--- title: "A large, solid disk of unknown rotational inertia \\(I\\) and known radius \\(R\\) is mounted on a fixed, horizontal, frictionless axle that passes through its center. A light string is wrapped tightly around the outer edge of the disk. A block of mass \\(m\\) is attached to the free end of the string. The block is released from rest and falls, unwinding the string and causing the disk to rotate without the string slipping. Students are tasked with designing an experiment to determine the rotational inertia \\(I\\) of the disk. They have access to the setup described above, a set of blocks of varying known masses, and standard physics laboratory equipment." description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]" url: "https://nerd-notes.com/ubq/110095/" date_modified: "2026-03-26T11:50:46+00:00" --- # A large, solid disk of unknown rotational inertia \(I\) and known radius \(R\) is mounted on a fixed, horizontal, frictionless axle that passes through its center. A light string is wrapped tightly around the outer edge of the disk. A block of mass \(m\) is attached to the free end of the string. The block is released from rest and falls, unwinding the string and causing the disk to rotate without the string slipping. Students are tasked with designing an experiment to determine the rotational inertia \(I\) of the disk. They have access to the setup described above, a set of blocks of varying known masses, and standard physics laboratory equipment. A large, solid disk of unknown rotational inertia \(I\) and known radius \(R\) is mounted on a fixed, horizontal, frictionless axle that passes through its center. A light string is wrapped tightly around the outer edge of the disk. A block of mass \(m\) is attached to the free end of the string. The block is released from rest and falls, unwinding the string and causing the disk to rotate without the string slipping. Students are tasked with designing an experiment to determine the rotational inertia \(I\) of the disk. They have access to the setup described above, a set of blocks of varying known masses, and standard physics laboratory equipment. ![A horizontal axle passes through the center of a large solid disk, oriented vertically so the circular face is visible. A string is wrapped around the circumference of the disk and hangs vertically downward from the right side. A rectangular block labeled 'm' is attached to the bottom end of the string. A downward-pointing dashed arrow is drawn below the block to indicate its direction of motion. The radius of the disk is marked with a solid line from the center to the edge, labeled 'R'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774525846-ODTJCg.jpg) **Part a)** Identify the quantities that would be measured to determine the rotational inertia \(I\) of the disk and the equipment that would be used to measure them. Record your choices in the table below. You do not need to fill every row. If you need additional rows, you may add them to the space just below the table. | Quantity to be Measured | Symbol | Equipment for Measurement | |-------------------------|--------|---------------------------| | | | | | | | | | | | | **Describe** an experimental procedure to determine the rotational inertia \(I\) of the disk. Provide enough detail so that another student could replicate the experiment, including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table above. *(3 points)* **Part b)** To analyze the experimental data, the students must understand the theoretical relationship between the forces and the acceleration of the system. i. On the dot and circle below, **draw and label** the forces (not components) that are exerted on the falling block and the rotating disk at an instant after the block is released. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot or the center of the circle. ii. **Derive** an expression for the linear acceleration \(a\) of the block in terms of \(m\), \(I\), \(R\), and physical constants, as appropriate. *(5 points)* **Part c)** The students calculate the linear acceleration \(a\) of the block for each different mass \(m\) used in the experiment. They want to create a linear graph to determine the rotational inertia \(I\) of the disk. **Indicate** what quantity should be plotted on each axis to produce a graph with a linear trend where the slope could be used to directly or indirectly determine the rotational inertia \(I\) of the disk. Vertical axis: ______________________ Horizontal axis: ______________________ **Explain** how the slope of the best-fit line to this graph can be used to determine \(I\). *(2 points)* **Part d)** In a later trial, the string completely unwinds and detaches from the disk. At the moment the string detaches (\(t = 0\)), the disk is spinning freely with a large initial angular speed \(\omega_0\). A student then pushes a friction pad against the outer edge of the disk to slow it down. The force applied by the student steadily increases, such that the frictional torque \(\tau\) exerted on the disk increases linearly with time until the disk comes to rest at time \(t_f\). i. **Sketch** a graph of the magnitude of the net torque \(\tau\) exerted on the disk as a function of time \(t\) from \(t = 0\) to \(t = t_f\). ii. Using the principles of angular impulse and momentum, **explain** how the students could use the graph sketched in part (d)(i) to derive an expression for the time \(t_f\) it takes for the disk to come to rest, in terms of \(I\), \(\omega_0\), and the maximum torque \(\tau_{max}\) exerted on the disk just before it stops. *(3 points)* *The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110095/*