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Torque, angular momentum and rotational power questions
The rotational twins of force, momentum and power, and the bookkeeping that ties them together. A skater's spin, a lorry's clutch and a neutron star all obey the same line: with no external torque, Iω does not change.
17 original questions · 51 marks · the torque, angular momentum and rotational power notes · Engineering physics
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State what is meant by the torque of a force about an axis, and give its unit.
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Torque is the turning effect of a force about the axis, the product of the force and its perpendicular distance from the axis, T = Fr for a force applied tangentially at radius r (1). Its unit is N m (1).Write the equation linking the net torque on a rotating body to its angular acceleration, and define each symbol.
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T = Iα (1), where T is the net torque in N m, I the moment of inertia about that axis in kg m2 and α the angular acceleration in rad s−2 (1). It is the rotational form of F = ma, and it is the net torque, not the drive torque, that belongs in it.State the principle of conservation of angular momentum.
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If no resultant external torque acts on a system, its total angular momentum Iω stays constant (1). Internal forces, muscles, clutches or a collapsing structure, cannot change it, however much they rearrange the mass (1).Explain why an ice skater spinning with her arms outstretched speeds up when she pulls her arms in.
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No external torque acts, so her angular momentum Iω is conserved (1). Pulling her arms in moves mass closer to the axis, reducing I, so ω must rise to keep the product constant (1). Nothing gives her an extra twist: she is not pushing against anything outside herself.Explain how a heavy flywheel mounted on the crankshaft of a piston engine smooths out the fluctuations in torque from the power strokes.
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Its large moment of inertia means a given torque fluctuation produces only a tiny change in ω, since α = T/I (1). It absorbs angular momentum during each power stroke and returns it during the gaps between them, so the shaft turns almost uniformly (1).Give the unit of angular momentum.
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kg m2 s−1, equivalently N m s (1), following straight from Iω with I in kg m2 and ω in rad s−1.Calculate the torque produced by a force of 250 N applied tangentially at the rim of a wheel of diameter 0.60 m.
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The radius is half the diameter: r = 0.60/2 = 0.30 m (1). T = Fr = 250 × 0.30 = 75 N m (1). Using the diameter gives 150 N m, double the true torque, and it is the commonest slip in this calculation.A flywheel of moment of inertia 8.0 kg m2 is acted on by a constant net torque of 20 N m, starting from rest. Calculate its angular acceleration and its angular velocity after 15 s.
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α = T/I (1) = 20/8.0 = 2.5 rad s−2 (1). ω2 = ω1 + αt = 0 + 2.5 × 15 = 37.5 rad s−1 (1).An engine develops an output power of 45 kW at 2400 revolutions per minute. Calculate the torque at the output shaft.
A disc of moment of inertia 0.75 kg m2 spins at 24 rad s−1. A brake applies a steady friction torque of 3.0 N m. Calculate the time taken for the disc to stop.
A skater spinning at 1.5 revolutions per second has a moment of inertia of 5.6 kg m2. She pulls her arms in, reducing it to 2.0 kg m2. Calculate her new rate of spin and the factor by which her kinetic energy changes.
A motor applies a constant torque of 12 N m to a shaft, turning it through 500 rad in 25 s. Calculate the work done and the average power delivered.
A drum of moment of inertia 4.0 kg m2 is driven by a torque of 30 N m and opposed by a friction torque of 6.0 N m. Starting from rest, calculate its angular acceleration and the number of revolutions it makes in the first 10 s.
A disc of moment of inertia 1.8 kg m2 spinning at 120 rad s−1 is lowered onto a stationary disc of moment of inertia 1.2 kg m2 mounted on the same axis. Friction between them brings the pair to a common angular velocity, the friction torque between the faces being 9.0 N m. Calculate the common angular velocity, the kinetic energy before and after, the energy lost, and the time the coupling takes.
A turbine rotor of moment of inertia 220 kg m2 is run up from rest to 3000 revolutions per minute by a constant torque of 1100 N m. Calculate the time taken and the angle turned through, and show that the work done by the torque equals the rotor's final kinetic energy.
A spinning skater pulls her arms inward and her rate of spin rises. Explain why her angular momentum is unchanged while her kinetic energy increases, and state where the extra energy comes from.
A wheel of moment of inertia 0.40 kg m2 spinning at 30 rad s−1 is brought to rest in 2.5 s by a brake pad pressed against its rim at a radius of 0.25 m. Calculate the friction force at the rim and the heat generated at the pad.
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