Torque

Kinetic art by Anthony Howe, 2010

Torque, τ

• Force is the action that creates �changes in linear motion.
• For rotational motion, the same force can cause very different results.
• A torque is an action that causes objects to rotate.
• A torque is required to rotate an object, just as a force is required to move an object in a line.

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Location, location, location!

• Torque is created by force, but it also depends on where the force is applied and the point about which the object rotates.
• For example, a door pushed at its handle will easily turn and open, but a door pushed near its hinges will not move as easily. The force may be the same but the torque is quite different.

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Center of rotation

• Center of rotation: the point or line about which an object turns.
• For example, a door’s center of rotation is at its hinges.
• A force applied far from the center of rotation produces a greater torque than a force applied close to the center of rotation.
• What happens when a force is applied at the hinges?

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Line of action

• Line of action: the location and direction of the force causing the torque.

Force applied must be perpendicular

• The lever arm, r (also called a moment arm) is the perpendicular distance between the line of action of the force and the center of rotation

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Calculating torque�

The torque (τ) created by a force is equal to the lever arm length (r) times the magnitude of the force (F).

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• The units of torque are force times distance, or newton·meters (N·m)
• Forces applied at the center of rotation create zero torque.

Torques can be combined

Net torque practice

τ_net = τ_A + τ_B

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= F_Ar_A + F_Br_B

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= +(10N)(.2m) + -(10N)(.1m)

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= 1 N·m

What is the net torque acting on the freely rotating object shown below.

You try…

• Note: When considering the torque exerted by the mass of the lever itself, treat the mass of the lever as if it is concentrated at its center �of mass.

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A force of 50 newtons is applied to a 0.50 kg wrench that is 30 centimeters long. Calculate the net torque acting on the wrench if the applied force is perpendicular to the wrench at the end of the wrench.

You try…

A force of 50 newtons is applied to a 0.50 kg wrench that is 30 centimeters long. Calculate the net torque acting on the wrench if the applied force is perpendicular to the wrench at the end of the wrench.

τ_net = τ_applied + τ_{wrench mass}

= F_appliedr_applied + F_wrenchr_CMwrench

= -(50 N)(0.3m) + -(.5kg)(9.8m/s²)(.15 m)

= -16 Nm

Statics

Any object that is not moving is in

• rotational equilibrium �(τ_net = 0 Nm)

and

• translational equilibrium �(F_net = 0 N)

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Example

• Consider a 10-meter bridge that weighs 500 newtons supported at both ends. A person who weighs 750 newtons is standing 2 meters from one end of the bridge.
• What are the forces (F_A, F_B) holding the bridge up at either end?

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Example

Example

Example

Example

Example

Any point can be considered the center of rotation since the bridge is static, so pick a point that eliminates one of the torques, such as point A. So now, τ_A = 0 Nm.

Example

Any point can be considered the center of rotation since the bridge is static, so pick a point that eliminates one of the torques, such as point A. So now, τ_A = 0 Nm.

Example

Example

Example

Force and lever arm are not always�perpendicular�

When the force and lever arm are not perpendicular, an extra step is required to calculate torque.

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Original equation: τ = rF

New equation: τ = r_┴F or

τ = rFsinθ

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θ

θ = the angle from the lever arm to the line of action

r

F

You try…

• A 20-cm wrench with a mass of 0.50 kg is used to loosen a bolt. A force is applied 0.20 m from the bolt at an angle of 30^o from the perpendicular. If it takes 50 N to loosen the bolt when the force is applied perpendicularly to the wrench, how much force must be applied at the angle to loosen the bolt?

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Three forces labeled A, B, C are applied to a rod which pivots on an axis through its center.

Which force causes the largest magnitude torque?

1. A C) C
2. B D) two or more forces tie for largest size torque.

AP Multiple Choice Practice

A door is pushed on by two forces, a smaller force at the door knob and a larger force nearer the hinge as shown. The door does not move. The force exerted on the door by the hinge ____.

A) is zero C) points ↓ (along -y)

B) points ↑ (along +y) D) points (lower right, in diagram)

AP Multiple Choice Practice

What is the ratio ?

(Hint: consider the torque about the mass M).

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A) 2/3 B) 1/3 C) 1/2 D) 2

A mass M is placed on a very light board supported at the ends, as shown. The free-body diagram shows directions of the forces, but not their correct relative sizes.

AP Multiple Choice Practice