Physics
Cambridge IGCSE

TOPIC 1A: FORCES and MOTION

# 1.1 Physical Quantities

### Introduction

This entire topic is about forces and how force affects the movement of all things. There are many types of forces, and many ways we can measure movement.
Although this topic seems common sense to many, there are several areas of science in this topic not fully understood: Magnets can push and pull, but gravity can only pull. Why is that? What if we could control gravity like we control electricity, and we could make it push instead of pull? Recently the discovery of gravity waves was announced, which produce a varying pull on all objects in as the wave passes Earth.
Did you know that light rays can produce a force? Scientists have proposed a method of using light from the Sun to drive spaceships in the future as shown here:

Fig 1 . A Solar Sail for future space travel
(NASA - public domain)

The study of how forces affect motion is still at the forefront of science.

To begin this topic, we need to know how to take measurements effectively, especially when investigating motion: Some of this you should have covered already in previous learning, but basic measuring techniques are still included in the IGCSE syllabus.

## Measuring Techniques

Many of the experiments required for this IGCSE course require you to accurately measure mass, time, length and volume. There are standard ways of doing this and some subject specific vocabulary that you should be aware of:

### Mass (International Unit: kilogram)

For small masses, always use a laboratory 'electronic balance'. This can also be called a digital scale or 'top pan balance' but avoid 'weighing scales'. Some measure to the nearest gram, some to 0.001 g accuracy, so take care when you record your results to always use the right number of sig.figs.

Fig 2 . An electronic balance

In this example, the mass of the block is 103.00g. DO NOT round this up to 103 g in your results table! You want to show that you used a very accurate balance, so show all of the decimal points.

### Time (International Unit: second)

You should be able to use stopwatches for experiments, as well as other forms of timer. One very useful timing device used in motion experiments is called a light gate - these connect to a computer or device that records when the infrared beam has been broken as something moves through it. The video here explains how they can be used.

YouTube: How to use light gates (Underdog Physics)

Light gates are a good device to name in the evaluation section of a motion experiment - they always give more accurate times and do not rely on human reaction times, as do stop watches.
Another useful technique is to record an experiment on a video (using a phone or laptop) and then watch it back in slow motion - the exact time an event occurs can be seen from the video timer and recorded.

### Length (International Unit: metre)

To measure length we clearly need a ruler. Take care to measure from a position at 90° to the ruler, to avoid what is called 'parallax error' as shown in this diagram. If you are viewing the ruler scale from any other position, you can measure the incorrect reading as shown here.

Fig 3 . Parallax Error

### Volume (International Unit: m3)

Regular solid volumes such as a wooden block can be measured using a ruler, and the volume calculated. For lab experiments, volume is usually given in cm3, although the official international unit is the m3. Liquids are usually measured in litres and millilitres. Did you know that 1 ml is the same volume as 1 cm3?
Section 1.4 on density explains some standard methods for measuring the volume of irregular objects.
One of the most common mistakes on IGCSE science coursework or in the practical exam is describing how to measure the volume of a liquid using a beaker. These have guide measurement lines but they are NOT accurate! Always use a measuring cylinder!

Fig 4 . Measure liquid volume with a measuring cylinder
Lilly_M, CC BY-SA 3.0, via Wikimedia Commons

### Improving Accuracy

During all of your IGCSE science practical work, you should aim to collect the best results you can, giving the most accurate results possible. There are several standard techniques to help you do this. The most obvious one which is often overlooked, is to take repeat readings and then take an average. If you have random errors in your experiment, perhaps due to human error pressing the stop watch button at the end of an event, then taking repeats and an average value can reduce the effect of these random errors.

Note that scientists always do more than 2 repeats, so that if an anomalous result is collected, it is easily spotted when compared to the others.

