--- title: "A rigid T-shaped structure lies at rest on a horizontal, frictionless surface. The structure is constructed from two uniform rods: a handle of mass \\(M\\) and length \\(L\\), and a crossbar of mass \\(m\\) and total length \\(2D\\). The handle is attached at one end to the exact center of the crossbar. The structure is pinned to the surface at the junction of the two rods by a frictionless vertical axis, allowing it to rotate freely in the horizontal plane. The rotational inertia of a uniform rod of mass \\(m_0\\) and length \\(\\ell\\) is \\(\\dfrac{1}{12}m_0\\ell^2\\) about its center and \\(\\dfrac{1}{3}m_0\\ell^2\\) about its end." 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/111289/" date_modified: "2026-04-10T01:22:06+00:00" --- # A rigid T-shaped structure lies at rest on a horizontal, frictionless surface. The structure is constructed from two uniform rods: a handle of mass \(M\) and length \(L\), and a crossbar of mass \(m\) and total length \(2D\). The handle is attached at one end to the exact center of the crossbar. The structure is pinned to the surface at the junction of the two rods by a frictionless vertical axis, allowing it to rotate freely in the horizontal plane. The rotational inertia of a uniform rod of mass \(m_0\) and length \(\ell\) is \(\dfrac{1}{12}m_0\ell^2\) about its center and \(\dfrac{1}{3}m_0\ell^2\) about its end. A rigid T-shaped structure lies at rest on a horizontal, frictionless surface. The structure is constructed from two uniform rods: a handle of mass \(M\) and length \(L\), and a crossbar of mass \(m\) and total length \(2D\). The handle is attached at one end to the exact center of the crossbar. The structure is pinned to the surface at the junction of the two rods by a frictionless vertical axis, allowing it to rotate freely in the horizontal plane. The rotational inertia of a uniform rod of mass \(m_0\) and length \(\ell\) is \(\dfrac{1}{12}m_0\ell^2\) about its center and \(\dfrac{1}{3}m_0\ell^2\) about its end. ![Overhead view of a T-shaped object on a flat surface. The T is oriented like an inverted capital 'T', with the horizontal line at the top and the vertical line extending downward. The horizontal line is labeled 'Crossbar' with total length '2D'. The vertical line is labeled 'Handle' with length 'L'. The intersection of the two lines is marked with a black dot labeled 'Pivot'. Three force arrows are shown: 1. At the bottom free end of the handle, an arrow labeled F_1 points horizontally to the right, with a right-angle symbol indicating it is perpendicular to the handle. 2. At the far left end of the crossbar, an arrow labeled F_2 points diagonally upward and to the left. A dashed line extends horizontally to the left from the crossbar, and the angle between this dashed line and the F_2 arrow is labeled \theta. 3. At the far right end of the crossbar, an arrow labeled F_3 points straight upward, parallel to the handle, with a right-angle symbol indicating it is perpendicular to the crossbar.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775784126-9902l9.jpg) **Part a)** **Derive** an expression for the total rotational inertia \(I_{sys}\) of the T-shaped structure about the pivot. Express your answer in terms of \(M\), \(m\), \(L\), \(D\), and fundamental constants. *(2 points)* **Part b)** Three horizontal forces are applied simultaneously to the structure to initiate rotation, as shown in Figure 1: - A force of magnitude \(F_1\) is applied to the free end of the handle, perpendicular to the handle. - A force of magnitude \(F_2\) is applied to the left end of the crossbar at an angle \(\theta\) relative to the axis of the crossbar. - A force of magnitude \(F_3\) is applied to the right end of the crossbar, perpendicular to the crossbar. Assume counterclockwise is the positive direction for rotational motion. **Derive** an expression for the initial angular acceleration \(\alpha\) of the structure. Express your answer in terms of \(M\), \(m\), \(L\), \(D\), \(F_1\), \(F_2\), \(F_3\), \(\theta\), and fundamental constants. *(4 points)* **Part c)** **Derive** an expression for the magnitude of the linear acceleration of the right end of the crossbar immediately after the forces are applied. Express your answer in terms of the initial angular acceleration \(\alpha\) and appropriate given variables. *(1 points)* **Part d)** Suppose the structure is redesigned such that the handle is replaced by a new handle of the same total mass \(M\) and length \(L\), but all of its mass is concentrated as a small, dense point mass at the very free end of the handle. The mass and dimensions of the crossbar, as well as the magnitudes and locations of all three applied forces, remain unchanged. **Predict** whether the new initial angular acceleration of the structure will be greater than, less than, or equal to the original angular acceleration derived in part (b). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your reasoning using physical principles. *(3 points)* *The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/111289/*