Moments & Machines

Moments & Machines

Push a door near its hinge and it barely moves; push near the handle and it swings easily. Same force, very different results, because turning depends on where you push. That turning effect is called a moment. It is the secret behind levers, spanners and gears, and this module also asks the question those explanations usually skip: if a lever makes a job easier, what are you giving up in exchange?

The words for it

Five terms. The third is the one exam questions quietly test, because it is easy to read past:

Find the moment

A force of 20 N is applied to a spanner at a perpendicular distance of 0.5 m from the pivot. What is the moment, in newton-metres (Nm)?

The push that does nothing

You push on a door as hard as you can, but exactly along a line pointing straight at the hinge. The door does not turn at all. Why not?

• The line of your force passes through the pivot, so the perpendicular distance is zero, and M = F × 0 whatever the force is
• The force is not large enough to overcome the friction in the hinge
• You are pushing towards the hinge instead of away from it, so the moment acts in the wrong direction
• Your hand is too close to the hinge for the distance to matter

What a lever gives, and takes

A lever applies a force at a distance from a pivot. Applying the same force further out gives a bigger moment, which is why a long spanner beats a short one and why door handles are fitted far from the hinges. But nothing is free:

What does it cost you?

You swap a short crowbar for one twice as long and lift the same rock with half the effort. What happens to the distance your hands move?

• They move about twice as far, because the lever multiplies force but not energy: half the force over twice the distance is the same work
• They move about half as far, since everything about the job has become easier
• The same distance: only the force has changed
• It depends entirely on how heavy the rock is

The principle of moments

When an object is balanced, meaning it is not turning, the moments about the pivot cancel out exactly: total clockwise moment = total anticlockwise moment A balanced seesaw obeys this, and so does a shelf bracket, a crane with a counterweight and a set of scales. It is the tool for finding an unknown force or an unknown distance. One habit worth building: identify the pivot first and write it down. Almost every mistake in this topic comes from measuring a distance from the wrong point.

Solving a balance problem

Put the steps for solving a balanced-moments problem in order.

• Identify the pivot
• Work out the anticlockwise moment (force × distance)
• Work out the clockwise moment (force × distance)
• Set the clockwise moment equal to the anticlockwise moment
• Solve for the unknown force or distance

Balance the seesaw

A seesaw is balanced. On the left, a 30 N weight sits 2 m from the pivot. On the right, a weight sits 3 m from the pivot. What is the force on the right, in newtons (N)?

Find the distance

A balanced beam has a 40 N force acting 1.5 m from the pivot on one side. A 20 N force acts on the other side. How far from the pivot must the 20 N force be, in metres (m)?

Gears

Gears are toothed wheels that pass a turning effect from one wheel to another. Two meshed gears turn in opposite directions, because their teeth push each other along. A larger gear turns more slowly but with a greater moment, and the reason is worth having rather than memorising. Where the teeth mesh, they must move at the same speed, so a bigger wheel makes fewer turns for the same tooth movement. And the force at the teeth acts at a larger distance from the axle, which by M = F d is a larger moment. A bicycle is the clearest example. Low gear puts a large sprocket at the back: plenty of turning effect for a hill, and not much speed. High gear does the reverse.

Gears, hills and wheelbarrows

Four everyday machines, and the same physics under all of them.

• You reach a steep hill on a bicycle. Which gear do you want at the rear wheel, and what does it give you?
• Why can no arrangement of gears give you more turning effect AND more speed at the same time?
• A driving gear turns clockwise. The gear meshed directly with it turns which way?
• A wheelbarrow is easier to lift when the load is packed close to the wheel. Why?

Which are true?

Select ALL THREE statements that are TRUE.

• A force can be enormous and still produce no moment at all, if its line of action passes through the pivot
• A lever lets you use a smaller force, but your end has to travel further, because a lever multiplies force and not energy
• Two meshed gears turn in opposite directions, and the larger of them turns more slowly
• A longer lever multiplies the energy you put in as well as the force
• A shorter lever gives a bigger moment for the same force
• The moment depends on the distance from the pivot to where you are standing

Machines summary

The moment of a force is its turning effect: M = F _____, where the distance is measured from the pivot to the _____ the force acts along. A longer lever gives a bigger moment for the same force, but your end must move further, because a lever multiplies force and not _____. When an object is balanced, the total clockwise moment equals the total anticlockwise moment. Two meshed gears turn in _____ directions.

Explain the long spanner

Exam practice. In about 50 words, explain why a long spanner undoes a stiff bolt more easily than a short one, and what you give up in exchange. Include:

• which quantity in M = F d the longer spanner changes, and what that does to the moment
• what happens to the distance your hand has to move
• why that trade is unavoidable, in terms of the work done