) | Blank Column | Blank Column | |-------|----------------------|------------------------------|--------------|--------------| | 1     | 0.50                 | 1.76                         |              |              | | 2     | 1.00                 | 3.10                         |              |              | | 3     | 1.50                 | 4.06                         |              |              | | 4     | 2.00                 | 4.78                         |              |              | | 5     | 2.50                 | 5.33                         |              |              | **Indicate** which quantities could be graphed to yield a straight line whose slope could be used to determine the experimental rotational inertia \(I_{exp}\) of the pulley. Vertical axis: Horizontal axis: *(1 points)*

**Part d)** Use the blank columns in the data table to record any calculated values you will use to create your graph. **Plot** the data points for the quantities indicated in part (c) on the grid below. Clearly scale and label all axes, including units if appropriate. **Draw** a straight line that best represents the data. *(4 points)*

**Part e)** Using the slope of your best-fit line, **calculate** an experimental value for the rotational inertia \(I_{exp}\). *(2 points)*

**Part f)** The students notice that if they extrapolate their best-fit line, it does not pass through the origin. A student hypothesizes that this is caused by a constant frictional torque \(\tau_f\) in the pulley's axle, which they initially ignored. Assuming the students graphed the quantity \(m_0(g-a)\) on the vertical axis and \(a\) on the horizontal axis, **justify** how the presence of a constant frictional torque explains a non-zero \(y\)-intercept without significantly affecting the slope of the line. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117790/*