Torque and Moment Arm

More than just force matters

Distance matters, too.

But what about angle?

Only the force perpendicular to the distance provides torque Only the force perpendicular to the distance provides torque. For proof, see video of fan on rotating platform.

This quantity that depends on force, distance from pivot, and angle we call “torque”.

Torque is why we use larger wrenches when we need to loosen a tight nut or bolt.

Torque is the rotational equivalent of force. Just as a force causes acceleration, torque causes angular acceleration (α), in rad/sec². Torque causes a rotating object to change its speed of rotation.

What about mass?

Mass is an object’s “Resistance to acceleration” Mass is an object’s “Resistance to acceleration”. “Resistance to rotational acceleration” is moment of inertia, I.

Easy moment of inertia demo: Stand up Easy moment of inertia demo: Stand up. Hold your hands near your chest and twist at the waist back and forth. Continue to twist as you move extend your arms. As the mass of your hands get farther from your body, your moment of inertia increases and it is harder to twist your body.

Moment of inertia, I, depends on mass and its placement. Greater Moment of Inertia

Let’s do some math and see where it leads us. If τ = (Fsinθ)d, Then (by math): τ = F(dsinθ) But what would that LOOK like, physically?

We call this r , the “moment arm”.

This r is a “mathematical construct”. It is a “theory” This r is a “mathematical construct”. It is a “theory”. For it to be valid, it must correctly predict results in the real world.

Do the “pegboard lab”. Or watch the pegboard video.

Both are easy using moment arm. Examples of moment arm. A ladder is leaning against a frictionless wall. To calculate the coefficient of friction of the ground, we need the torque due to the wall and due to the ladder’s weight. Both are easy using moment arm.

Torque due to the ladder: τladder = r × F = ((L/2)cosθ)mg

τ wall = r × F = (Lcosθ)Fwall Torque due to the wall: τ wall = r × F = (Lcosθ)Fwall

Either of these last examples the torque could be calculated by first finding the perpendicular force. But the next example proves the reason for moment arm.

For the lamp to be in static equilibrium the torque due to the lamp must equal the torque due to the rope.

τlamp = r × F = (3.2sin40°)200

Moment arm also works for angular momentum: L = r x p. The rocket only changes its angular momentum when it strikes the rod. So the straight line distance does not matter, only the distance from the pivot: the moment arm.

Q: Calculate the angular momentum of a mass moving at constant velocity relative to a reference point

A: L = r x p r = dsinθ L = (dsinθ)mv