## Physics-SchoolUK.com

### The UK site for KS3 and KS4 Physics

KS4 Forces and their Interactions

Some people say that Physics is the study of forces and energy, so the topic of Forces is a good place for us to start.

The first section of the Forces chapter of the AQA KS4 Physics Specification is Forces and their Interactions.

### KS4 Forces - Introduction

What is a Force?

"A force is a push or a pull that acts on one object due to its interaction with some other object."

For example:

1) The force between a shopping trolley and a customer when the customer pushes the trolley along the aisle.

2) The force between a strong man and a car when the strong man pulls the car along a road.

3) The force felt between two magnets.

4) The force felt between a ball and the ground (the Earth) which is particularly noticeable when the ball is released from a height above the ground.

For each of the examples mentioned you should be able to identify the 2 objects that are involved and whether the force is a push or a pull.

Can you do it?

A tiny comet is observed in orbit, in the Oort Cloud, perhaps a billion kilometres from an observer on Earth. What force is involved and what are the two objects?

Answer: the force is gravity and the two objects are the comet and our Sun.

The point is: there are always TWO objects involved whenever a force acts, even when the distance between them is so great that it is difficult to identify both objects; and forces are just pushes or pulls.

### 4.5.1.2 Contact and Non-Contact Forces

In the first two examples in the section above, the forces required a contact to be made between the two objects, so they were examples of contact forces; simple eh?

In the third and fourth examples above, the forces involved were magnetism and gravity which required no contact, so they were examples of non-contact forces; again, really simple.

Let’s define these clearly:

Contact Force- a force that acts by direct contact.

Non-contact Force- a force that acts without direct contact.

Contact Forces are the forces that we see happening all around us day by day.
Examples include kicking a ball, picking up a bag, pedalling a bike, pushing a skateboard. Sometimes the force is applied continuously, like pedalling the bike, but other times it is just a short impulse, like kicking the ball, but in each case contact is made between two objects.

Sometimes the force is considered to be so special that it is given its own name, for example if we push a book across a table we notice that the force called friction is involved. Friction is just the name we give to a particular contact force that occurs between two objects eg the book and the table, when they move against each other. Other named contact forces include Tension, Compression and the rather curiously named Normal Force! (more on these three later).

The only Non-Contact Forces are Magnetism, Gravity and Electrostatic Forces, so quite an exclusive club and therefore an easy list to learn, so make sure you do. More on Gravity shortly.

### Drawing Forces - Force Arrows and the unit for Force

Most of you will be familiar with the way we illustrate the action of a force.

A force is drawn using a Force Arrow.

But it is worth pointing out that a force arrow is a very particular thing! It is a straight line with an arrow head at ONE end.

Here are some typical CORRECT force arrows.

Just a straight line and one arrow head.

And here are examples of INCORRECT force arrows.

I am sure you can now see why they are incorrect; they are either not straight or they have two arrow heads!

So, the single arrow head tells us the DIRECTION of the force and the length of the straight line indicates the SIZE (also known as the MAGNITUDE) of the force.

So, when understood, force arrows are easy to draw and explain.

Now for a little check up.

1) which force arrow represents the smallest force to the left?

2) Which force arrow represents the largest force to the right?

The arrow which represents the smallest force to the left is 1; the arrow which represents the largest force to the right is 2.
Notice how the thickness of the arrow is irrelevant.

Most of the time when we draw a force arrow we will put a value beside it rather than just say it is a large or a small force. We can do this because we measure forces in a unit named after one of the most famous scientists of all time; the English Physicist Sir Isaac Newton.

So, the unit for force is the “newton" and the symbol for it is the capital letter N.

So, our humble little Force Arrow is really important and it conveys a lot of information.

Always take care when you draw them. OK.

Now that we have really grasped the importance of the Force Arrow we can advance to the exotically named Scalar and Vector Quantities – wow!

### 4.5.1.1 Scalar and Vector Quantities

A Quantity is something we can measure like time, mass, voltage, speed or force, amongst many more examples.

A Scalar Quantity is a basic quantity that reveals to us only ONE piece of information when we measure it.

