**Scale Drawings**

It was fairly easy to work out the resultant force acting on an object in the examples in the previous section. This is because the forces were acting in opposite directions; either left and right, or up and down.

However, you may be asked to work out the resultant force when there are two forces that are acting on an object along different planes. For example, you may be asked to work out the resultant force where there is an upwards and a leftwards force. We are able to work out the resultant force for these types of questions by using scale drawings. We do a scale drawing by drawing a force going from a point. We then draw the next force starting from the end of the first force. It does not matter which force we draw first. Let’s suppose that we are given two forces; one force is upwards, and the other force is rightwards. We can find the resultant force by drawing the upwards force and then the rightwards force. Or we can draw the rightwards force and then the upwards force.

We need to use an appropriate scale when we are drawing the forces. For example, if we are working with fairly small forces, 1 cm on our diagram could represent 1 N. However, if we are working with larger forces, 1 cm on the diagram could represent a greater force such as 50 N. We will usually have a centimetre grid when answering questions like this in the exam.

After we have drawn the final force, we are able to find the resultant force by measuring the length of the line from where the first force started to where the final force finished. We then use our scale to convert it into a force. We also measure the bearing of the resultant force line as well. There are a few different rules that we need to remember when we are working with bearings. These rules are:

The best way to understand scale diagrams is to go through a few examples.

However, you may be asked to work out the resultant force when there are two forces that are acting on an object along different planes. For example, you may be asked to work out the resultant force where there is an upwards and a leftwards force. We are able to work out the resultant force for these types of questions by using scale drawings. We do a scale drawing by drawing a force going from a point. We then draw the next force starting from the end of the first force. It does not matter which force we draw first. Let’s suppose that we are given two forces; one force is upwards, and the other force is rightwards. We can find the resultant force by drawing the upwards force and then the rightwards force. Or we can draw the rightwards force and then the upwards force.

We need to use an appropriate scale when we are drawing the forces. For example, if we are working with fairly small forces, 1 cm on our diagram could represent 1 N. However, if we are working with larger forces, 1 cm on the diagram could represent a greater force such as 50 N. We will usually have a centimetre grid when answering questions like this in the exam.

After we have drawn the final force, we are able to find the resultant force by measuring the length of the line from where the first force started to where the final force finished. We then use our scale to convert it into a force. We also measure the bearing of the resultant force line as well. There are a few different rules that we need to remember when we are working with bearings. These rules are:

- Bearings are measured from the North line
- Bearings are measured in a clockwise direction
- Bearings are measured in degrees and we always give them as 3 digits

The best way to understand scale diagrams is to go through a few examples.

**Example 1**

*Click here*

*for a printable version of all of the diagrams in this section so that you can give the examples a go as well.*

There are two forces acting on a ball; one of the forces is a 3 Newton force in a northwards direction and the other force is a 4 Newton force in an eastwards direction. Find the resultant force acting on the ball. Use the diagram below with the scale 1 cm = 1 N to help you. The blue dot is where the ball starts.

The first step in answering this question is to draw a starting point for the ball. The next step is to draw on one of the forces (it does not matter which force we draw on first). I am going to draw the 3 N force in a northwards direction first (north on the diagram will be upwards). The question tells us to use the scale 1 cm = 1 N. The northwards force is 3 N, which means that the northwards force will be 3 cm long. I have added this force to the scale diagram.

The next step is to draw the second force on the diagram starting from where the first force ended. The second force is a 4 N force in an eastwards direction, which will be towards the right on the scale diagram. The scale on the scale diagram is still 1 cm = 1 N, so this force will be 4 N. I have added this force to the scale diagram.

We now have both of the forces on the diagram. The resultant force goes from the initial starting point to the final position of the force. The resultant force is shown on the diagram below.

We need to work out the magnitude and the bearing of the resultant force.

We work out the magnitude of the resultant force by measuring the length of the line and converting the length into a force by using the scale. The length of the resultant force on our scale diagram is 5 cm. The scale that we used is 1 cm = 1 N, and this means that our resultant force is 5 N.

The next step is to work out the bearing of the resultant force. Bearings are always measured clockwise from the north line (up) and are given as three-digit numbers. The working for the bearing is shown below.

We work out the magnitude of the resultant force by measuring the length of the line and converting the length into a force by using the scale. The length of the resultant force on our scale diagram is 5 cm. The scale that we used is 1 cm = 1 N, and this means that our resultant force is 5 N.

The next step is to work out the bearing of the resultant force. Bearings are always measured clockwise from the north line (up) and are given as three-digit numbers. The working for the bearing is shown below.

The bearing is 053° (remember, we always give bearings as three-digit numbers).

So, the resultant force is 5 N on a bearing of 053°.

So, the resultant force is 5 N on a bearing of 053°.

**Example 2**

Two forces are acting on a ship; one of the forces is a 500 Newton force in a southwards direction and the other force is a 700 Newton force in a westwards direction. Find the resultant force acting on the ship. Use the diagram below to help you.

The first step in answering this question is to draw a starting point for the ship. The next step is to draw on one of the forces (it does not matter which force we draw on first). I am going to draw the 500 N force in a southwards direction first (south on the diagram will be downwards). The question does not give us a scale to use, which means that we must come up with an appropriate one; I am going to use the scale 1 cm = 100 N. This means that the 500 N southwards force will be 5 cm on the scale diagram (500 ÷ 100 = 5). The 500 N southwards force is shown on the diagram below.

