Problem Statement
Solve the rotational mechanics problem: Solve the kinematics problem: A wheel starts from rest with angular acceleration $\alpha = 3$ rad/s². Find the angular displacement in 4 s and the angular velocity at that time. $\omega = \alpha t$; $\theta = \frac{1}{2}\alpha t^2$ Step 1: $\omega = 3 \times 4 = 12$ rad/s. Step 2: $\theta = \frac{1}
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Rotational kinematics mirrors linear kinematics with $\theta \leftrightarrow x$, $\omega \leftrightarrow v$, $\alpha \leftrightarrow a$. The angular velocity vector $\boldsymbol{\omega}$ points along the rotation axis (right-hand rule). For a point at distance $r$ from the axis: $v = r\omega$ and $a_\tau = r\alpha$, $a_n = r\omega^2 = v^2/r$.
- $v = r\omega$ — tangential speed from angular velocity
- $a_\tau = r\alpha$ — tangential acceleration
- $a_n = r\omega^2 = v^2/r$ — centripetal acceleration
- $\omega = d\theta/dt$, $\alpha = d\omega/dt$
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
Full substitution shown in the steps above.
Answer
$$\boxed{v = \sqrt{\dfrac{4gh}{3}}\text{ (solid cylinder rolling)}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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