Newton's laws - OCR GatewayFree body diagrams and vector diagrams - Higher

Isaac Newton’s laws surrounding forces were formulated hundreds of years ago, but are still used today – they help to describe the relationship between a body and the forces that act upon it.

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Free body diagrams and vector diagrams - Higher

are used to describe situations where several forces act on an object. Vector diagrams are used to resolve (break down) a single force into two forces acting at right angles to each other.

Free body diagrams

A free body diagram models the forces acting on an object. The object or 'body' is usually shown as a box or a dot. The forces are shown as thin arrows pointing away from the centre of the box or dot.

Two free body diagrams show a cross with arrows on each end and a box in the middle. The second is a straight line with arrows on each end and a black dot in the middle. — Figure caption,
Representing an object in a free body diagram as a box or a dot

Free body diagrams do not need to be drawn to scale but it can sometimes be useful if they are. It is important to label each arrow to show the of the force it represents. The type of force involved may also be shown.

Examples of free body diagrams

Weight and reaction force for a resting object:

Drawing of situation

Free body diagram

A box rests on a table. Two arrows pointing in opposite directions act upwards and downwards from the point at which they meet on the table.

There is a black dot with equally sized arrows pointing upwards and downwards away from the dot.

Weight, reaction force and friction for an object moving at constant speed down a hill:

Drawing of situation

Free body diagram

A box rests on an incline. There are three arrows; one acting vertically downwards from box’s base, one arrow acting perpendicular to the incline and one up the incline.

Three arrows of varying sizes coming out of a black dot. Each one is labelled with the relevant force.

Weight, upthrust, and for an accelerating speedboat:

Drawing of situation

Free body diagram

A boat rests on water. There are four arrows of different lengths coming out of the boat, pointing in different directions.

Arrows point upwards and downwards away from a square. Large arrow points to the right and smaller arrow points left.

Question

A box is at rest on a table. Draw the free body diagram for this situation.

Question

A trolley is being pulled along a rough surface at a constant speed. Draw the free body diagram for this situation.

Vector diagrams

The resultant vector for two vectors at right angles to each other can be worked out using a scale diagram, or using a calculation.

Using a scale diagram

In the diagram below, two velocities are at right angles to each other. One is 4 m/s and the other is 3 m/s.

Scale diagram, 3m/s on the x-axis going right, 4m/s on the y-axis going upwards, resultant going diagonally upwards at 5m/s. — Figure caption,
The resultant vector of two vectors at right angles

If the diagram is drawn to scale like this, the magnitude of the resultant vector can be found by measuring the length of the diagonal vector arrow.

Pythagoras' theorem can be used to calculate the resultant vector.

In any right-angled triangle, the square of the longest side is the sum of the squares of the other two sides. This can be written in the formula:

a² + b² = c²

This is where c is the longest side.

In the example above, a = 4 m/s and b = 3 m/s.

c² = 4² + 3²

c² = 16 + 9 = 25

\(c = \sqrt{25}\)

\(c = 5~m/s\)

Here, c is equal to the line marked 5 m/s, as both dissect the rectangle in the same way.

Newton's second law

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