The next step is to draw the second force on the diagram starting from where the first force ended. The second force is a 700 N force in a westwards direction, which is towards the left on the scale diagram. The scale that we used earlier was 1 cm = 100 N, and this means that this force will be 7 cm (700 ÷ 100 = 7). I have added this force to the scale diagram.

We now have both of the forces on the diagram. The resultant force goes from the initial starting point to the final position of the force. The resultant force is shown on the diagram below.

We need to work out the magnitude and the bearing of the resultant force.

We work out the magnitude of the resultant force by measuring the length of the line and converting the length into a force by using the scale. The length of the resultant force on our scale diagram is 8.6 cm. The scale that we used is 1 cm = 100 N, and this means that our resultant force is 860 N (8.6 x 100 = 860).

The next step is to work out the bearing of the resultant force. Bearings are always measured clockwise from the north line (up) and are given as three-digit numbers. I have added a north line and an arrow showing the bearing for the resultant force.

We work out the magnitude of the resultant force by measuring the length of the line and converting the length into a force by using the scale. The length of the resultant force on our scale diagram is 8.6 cm. The scale that we used is 1 cm = 100 N, and this means that our resultant force is 860 N (8.6 x 100 = 860).

The next step is to work out the bearing of the resultant force. Bearings are always measured clockwise from the north line (up) and are given as three-digit numbers. I have added a north line and an arrow showing the bearing for the resultant force.

The easiest way to work out the bearing for bearings above 180°, is to measure the angle anticlockwise between the north line and the bearing line. We then take this angle from 360° (a full turn) to work out the bearing.

The anticlockwise angle between the north line and the resultant force is 126°. We then take this off of 360°.

The bearing is 234°.

We now have our answer; the resultant force is 860 N on a bearing of 234°.

There will be a range of acceptable answers for resultant forces and bearings from scale diagrams. For this question the range of acceptable answers could have been a resultant force between 830 N and 890 N, and the bearing could have been between 229° and 239°.

We now have our answer; the resultant force is 860 N on a bearing of 234°.

There will be a range of acceptable answers for resultant forces and bearings from scale diagrams. For this question the range of acceptable answers could have been a resultant force between 830 N and 890 N, and the bearing could have been between 229° and 239°.

An object is in an equilibrium if there is a resultant force of zero. When an object is in equilibrium, it means that the object will continue with the same motion. A resultant force of zero for a stationary object will result in the object remaining stationary. A resultant force of zero for a moving object will result in the object continuing to move at exactly the same velocity (speed in a particular direction).

We can work out if the resultant force is zero by using scale diagrams. We draw all of the forces tip to tail that are occurring on an object (the order that we draw the forces does not matter). If all of the forces lead back to the same starting point, it means that there is a resultant force of zero, thus meaning that the object is in an equilibrium. Here is an example.

There are three forces acting on the object below. Show that the resultant force is zero.

**Equilibrium**An object is in an equilibrium if there is a resultant force of zero. When an object is in equilibrium, it means that the object will continue with the same motion. A resultant force of zero for a stationary object will result in the object remaining stationary. A resultant force of zero for a moving object will result in the object continuing to move at exactly the same velocity (speed in a particular direction).

We can work out if the resultant force is zero by using scale diagrams. We draw all of the forces tip to tail that are occurring on an object (the order that we draw the forces does not matter). If all of the forces lead back to the same starting point, it means that there is a resultant force of zero, thus meaning that the object is in an equilibrium. Here is an example.

There are three forces acting on the object below. Show that the resultant force is zero.

We are able to show that the resultant force is zero by drawing the forces tip to tail to each other on the diagram; the order that we draw the forces does not matter.

From the above diagram, we can see that the forces lead back to the starting point. This means that the resultant force is zero and the object is in an equilibrium.

**Splitting a Resultant Force**

We are able to split a resultant force at an angle into two separate forces; a horizontal force and a vertical force. These two separate forces acting together have the same effect as the single resultant force. The reason why we may want to split the resultant force into two separate forces is because having two separate forces makes it easier to solve certain calculations. We are able to find the separate horizontal and vertical force by drawing a scale diagram. Let’s have an example.

Split the resultant force on the scale diagram below into a horizontal and vertical force. The scale is 1 cm = 2 N and each square is 1 cm.

I am going to draw the horizontal and vertical components on the scale diagram below.

The scale for the diagram is 1 cm = 2 N.

The horizontal component is 6 cm long, which represents a force of 12 N (6 x 2 = 12). The horizontal force is going towards the right, which means that it is acting in an eastwards direction.

The vertical component is 5 cm long, which represents a force of 10 N (5 x 2 = 10). The vertical force is going upwards, which means that it is acting in a northwards direction.

The horizontal component is 6 cm long, which represents a force of 12 N (6 x 2 = 12). The horizontal force is going towards the right, which means that it is acting in an eastwards direction.

The vertical component is 5 cm long, which represents a force of 10 N (5 x 2 = 10). The vertical force is going upwards, which means that it is acting in a northwards direction.

Therefore, when we split the resultant force into two different forces, the two forces are a force of 12 N in an eastwards direction and a force of 10 N in a northwards direction.