Forces and Newton’s laws of motion
Nature and Nature’s Laws lay hid in Night.
God said, Let Newton be! and All was Light.
The picture shows crates of supplies being dropped into a remote area by parachute. What forces are acting on a crate of supplies and the parachute?
One force which acts on every object near the earth’s surface is its own weight. This is the force of gravity pulling it towards the centre of the earth. The weight of the crate acts on the crate and the weight of the parachute acts on the parachute.
The parachute is designed to make use of air resistance. A resistance force is present whenever a solid object moves through a liquid or gas. It acts in the opposite direction to the motion and depends on the speed of the object.
The crate also experiences air resistance, but to a lesser extent than the parachute. Other forces are the tensions in the guy lines attaching the crate to the parachute. These pull upwards on the crate and downwards on the parachute. All these forces can be shown most clearly if you draw force diagrams for the crate and the parachute.
Force diagrams are essential for the understanding of most mechanical situations.
A force is a vector: it has a magnitude, or size, and a direction. It also has a line of action. This line often passes through a point of particular interest. Any force diagram should show clearly
● the direction of the force
● the magnitude of the force
● the line of action.
Example: A body of mass 5 kg is acted upon by two forces of 6N and 8N which are perpendicular. Find the resultant force and the direction that it makes with the horizontal.
In figures 3.1 and 3.2 each force is shown by an arrow along its line of action. The air resistance has been depicted by a lot of separate arrows but this is not very satisfactory. It is much better if the combined effect can be shown by one arrow.
When you have learned more about vectors, you will see how the tensions in the guy lines can also be combined into one force if you wish. The forces on the crate and parachute can then be simplified.
Centre of mass and the particle model
When you combine forces you are finding their resultant. The weights of the crate and parachute are also found by combining forces; they are the resultant of the weights of all their separate parts. Each weight acts through a point called the centre of mass or centre of gravity.
Think about balancing a pen on your finger. The diagrams show the forces acting on the pen.
So long as you place your finger under the centre of mass of the pen, as in figure 3.5, it will balance. There is a force called a reaction between your finger and the pen which balances the weight of the pen. The forces on the pen are then said to be in equilibrium. If you place your finger under another point, as in figure 3.6, the pen will fall. The pen can only be in equilibrium if the two forces have the same line of action.
If you balance the pen on two fingers, there is a reaction between each finger and the pen at the point where it touches the pen. These reactions can be combined into one resultant vertical reaction acting through the centre of mass.
The behaviour of objects which are liable to rotate under the action of forces is covered in Mechanics 2 Chapter 11. In Mechanics 1 you will only deal with situations where the resultant of the forces does not cause rotation. An object can then be modelled as a particle, that is a point mass, situated at its centre of mass.
Newton’s third law of motion
Sir Isaac Newton (1642–1727) is famous for his work on gravity and the mechanics you learn in this course is often called Newtonian Mechanics because it is based entirely on Newton’s three laws of motion. These laws provide us with an extremely powerful model of how objects, ranging in size from specks of dust to planets and stars, behave when they are influenced by forces.
We start with Newton’s third law which says that
● When one object exerts a force on another there is always a reaction of the same kind which is equal, and opposite in direction, to the acting force.
You might have noticed that the combined tensions acting on the parachute and the crate in figures 3.3 and 3.4 are both marked with the same letter, T. The crate applies a force on the parachute through the supporting guy lines and the parachute applies an equal and opposite force on the crate.
When you apply a force to a chair by sitting on it, it responds with an equal and opposite force on you. Figure 3.8 shows the forces acting when someone sits on a chair.
The reactions of the floor on the chair and on your feet act where there is contact with the floor. You can use R1, R2 and R3 to show that they have different magnitudes. There are equal and opposite forces acting on the floor, but the forces on the floor are not being considered and so do not appear here.
Why is the weight of the person not shown on the force diagram for the chair?
Gravitational forces obey Newton’s third law just as other forces between bodies.
According to Newton’s universal law of gravitation, the earth pulls us towards its centre and we pull the earth in the opposite direction. However, in this book we are only concerned with the gravitational force on us and not the force we exert on the earth.
All the forces you meet in mechanics apart from the gravitational force are the result of physical contact. This might be between two solids or between a solid and a liquid or gas.
Friction and normal reaction
When you push your hand along a table, the table reacts in two ways.
● Firstly there are forces which stop your hand going through the table. Such forces are always present when there is any contact between your hand and the table. They are at right angles to the surface of the table and their resultant is called the normal reaction between your hand and the table.
● There is also another force which tends to prevent your hand from sliding. This is the friction and it acts in a direction which opposes the sliding.
Figure 3.9 shows the reaction forces acting on your hand and on the table. By Newton’s third law they are equal and opposite to each other. The frictional force is due to tiny bumps on the two surfaces (see electronmicrograph below). When you hold your hands together you will feel the normal reaction between them. When you slide them against each other you will feel the friction.
(i) at the instant it is hit by the racket
(ii) as it crosses the net
(iii) at the instant it lands on the other side.
Newton’s first law can be stated as follows.
● Every particle continues in a state of rest or uniform motion in a straight line unless acted on by a resultant external force.
Newton’s first law provides a reason for the handles on trains and buses. When you are on a train which is stationary or moving at constant speed in a straight line you can easily stand without support. But when the velocity of the train changes, a force is required to change your velocity to match.
(i) The coin is stationary.
(ii) The coin is moving upwards with a constant velocity.
(iii) The speed of the coin is increasing as it moves upwards.
(iv) The speed of the coin is decreasing as it moves upwards.
(i) When the coin is stationary the velocity does not change. The forces are in equilibrium and R = W.
