Circular Motion, Moments, Levers and Gears

Constant speed can still mean changing velocity

Speed tells us only how fast an object is moving. Velocity includes both speed and direction. An object travelling around a circle at constant speed therefore does not have constant velocity: its direction changes continuously, so its velocity changes continuously.

At every point on the circle, the instantaneous velocity points along a tangent to the path. A tangent touches the circle at that point. If the force maintaining the circular motion disappeared, the object would continue in this tangential direction rather than travelling radially outwards.

Changing velocity means acceleration, even when speed remains constant. For uniform circular motion, the resultant force is directed inwards, towards the centre of the circle, and is perpendicular to the instantaneous direction of motion. It changes the direction of the velocity rather than its magnitude.

For example, gravity pulls an orbiting planet towards the Sun, along the radius of its approximately circular orbit. The planet’s instantaneous motion is tangential to the orbit, so gravity continually changes its direction. A turning car similarly needs a sideways resultant force towards the centre of its curved path.

Forces can make objects rotate

A force whose line of action passes through a pivot does not produce a turning effect about that pivot. When the line of action misses the pivot, the force can make the object rotate. The turning effect is called the moment of the force.

A force on one side of a pivot may cause a clockwise moment, while a force on the other side may cause an anticlockwise moment. The overall rotational effect depends on the moments in both directions. If the total clockwise moment equals the total anticlockwise moment, there is no resultant moment and the object has no rotational acceleration.

The size of a moment depends on both the force and how far its line of action is from the pivot:

$$M=Fd$$

where $M$ is the moment in newton metres ($\text{N m}$), $F$ is the force in newtons ($\text{N}$), and $d$ is the perpendicular distance in metres ($\text{m}$) from the pivot to the force’s line of action.

The distance must be measured normal to the direction of the force—in other words, at $90^\circ$ to its line of action. If a force acts at right angles to a straight lever, this may equal the distance along the lever. If the force is angled, the full lever length is not generally the correct distance.

To calculate a moment, first identify the pivot and draw or imagine the force’s line of action. Find the shortest perpendicular distance to that line, convert it to metres, and then multiply it by the force.

Calculating a moment

A downward force of 40 N acts on a horizontal lever. Its line of action is 0.35 m from the pivot, measured perpendicular to the force. Calculate the moment about the pivot.

$$M = Fd$$ $$40 \times 0.35 = 14\,\text{N m}$$

The result is the turning effect produced by that force about the stated pivot. Its direction should also be described as clockwise or anticlockwise when the diagram makes this clear. Increasing either the force or the perpendicular distance increases the moment.

Levers transmit and change rotational effects

A lever is a rigid object that turns about a pivot. An input force, often called the effort, creates a moment that is transmitted through the lever to a load. Everyday examples include a crowbar, a bottle opener and the handles of scissors.

Placing the effort farther from the pivot increases its moment without increasing the effort force. A long effort arm can therefore make a lever a force multiplier: a smaller effort can balance or move a larger load acting closer to the pivot. The trade-off is distance—the effort end must move farther than the load. A lever arranged the other way can act as a distance multiplier, making the load move farther or faster while requiring a larger effort.

Gears transmit moments between rotating axles

Gears are toothed wheels. When two gears mesh, their teeth push on one another and transfer a rotational effect from the driving gear to the driven gear. The two gears rotate in opposite directions.

Gear size changes the balance between turning effect and rotational speed. If a small gear drives a larger gear, the larger gear turns more slowly but can provide a greater moment. If a large gear drives a smaller gear, the smaller gear turns faster but provides a smaller moment. Gears can therefore change rotational speed, direction and moment while transmitting rotation between axles; they do not create energy.

In the OCR specification, the assessed outcomes on rotational forces, moments, levers and gears are identified as separate-science content.

Circular motion

• Uniform circular motion has constant speed but continuously changing velocity.
• Velocity is tangential to the path.
• The resultant force acts towards the centre, perpendicular to the instantaneous motion, and changes its direction.

Moments

$M=Fd$

• $M$: moment in $\text{N m}$
• $F$: force in $\text{N}$
• $d$: perpendicular distance from the pivot to the force’s line of action in $\text{m}$
• Identify the moment as clockwise or anticlockwise where possible.
• No resultant moment means total clockwise moment equals total anticlockwise moment.

Levers and gears

• A lever transmits a rotational effect about a pivot.
• Force multiplier: smaller effort, but the effort moves farther.
• Distance multiplier: load moves farther or faster, but greater effort is needed.
• Meshed gears rotate in opposite directions.
• Small driver to large driven gear: lower speed and greater moment.
• Large driver to small driven gear: higher speed and smaller moment.