Problem Statement
A uniform disc of mass m and radius R rotates about its axis. A braking torque $M = \alpha$ $\omega$ acts (proportional to angular velocity). Find $\omega$ (t) starting from $\omega$ $_0$.
Given Information
- All numerical data are stated in the problem above; symbols are defined as they appear.
Physical Concepts & Formulas
These problems apply the rotational analogue of Newton’s laws about a chosen axis, using moment of inertia and torque.
- $M = I\beta$ — equation of rotational dynamics
- $I = \sum m_i r_i^2$ — moment of inertia
- $E_k = \tfrac{1}{2}I\omega^2$ — rotational kinetic energy
- $L = I\omega$ — angular momentum of a rigid body
Step-by-Step Solution
Step 1 — Identify the governing principle: We begin by recognising which physical law controls the situation and why it is the correct starting point for this problem.
$$I d \omega /dt = – \alpha \omega$$
Step 2 — Set up the relevant equations: Next we write down the equations that follow from that principle, introducing the symbols we will carry through the algebra.
$$Separate: d \omega / \omega = -( \alpha /I)dt$$
Step 3 — Apply the given conditions: We now substitute the specific conditions and constraints given in the problem so the equations describe this particular situation.
$$Integrate: \ln ( \omega / \omega _0) = – \alpha t/I$$
Step 4 — Solve for the required quantity: With the equations specialised, we isolate and solve for the unknown the problem asks us to find.
$$\omega (t) = \omega _0 \cdot \exp (- \alpha t/I) = \omega _0 \cdot \exp (-2 \alpha t/(mR^{2}))$$
Worked Calculation
$$\omega (t) = \omega _0 \cdot e^{- \alpha t/I} = \omega _0 \cdot e^{-2}\alpha t/(mR^{2})$$
Answer
$$\boxed{\omega (t) = \omega _0 \cdot \exp (- \alpha t/I)}$$
This is the quantity the problem asked for, expressed in terms of the given data: $\omega (t) = \omega _0 \cdot \exp (- \alpha t/I)$.
Physical Interpretation
Velocity-proportional braking gives exponential decay — the disc never truly stops but approaches zero asymptotically. This is identical in form to RC circuit discharge. The magnitude of the answer is consistent with everyday physical experience for this class of problem in Irodov’s Part 1 — the result shows how the answer scales with the given quantities. If we doubled the dominant input, the boxed formula tells us exactly how the output would respond, and that scaling is the key physical insight this problem trains. Comparing the answer with the appropriate limiting cases (very small or very large values of the dominant parameter) recovers the familiar Newtonian or intuitive expectation, which is a useful sanity check.
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