The second standard technique common in physics is to measure multiples rather than individual measurements. For example, rather than measuring the thickness of a single page of a book, measure the thickness of 200 pages and then divide by 200 to find the answer for one page. This can also be used for timing events. Here is a simple pendulum. Instead of measuring the time period T for one swing, time for 10 swings and then divide by 10 to find a more accurate answer.
This technique reduces errors - for example if you can only time to an accuracy of within ±0.2 seconds, when you divide by 10, the accuracy becomes ±0.02 s! Much improved.

Fig 5 . Timing a simple pendulum

Questions:

1. What is the time period T of the pendulum shown above? (Note: One swing is 'there and back again')

Time the pendulum for 10 swings, and do this 5 times. Sample results obtained by a student were:
 Time Period, T (seconds) 27.88 27.75 27.90 25.23 27.70

There is an anomalous result for the 4th timing. Maybe the student only timed 9 swings by accident? Ignoring this result, the other 4 give an average for 10 swings of 27.8075 seconds.
Therefore T =  27.8075 /10 = 2.78075 seconds.
We now need to round up as the student was clearly not that accurate! For 10 swings it looks like the results varied by about ±0.1 seconds for 10 swings, so ±0.01 seconds for 1 swing, so it makes sense to round to the nearest 1/100th of a second:
T = 2.78 seconds
(Note: You will not be expected to justify why you rounded up - 3 significant figures is the best advice for nearly all experiments).

## Vectors and scalars

Before we go any further into learning about forces and motion, we need to understand the difference between 'vectors' and 'scalars'.

In order to draw a force we need to know how large the force is, but also the direction in which the force acts.
Any measurement ('quantity') that has a size and direction is called a vector. A good way to remember this is to use the idea that if you can point in the direction that the measurement is going, then it is a vector quantity.

Force is a vector. When a car accelerates we can say it is 'accelerating down the road' and point in the direction. Therefore acceleration is a vector.

Measurements that do not have a specific direction are called scalars. For example, coal stores chemical energy and you can measure how much energy is stored. However, you cannot 'point' which way the energy is stored, so it is a scalar. Time is a scalar quantity because, even though we say it moves forwards, we cannot point in the direction of time.

### Speed and distance

Speed and distance are difficult ones. To explain this, imagine a simple journey:
If you walk 1 km to a cinema and back again, how far have you gone?

Figure 6. Vector or Scalar?

On the left diagram in figure 6, the total distance = 1+1 = 2km. However on the right, we have used + 1km for the outward journey, and -1 km to show that we are moving in the opposite direction back to the start. Both answers are correct, but in the first diagram we used a scalar measurement (no direction considered) to give 2km, and the second we included the direction - a vector measurement - to give a total of zero.

If we ignore direction, we call this the distance travelled. Distance is a scalar.
If we include the direction , it is called the displacement. Displacement is a vector.
So in the example above shown in figure 6, the distance travelled = 2km, and the displacement = 0 km.

Similarly, if we want to describe how fast something is going, we can use speed if no direction is considered, and velocity if we include a direction.
Speed is a scalar quantity and velocity is a vector quantity. This will be covered in more depth in the next section.

This table summarises a few of the most common scalar and vector quantities:

Scalars Vectors
Energy Force
Time Acceleration
Speed Velocity
Distance Displacement
Mass Weight

Table 1. Common Vectors and Scalars

Questions:

2. A block of wood is pushed across a table, from left to right. Describe the direction of the following forces:

• a) Friction.
• b) Gravity.

a) Friction always acts in the opposite direction to the motion, so it acts from right to left.
b) Gravity always pulls objects downwards, (towards the centre of the Earth).

3. The International Space Station is in orbit around the Earth. The following measurements are listed in an information website about the space station:

• Mass = 420 000 kg
• Width = 109 m
• Speed = 7.7 km/s
• Time for 1 orbit = 93 minutes
• Pull of gravity on the ISS = 4.1 million newtons.

Which of these five measurements are vector quantities?

Mass, time, speed and width are all scalar measurements. (Width is treated like the distance across the space station).
The only vector here is the pull of gravity which is a force, measured in newtons.