In fact four of the five examples just mentioned are scalar quantities. Can you think which of the five are the scalar quantities?
Or, which one is not a scalar quantity – which one reveals TWO pieces of information when we measure it?

The first four are scalar quantities; when we measure them we get just one piece of information, a time, a mass, a voltage, a speed. But when we measure a force we get two pieces of information; we get its size and its direction.

So, if force is not a Scalar Quantity, then what type of quantity is it?

Force is a Vector Quantity; it reveals TWO pieces of information when we measure it.

OK, we can now define these terms carefully:
Scalar Quantity – this is a quantity that has size (or magnitude) ONLY. Examples include mass, speed, distance, time.

Vector Quantity – this is a quantity that has size (or magnitude) AND direction. Examples include force, velocity, weight, acceleration, displacement.

So FORCE is a Vector Quantitly because it conveys TWO pieces of information, a size and a direction.

Finally, in this short section: the force arrows that we have been drawing, that are used to describe forces, are often known as “Vector Arrows”. We will meet them again shortly. But now we need to look in detail at two common quantities; one is a Scalar and the other is a Vector. They are Mass and Weight. People are always getting them mixed up; but you must not!

### 4.5.1.3 Gravity

#### Mass and Weight

First of all, let’s agree that no matter whether you ever understand the difference between mass and weight we are NEVER going to talk about “stones”, “pounds” or “ounces” !!

Only people who are lost in time or who are literally in the “stone” age still use stones, pounds and ounces; apart, of course from Americans, who don’t use stones but do use pounds!

In the UK, for a long, long time we have used Kilograms for our measurement of…………(mass or weight?)

What are we measuring when we stand on our bathroom scales? Mass or weight?

When a person stands on a bathroom scale and announces to the world “my weight is 75kg”, they have got it..….. WRONG !

They should announce: “My mass is 75kg.”

Bathroom scales tell you your Mass, in Kilograms. That’s a fact. So, learn it.

They do not do not tell you your Weight.

OK, so we need to know what is really meant by Mass and by Weight.

Time for definitions again:

Mass is a measure of the amount of matter or stuff in an object, measured in Kg.

So Mass is purely a property of the object and it will NOT change if the object is moved from place to place in the universe (eg from Earth to the Moon). If your mass is 75kg on Earth, it will be 75kg on the Moon or if you were floating about on the International Space Station.

Weight is the force acting on an object due to gravity, measured in newtons (N).

So Weight depends on both the mass of the object and on the local gravitational field strength. It WILL change if the object is moved from place to place in the universe. We would all weigh less on the Moon, but we would all weigh a lot more on Jupiter (not that anyone plans to go there!).

Mass is easy, its just a measurement in Kg.

Weight, on the other hand, depends on two things.
The first is the mass of the object.

Not surprisingly, we find that the weight of an object is greater for objects with large masses; in fact weight and mass are directly proportional to each other. We often write this using the following shorthand:

weight ∝ mass

(The symbol in the middle means “proportional”, ∝ . You will meet it again in other areas of Physics or Maths.)

The second factor on which weight depends is the size of the local gravitational field strength which is all about how massive the planet is on which you are standing or are close to.

Most of the time we are on the surface of the Earth and the size of our local gravitational field strength is about 10 newtons of force on every 1kg of mass or to put it more neatly,

the gravitational field strength at the Earth’s surface is 10N/Kg.

The two quantities, mass and gravitational field strength, come together in the following formula which we use to calculate the weight of an object.

Notice that the unit for Weight is the newton.

This shouldn’t surprise you because we have said that weight is a Force, so never again shall you say that your weight is so many kilograms!

Hopefully you managed to do all of these calculations. You MUST learn the equation for weight if you are at KS4 and be ready to use it to answer examination questions.

Let’s move on now to consider how we measure weight?

#### Measuring Weight

To measure weight we make use of the fact that weight is due to the force of gravity and due to the mass of an object.

So, the simplest way of measuring weight is to dangle an object (a mass) from a spring and let gravity pull it downwards; the further it gets pulled downwards the greater we say is its weight.

