Quick answer: Torque is the turning effect of a force, τ = r F sin θ, measured in newton metres. Rotation uses angle in radians, angular velocity in rad/s and angular acceleration in rad/s², with equations that mirror straight-line motion. Also, an object is in static equilibrium when the net force and the net torque are both zero.
Level: AP Physics 1 (algebra-based) · Curriculum: AP P1 Unit 5
How do you describe rotation?
Rotation uses angle instead of distance. In addition, an object turning about a fixed axis has an angular position, angular velocity and angular acceleration, each with a straight-line partner.
Angular displacement θ is the angle that the object turns through, in radians (rad). Then angular velocity ω = Δθ ÷ Δt, in rad/s. Finally, angular acceleration α = Δω ÷ Δt, in rad/s². one revolution is 2π rad = 360°, and 1 rpm = 2π ÷ 60 rad/s.
| Quantity | Straight-line motion | Rotation about a fixed axis |
|---|---|---|
| position change | x (m) | θ (rad) |
| rate of change | v = Δx ÷ Δt | ω = Δθ ÷ Δt |
| acceleration | a = Δv ÷ Δt | α = Δω ÷ Δt |
| constant acceleration | v = u + at; x = ut + ½at² | ω = ω₀ + αt; θ = ω₀t + ½αt² |
| link for a point at radius r | s = rθ | v = rω, at = rα |
ω = ω₀ + α t, θ = ω₀ t + ½ α t², ω² = ω₀² + 2 α θ
ω₀ = starting angular velocity (rad/s), α = constant angular acceleration (rad/s²), t = time (s), θ = angle (rad).
A ceiling fan in Nairobi starts from rest and speeds up at 2.5 rad/s² for 4.0 s. Its blade tip is 0.60 m from the axis.
ω = 0 + 2.5 × 4.0 = 10 rad/s.
θ = ½ × 2.5 × 4.0² = 20 rad, which is 20 ÷ 2π = 3.2 revolutions.
Tip speed v = rω = 0.60 × 10 = 6.0 m/s.
What is torque and how do you calculate it?
A force changes rotation only through its turning effect. That effect grows with the size of the force and with how far from the axis it acts. Similarly, only the part of the force at right angles to the line from the axis counts.
Torque is the turning effect of a force about an axis: τ = r F sin θ, where r is the distance from the axis to the point of application and θ is the angle between r and F. Also, the perpendicular distance from the axis to the line of the force is the lever arm, r sin θ.
τ = r F sin θ
τ = torque (N m), r = distance from the axis (m), F = force (N), θ = angle between r and F. Take anticlockwise torque as positive and clockwise as negative.
Worked example: torque on a spanner
Wanjiru pulls on a spanner with F = 80 N. In addition, the force is 0.25 m from the bolt and makes an angle of 60° with the handle.
τ = r F sin θ = 0.25 × 80 × sin 60° = 0.25 × 80 × 0.866 = 17 N m.
If she pulled at 90° she would get 20 N m, the largest possible with this force.
When is an object in static equilibrium?
An object that is not accelerating and not rotating is in static equilibrium. Moreover, two conditions must hold together, and you can apply each about any point you choose. Choosing the hinge as the axis removes the unknown hinge force from the torque equation.
Static equilibrium: the net force is zero (ΣF = 0 in each direction) and the net torque about any axis is zero (Στ = 0). Similarly, the weight of a uniform object acts at its centre.
Worked example: a hinged pole in static equilibrium
A uniform pole of mass 8.0 kg and length 2.0 m hangs on a hinge fixed to a wall. Also, a cable at 30° above the pole holds the free end, and a 5.0 kg sign hangs from that end. Take torques about the hinge.
T × (2.0 sin 30°) = (8.0 × 9.81) × 1.0 + (5.0 × 9.81) × 2.0 = 78.5 + 98.1 = 176.6 N m.
T = 176.6 ÷ 1.0 = 177 N.
Hinge force: horizontal = T cos 30° = 153 N away from the wall; vertical = 127.5 − 88.3 = 39 N upward.
Do not multiply a force by the full distance to the axis when the force is not perpendicular. Use either r F sin θ, or the perpendicular component of the force times r. Check that the torques you add have the right signs.
Choose the axis where the most unknown forces act, usually the hinge. Those forces have zero lever arm, so they vanish from the torque equation.
- Angle in radians: 2π rad = 360°; ω = Δθ ÷ Δt; α = Δω ÷ Δt.
- Constant α: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ.
- For a point at radius r: s = rθ, v = rω, a_t = rα.
- Torque τ = rF sin θ (N m); anticlockwise positive.
- Static equilibrium: ΣF = 0 and Στ = 0 about any axis.
Angular motion, torque and balance: frequently asked questions
What is the difference between torque and force?
Force causes linear acceleration, while torque causes angular acceleration about an axis. Torque depends on the size of the force, where it acts and its direction relative to the axis.
Why do rotation equations use radians?
Radians make arc length equal to rθ, so v = rω and a = rα hold without conversion factors. With degrees these links would need an extra factor of π ÷ 180.
How do you find the direction of a torque?
Imagine the force turning the object about the axis. Anticlockwise rotation counts as positive and clockwise as negative. In addition, the sign lets you add torques correctly in an equilibrium equation.
Where does the weight of a uniform beam act?
At its geometric centre. Torque calculations treat the entire weight as a single force there, at half the length from either end.
Why can you choose any axis in an equilibrium problem?
If the net torque is zero about one point and the net force is zero, it is zero about every point. Moreover, a good choice removes unknown forces and makes the algebra shorter.
