5.1 An introduction to moments and mechanical forces of rotation - defining a moment

This page will help you answer questions such as: What is a moment? What is a mechanical advantage?  How do you calculate the turning effect of a force? Why are the turning effects of a force so important?  Where do we apply the advantages of the turning effect of a force?

Forces can cause an object to rotate and the turning effect of the force is called a moment.

If a resultant force acts on an object about a fixed turning point (the pivot) it will cause the object to rotate e.g. turning a nut with a spanner, applying a screwdriver, opening a door fixed on hinges. The pivot might also be called the fulcrum.

The rotational or turning force effect, the moment, has a magnitude easily calculated from the simple formula:

M = F x d, where M = the moment of a force (Nm), F = force applied (N)

and d (m) is the perpendicular distance from the pivot point to the line of action of the force.

You get the maximum moment by pushing/pulling the spanner at a right angle (at 90^o) to the line (d) between the pivot point and the line of action where the force is applied.

Screwing or unscrewing a nut on a bolt

This is illustrated by the simple diagram of a spanner above.

The turning force or moment is F x d.

See Part 5.2 Some simple moment calculations with worked out answers

The pivot point is the central axis of the bolt on which the nut is being turned by the spanner.

To tighten or loosen the nut you apply a force, to the best mechanical advantage, at 90^o to the spanner arm itself.

Applying the force at any other angle less than 90^o reduces d and so reduces the effective moment of the force.

You determine the force F by how hard you push/pull the end of the spanner, but d is a fixed distance for a given spanner.

This is one of many situations where you are applying a force to increase the effect of your hand muscles.

The size of the moment increases with increase in distance d or applied force F.

The longer the spanner, the greater is d, hence the greater the turning force generated - the greater the mechanical advantage of the lever system.

All sorts of machines use turning forces to increase a force (or pressure) effect to gain a mechanical advantage from a printer's press to a bicycle.

More on the physics of unscrewing a tight nut!

Spanner 'situations' A to D

Relative comments on the three 'moment' situations A to C

Situation A

With the longer spanner, and applying the force perpendicular (at 90^o) to the line from the point of applying the force (end of 'handle') to the pivot point (centre of the nut or bolt), you generate the maximum moment (F x d).

The perpendicular distance d, is the shortest distance between the pivot and the line of action of the force

Situation B

If you apply the force at any other angle than at 90^o to the perpendicular distance line (d), d will always be shorter and hence a smaller moment is generated by applying the same force as in situation A.

Note that when the line of action of the force is down the 'spine' of the spanner, d is zero and the moment is zero.

This is shown by the blue force arrow at situation D.

Situation C

With the shorter spanner, despite applying the force at 90^o (perpendicular), d is smaller and you cannot generate as large a moment as in situation A.

So, for the same applied force, the moment is smaller for the smaller length spanner.

 

Situations A and B and riding a bicycle

You get situation B (from above) pedalling a bicycle when your foot and pedal are near the top or bottom of the crank's cycle - the pivot point is the crank axle. If you press downwards, you are creating the minimum moment - the minimum driving force forward.

At the top of the 'pedal cycle', if the direction of force is downwards through the crank, the turning force, the moment, is very small.

BUT, at the top of the 'pedal cycle', you soon learn to push forwards to utilise the maximum moment and generate the maximum force so that you mimic situation A where you are generating a force at 90^o to the pivoting axle and the line of action of the force - maximum moment = F x d on the above diagram.

You get the maximum force transferred when the crank and pedal are parallel to the ground and you exerting your maximum downward force at 90^o to the pivot point (axle) and the line of action of the force.

The centre of mass and the stability of a free standing object

The centre of mass is a single point in the object through which the whole weight of an object is considered to act.

Its quite easy to envisage where it is for a regular shape e.g. a rectangular block - shown in profile in the diagram below. It is coincident with what is termed the 'centre of gravity' of an object.

A standing object becomes unstable when the vertical line through its centre of mass falls outside its base, which effectively acts as a base - this happens if it is tilted over on one edge, thereby creating a moment - a turning force.

Under these conditions, the weight of the object causes a turning effect about the pivotal base.

The idea is illustrated by the diagram below of a regular shaped block, shown in profile, and tilted at various angles (but it could be a bus going round a corner!).

1. The vertical line from the centre of mass passes right through the centre of the block's base.

The object is completely stable - no moment (turning force) is generated.

2. The vertical line from the centre of mass still passes through the base, but not its centre, and the block is unstable, so it will wobble a bit from side to side, and eventually settle down in an stable upright position as in 1.

The edge of the block touching the surface acts as a pivot point.

The weight of the block creates an anticlockwise moment (turning force) that makes the block fall back in an anticlockwise direction, but not sufficient to topple the block over on its longer side.

There is a point between 2. and 3. where theoretically, the two moments are equal and the block will be balanced on one edge (an equilibrium situation).

BUT, this is not a very stable situation, any slight movement e.g. of the air, would cause it to topple over!

When two moments equal each other, we call it an equilibrium situation, which may or may not be stable.

See Part 5.3 Moment calculations and a balancing situation (equilibrium)

3. The vertical line from the centre of mass passes outside of the block's base. The block won't even wobble, it is highly unstable and will just topple over on its longer side (to become stable!).

Again, the edge of the block touching the surface acts as a pivot point.

Again, the weight of the block creates a clockwise moment (turning force) that makes the block fall over in a clockwise direction, and sufficient to topple the block over on to its longer side.

Tests on stability in terms of the centre of mass are important e.g. road vehicles like buses are safety tested to see the maximum angle allowed when tilted over without toppling over in an accident.