We can calibrate such a spring against known “Weights”.
(making use of: weight = mass X g ).

The following apparatus can be used to do this.

When completed, we end up with a calibrated spring-balance or a newtonmeter and we can use it to tell us the weight of any object that we then dangle from the spring.
The weight would be given in newtons, of course.

We don't have to make our own newtonmeter; we can buy ready made ones, but they are really just the same, consisting of a spring and a pointer and a scale. Objects are dangled from the spring. These might look like:

#### Weight and the Centre of Mass

Ok, before we leave Mass and Weight, just one more thing.

If you were asked to balance a ruler on your finger tip, where on the ruler would you locate the balance point?

If you cut out a cardboard circle and tried to balance it on a pen tip, where on the cardboard circle would you place the pen tip?

When one ballet dancer holds up another on one hand such that she balances, where is the hand placed?

OK, that’s enough of the questions!

The answer to each question is the same.

The balance point in each of the examples is at the point on the object that is known as the “centre of mass” of the object.

The Centre of Mass is the point at which all of the mass or weight of the object seems to act. (this is a good definition of C.O.M, centre of mass).

For regular shaped objects like the ruler and the cardboard circle, the C.O.M is at their symmetrical centre; so its easy to find their balance point and hence, their Centre of Mass. Just find their precise symmetrical centre; see the illustrations below.

M is the point of the Centre of Mass.

For humans the balance point, and the C.O.M, is just above the belt line, which is why the male ballet dancer is always seen to position his hand close to the female dancer’s waist region.

A more systematic way to find the C.O.M of an irregular shaped object is to do the following:

1. Let’s assume you have an irregular shaped piece of cardboard; first suspend it from a suitable pin so that it will pivot freely.
2. Hang a plumb-line (which is just a mass on the end of a piece of string) from the same pin; allow both the card and the plumb-line to come to rest.
3. Draw a line on the card behind the plumb-line.
4. Repeat all the above with the card suspended from another hole. The diagram below shows the irregular object suspended from two different holes.
5. You will now have 2 lines which should cross on your card; the point at which they cross will be the C.O.M of the card!
6. Test by balancing the card on your finger tip (or the tip of a pencil) at the point where the 2 lines cross.

Why does this method work?

The C.O.M will always lie below any point of suspension. So, with the piece of card, the C.O.M is somewhere along each line below the pin. So, it has to be precisely at the point where the two lines cross. This is why you have to draw 2 lines.

#### Isaac Newton’s Universal Law of Gravity

In about 1660 Sir Isaac wrote his famous law of Gravity and this is what it stated;

“Every object in the universe pulls every other object with a gravitational force which increases as their masses increases and decreases as their separation increases.”

To help to understand this force, Newton spoke of a “field of gravity” around each mass and whenever any other mass entered this field it experienced a force.

Now, we have already been using the idea of a “gravitational field” because our formula for Weight involved the Gravitational Field Strength; do you remember?

We could think of this “field of gravity” as a “force field” around the object. So, every object, including us, has a force field around it! The force field is very weak for objects with a small mass, like us, but it is large for objects with a large mass, like a star such as our Sun or a planet.

For example, there is a significant force due to gravity between the Sun and the Earth.

Notice, Newton's Law tells us that whenever two objects interact, BOTH feel the same pull due to gravity, so in this diagram the force arrow lengths are the same. Also notice that in each case the force acts from the C.O.M, which is at the centre of each sphere.

Between the Sun and Venus there is also a force due to gravity, but because Venus is closer to the Sun than the Earth the force is LARGER, in accordance with Newton’s Law – as the distance between the objects decreases, the force increases.

Venus is similar in mass to the Earth but is closer to the Sun so the force of gravity is greater, so the force arrows here are longer, but still a pair of force arrows.

Newton's idea of a "gravitational field" or a "force field" around an object like the Earth helps to explain the behaviour of other objects which come close to it. For example, as meteors or small asteroids pass close to the Earth, they enter its gravitational field and get attracted towards the Earth which might cause them to veer off their origninal path and to crash through the Earth’s atmosphere where they burn up and we see them as “shooting stars”.

