Problem Statement
A flywheel of moment of inertia I is driven by a motor providing constant torque N. A braking torque $N_0$ opposes rotation (due to friction). Find the maximum angular velocity.
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. $N – N_0$ = 0… but since $N_0$ might be velocity-dependent. For constant $N_0$: any $\omega$ gives same net torque $N-N_0$, so angular velocity increases indefinitely unless $N_0$ is speed-dependent.
- $M = I\beta$ — 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.
$$If N_0 = \beta \omega (viscous), at max \omega : N = \beta \omega _max → \omega _max = N/ \beta$$
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. For constant friction $N_0$: if N > $N_0$, $\omega$ grows indefinitely (no max)
But if $N_0$ increases with $\omega$ , equilibrium exists
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. Standard result for linear damping: $\omega$ $_max$ = (N $- N_0_static$)$/ \beta$
Worked Calculation
$$\omega _max = N/ \beta (if braking torque N_0 = \beta \omega is viscous)$$
Answer
$$\boxed{\omega _max = N/ \beta (viscous) or unbounded (constant dry friction if N > N_0_const)}$$
This is the quantity the problem asked for, expressed in terms of the given data: $\omega _max = N/ \beta (viscous) or unbounded (constant dry friction if N > N_0_const)$.
Physical Interpretation
Physical machines always have some velocity-proportional drag (air resistance, viscous bearing friction), giving a real terminal angular velocity. 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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