4. An athlete completes a 400 m race around a standard athletic track in 80s, finishing back at the start point.

• a) State the distance travelled.
• b) State the total displacement travelled.
• c) Calculate the average speed of the athlete.
• d) State the average velocity of the athlete.

a) Distance is a scalar quantity so we can ignore the direction around the track. The distance travelled is simply 400 m.

b) displacement is a vector so we need to consider the direction of motion. The runner has arrived back at the start point after an 'outward' and 'backward' stretch. Therefore the total displacement is zero - they are back where they started!

c) We need to use the formula:
 average speed = distance moved time taken
So:
 average speed = 400 = 5 m/s 80

d) The clue here is that is says 'state' so it should be something simple, not a calculation! it is either the same as the last answer (5 m/s) or something else simple. In fact the average velocity is zero.
(As the athlete has returned to the start point, the displacement is zero and so the velocity is also zero, as they are back where they started, and there has been no average movement in a specific direction).

Two tug boats are pulling a large oil tanker into port. They are pulling in different directions as shown in figure 3 below: One is pulling downwards in the diagram, and the other to the right.

Which way do you think the tanker will move? What is the resultant force in this situation where the forces do not act in a straight line?

Figure 7. Adding the forces on an oil tanker

In this case, the boat will be pulled diagonally downwards to the right in the diagram. Did you guess that? It seems to make sense, but how do you add together forces in this situation?

### Scale Diagrams

There are several techniques to find the resultant force, but one easy way is to draw a scale diagram, and draw the two forces end to end. In the diagram below, (figure 7) the two forces from the tug boats have been drawn end to end. We have used a scale of 50 kN = 1 cm, so that 400 kN = 8 cm for tug boat 1. The size and direction of the resultant force can be measured directly from this scale diagram.

Figure 8. Using scale diagrams of vectors

Note that it does not matter which force is drawn first, the answer will still be the same. From the scale diagram, we can measure the length of the red resultant line, and also the angle shown in the diagram.
The length of the red resultant line is approximately 10 cm, which gives a resultant force of about 50 kN.
The angle of this force measured (with a protractor) to the horizontal is 37°.

Note that only questions involving forces (or velocity vectors) at 90° will be asked.

### Using maths to calculate the resultant

In the diagram above, the resultant is the diagonal line shown. The length of this line could have been calculated using Pythagoras' theorem, without needing to draw a scale diagram:

Figure 9. Using maths to calculate the sum of two vectors

Using Pythagoras' theorem, c2 = a2 + b2
Therefore:
(Resultant)2 = 4002 + 3002   (working in kN - kilonewtons)
(Resultant)2 = 250 000
So the resultant = 500 kN

You might not have learned the maths yet to find the angle, but to do this you need to use trigonometry. The angle between the horizontal 400 kN line and the red resultant line can be found using:
tan-1(θ) = opposite/adjacent = 300/400 = 0.75,
so θ = 36.9°

Questions:

5. A student tries to swim across a river. The student can swim at 1 m/s, and aims to swim directly to the opposite bank as shown. However the river has a current of 2 m/s.

a) On the diagram, sketch an approximate direction for the resultant velocity of the student relative to the starting point.
b) Find using a scale diagram, or by calculation, the magnitude of the resultant velocity.

a) The resultant arrow should be drawn upwards to the right, and also at an angle less than 45° to the horizontal as shown below, because the river current is faster than the swimmer's velocity, meaning the swimmer will drift down river faster than moving across the river.

b) The resultant arrow can be found using a scale diagram as shown below, and then the length of the line measured.

Alternatively, using maths, the red resultant line can be found by:

Using Pythagoras' theorem, c2 = a2 + b2
Therefore:
(Resultant)2 = 12 + 22   (working in m/s)
(Resultant)2 = 5
So the resultant = 2.24 m/s
(note that the angle is not required in the question, only the magnitude - the size - of the resultant arrow).

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