A motor applies a constant torque of 50 N·m to a shaft. The shaft turns through 4π radians. How much work does the motor do?
A50π J ≈ 157 J
B200π J ≈ 628 J
C200 J
DThe work cannot be determined without knowing the angular velocity
W = τθ = 50 N·m × 4π rad = 200π J ≈ 628 J. Radians are dimensionless, so N·m × rad = N·m = J. Option A (50π) would arise from mistakenly using θ = π. Option C (200 J) omits the π factor. Option D confuses work with power — power requires angular velocity, but work requires only torque and angular displacement, which are both given here.
Question 2 Multiple Choice
A flywheel with moment of inertia I = 2 kg·m² is spinning at ω = 10 rad/s and is then brought to rest by friction. How much work did friction do on the flywheel?
A+100 J (friction added rotational energy to slow it down)
D−200 J (W = Iω_f² − Iω_i² = 0 − 2 × 100 = −200 J, omitting the ½)
By the work-energy theorem for rotation: W_net = ½Iω_f² − ½Iω_i² = ½(2)(0²) − ½(2)(10²) = 0 − 100 = −100 J. The negative sign reflects that friction removes energy from the flywheel. Option A has the wrong sign (friction doesn't add energy here). Option B forgets to square ω. Option D omits the ½ factor. The rotational work-energy theorem is structurally identical to the linear version — just substitute I for m and ω for v.
Question 3 True / False
The instantaneous power delivered to a rotating body is P = τω, the direct rotational analog of P = Fv in linear mechanics.
TTrue
FFalse
Answer: True
The analogy holds exactly: force F corresponds to torque τ, and linear velocity v corresponds to angular velocity ω. Since P = Fv (linear) and the substitution F→τ, v→ω is consistent throughout rotational mechanics, P = τω follows. Both express power as the product of the effort quantity and the rate of displacement. This is also consistent dimensionally: τ (N·m) × ω (rad/s) = N·m/s = W, since radians are dimensionless.
Question 4 True / False
Because angular displacement is measured in radians, which are dimensionless, the product of torque (N·m) and angular displacement (rad) does not have units of joules.
TTrue
FFalse
Answer: False
Radians are dimensionless — they are a ratio of arc length to radius, both in meters, so the meters cancel. Therefore torque (N·m) × angular displacement (rad) = N·m × (dimensionless) = N·m = J. Radians being dimensionless is precisely what makes the rotational formulas W = τθ and P = τω dimensionally consistent with their linear counterparts. The units work out correctly, and the product has units of joules.
Question 5 Short Answer
A car engine delivers constant power P. Using P = τω, explain why a low gear (high torque, low wheel angular velocity) and a high gear (low torque, high wheel angular velocity) can both transmit the same engine power to the wheels.
Think about your answer, then reveal below.
Model answer: Power is the product of torque and angular velocity: P = τω. At constant P, increasing one factor requires decreasing the other. In a low gear, the transmission multiplies torque (τ is large) at the cost of wheel angular velocity (ω is small) — useful for acceleration. In a high gear, wheel angular velocity is high (ω is large) but torque is reduced (τ is small) — useful for cruising speed. Since P = τω is constant in both cases, both gears can transmit the same engine power while trading off between torque and speed.
This is the fundamental tradeoff in mechanical power transmission. The gear ratio determines how torque and angular velocity are exchanged, but their product (power) is conserved (ignoring friction losses). This principle applies to all mechanical systems: bicycle gears, electric motors, turbines, and flywheels all exploit the τω = constant relationship at fixed power.