(ii) When the coin is moving upwards with a constant velocity the velocity does not change. The forces are in equilibrium and R = W.
(iv) When the speed of the coin is decreasing as it moves upwards there must be a net downward force to make the velocity decrease and slow the coin down as it moves upwards. In this case W > R and the net force is W−R.
In problems about such things as cycles, cars and trains, all the forces acting along the line of motion will usually be reduced to two or three: the driving force forwards, the resistance to motion (air resistance, etc.) and possibly a braking force backwards.
The lines joining the crate of supplies to the parachute described at the beginning of this chapter are in tension. They pull upwards on the crate and downwards on the parachute. You are familiar with tensions in ropes and strings, but rigid objects can also be in tension.
Now draw the forces acting on your hands and on the pencil when you push the pencil inwards.
Your first diagram might look like figure 3.18. The pencil is in tension so there is an inward tension force on each hand.
If each hand applies a force of 2 units on the pencil, the tension or thrust acting on each hand is also 2 units because each hand is in equilibrium.
You have already met the idea that a single force can have the same effect as several forces acting together. Imagine that several people are pushing a car. A single rope pulled by another car can have the same effect. The force of the rope is equivalent to the resultant of the forces of the people pushing the car. When there is no
resultant force, the forces are in equilibrium and there is no change in motion.
Draw diagrams showing the horizontal forces acting on the car and the trailer
(ii) when the speed of the car is increasing
(iii) when the car brakes and slows down rapidly.
(i) When the car moves at constant speed, the forces are as shown in figure 3.20 (overleaf). The tow bar is in tension and the effect is a forward force on the trailer and an equal and opposite backward force on the car.
For the trailer: T − S = 0
For the car: D − R − T = 0
(ii) When the car speeds up, the same diagram will do, but now the magnitudes of the forces are different. There is a resultant forward force on both the car and the trailer.
For the trailer: resultant = T – S
For the car: resultant = D – R – T
(iii) When the car brakes a resultant backward force is required to slow down the trailer. When the resistance S is not sufficiently large to do this, a thrust in the tow bar comes into play as shown in the figure 3.21.
Newton’s second law gives us more information about the relationship between the magnitude of the resultant force and the change in motion. Newton said that
● The change in motion is proportional to the force.
For objects with constant mass, this can be interpreted as the force is proportional to the acceleration.
Resultant force = a constant × acceleration
A force of 1 newton will give a mass of 1 kilogram an acceleration of 1 m s–2. The equation then becomes:
Resultant force = mass × acceleration
This is written: F = ma
The resultant force and the acceleration are always in the same direction.
The mass of an object is related to the amount of matter in the object. It is a scalar. The weight of an object is a force. It has magnitude and direction and so is a vector.
The mass of an astronaut on the moon is the same as his mass on the earth but his weight is only about one-sixth of his weight on the earth. This is why he can bounce around more easily on the moon. The gravitational force on the moon is less because the mass of the moon is less than that of the earth.
When the weight is the only force acting on an object, Newton’s second law means that
Anyone who says 1 kg of apples weighs 1 kg is not strictly correct. The terms weight and mass are often confused in everyday language but it is very important for your study of mechanics that you should understand the difference.
(i) a baby of mass 3 kg
(ii) a golf ball of mass 46 g?
In the remainder of this chapter weight will be represented by mg. You will learn to apply Newton’s second law more generally in the next chapter.
A pulley can be used to change the direction of a force; for example it is much easier to pull down on a rope than to lift a heavy weight. When a pulley is well designed it takes a relatively small force to make it turn and such a pulley is modelled as being smooth and light. Whatever the direction of the string passing over this pulley, its tension is the same on both sides.
The rope is in tension. It is not possible for a rope to exert a thrust force.
(i) Draw diagrams to show the forces acting on each of A and B.
(ii) If the block A does not slip, find the tension in the string and calculate the magnitude of the friction force on the block.
(iii) Write down the resultant force acting on each of A and B if the block slips and accelerates.
The masses of 2 kg and 5 kg are not shown in the force diagram. The weights 2g N and 5g N are more appropriate.
(ii) When the block does not slip, the forces on B are in equilibrium so
For A, the resultant horizontal force is zero so
(iii) When the block slips, the forces are not in equilibrium and T and F have different magnitudes.
The resultant horizontal force on A is (T – F) N towards the right.
The resultant force on B is (5g – T) N vertically downwards.
In mechanics you express the real world as mathematical models. The process of modelling involves the cycle shown in Figure 3.25 and this is used in the example that follows.
Model 2: Air resistance is constant and the same for all objects.
other than its mass which might affect its motion as it falls? How do people and
animals maximise or minimise the force of the air?
This contradicts the prediction of model 2. A large surface at right angles to the motion seems to increase the resistance.
Assume the air resistance is kA where k is constant and A is the area of the surface perpendicular to the motion.
For this experiment you will need some rigid corrugated card such as that used for packing or in grocery boxes (cereal box card is too thin), scissors and tape.
Cut out ten equal squares of side 8 cm. Stick two together by binding the edges with tape to make them smooth. Then stick three and four together in the same way so that you have four blocks A to D of different thickness as shown in the diagram.
Observe what happens when you hold one or two blocks horizontally at a height of about 2 m and let them fall. You do not need to measure anything in this experiment, unless you want to record the area and mass of each block, but write down your observations in an orderly fashion.
1 Drop each one separately. Could its acceleration be constant?
2 Compare A with B and C with D. �Make sure you drop each pair from the same height and at the same instant of time. Do they take the same time to fall? Predict what will happen with other combinations and test your predictions.
4 Now compare A with E, B with F, C with G and D with H. Compare also the two blocks whose dimensions are all in the same ratio, i.e. B and G.
injury to human beings.