Even smaller objects like artificial satellites (eg those used for GPS systems or satellite TV) and space stations such as the International Space Station, are kept in orbit around the Earth by the force of gravity, though the size of the force is much smaller than between, say, the Sun and any of the planets.

Here the force of gravity between the Earth and a satellite is tiny compared to other astronomical objects, so the force arrows are drawn shorter. NB Their thickness is unimportant.

Let’s also mention the most common example of Newton’s Law Of Gravity, the force of gravity between each one of us and the Earth; the force which we tend to call our “weight” which we have already learnt a lot about.

The force of gravity between the Earth and a person is very small compared to all the others mentioned above, so the force arrows are the smallest, but still notice that the Earth is pulled just as much by you as you are pulled by it! Hence the two arrows, always.

Why don’t we notice this?

The reason is that the Earth’s mass is so unbelievably huge compared to our mass that it barely moves in the interaction between it and us; so, although we get noticeably pulled towards the Earth, the Earth does not get noticeably pulled towards us!

### 4.5.1.4 Resultant Forces

The term “resultant force” is only relevant in a situation where two or more forces are involved, acting on one object

So, in the first example below, you don't really need to use the term "Resultant force" since there is only one force present. It is enough to say "there is a force of 5N pulling the object to the right."

But in the next diagram there are two forces acting on the one object, so we can now bring in the term “resultant force”,

...but only if we know what we are talking about!

So, what does the term mean?

Time for another definition:

A resultant force is a single force that has the same effect as all the original forces acting together.

To work out this “same effect” is quite simple.

For example in the 2nd diagram, the two forces each pull the object to the right, one with a force of 5N and the other with a force of 3N, so the Resultant Force is simply the “result” of adding these two values;

so the resultant force is 8N to the right. In other words: 5 + 3 = 8

The 3rd diagram shows the single “Resultant force which has the “same effect” as the original two forces.

Easy! But what if the forces acting on the one object act in different directions as shown in the 4th diagram?

If an object is pulled to the right by a force of 6N and to the left by a force of 10N, then the single force that has the “same effect” is a force of 4N to the left. See the 5th diagram.

The Resultant Force, which has the same effect as the original two forces, is a force of 4N acting to the left.

In other words: 10 - 6 = 4

We subtract the 6N force from the 10N because it acts in the opposite direction. (In the previous example, when the forces acted in the same direction, we simply added them.)

Notice in question 4 above that the resultant force became zero when the new member joined Jim’s team; the two teams are still pulling but their forces are now cancelling each other out. If the two teams keep pulling with these forces then they will remain fixed in position forever!

When the resultant force acting on an object is found to be zero, we say that the forces acting on the object are Balanced and that the object is in a state of Equilibrium.

The word “Equilibrium” is a good choice here because it means that everything concerning the motion of the object stays “the same” or “unchanged”.

So if the object WAS :
a) stationary before the pair of balanced forces act on it, then it remains stationary (because the forces have effectively cancelled out and disappeared!),

Here is an illustration:

Here, before any forces are applied, an object is stationary.

Now, forces are applied, but since they are BALANCED, the object remains in its state of EQUILIBRIUM and its motion does NOT change. It remains stationary.

And if the object WAS :
b) already moving before the pair of balanced forces act on it, then it just carries on moving at its previous speed and direction, totally unchanged. (Again, because the pair of forces have effectively cancelled out and disappeared!)

Here is an illustration:

This car is currently moving at a constant speed; it is in a state of EQUILIBRIUM.

Now, the engine produces an extra 7N of forward force, but there is also an additional 7N of air resistance. The resultant of these two forces is zero, so these forces BALANCE and the car remains in EQUILIBRIUM. Its motion does NOT change. It ontinues at its constant speed of 2m/s.

So, just to repeat - if the resultant force on an object is found to be ZERO, then the object is in a state of EQUILIBRIUM and its motion does NOT change. So whatever it WAS doing, it carries on doing.

Have you got it?

Also, a great consequence of this idea of "Equilibrium" is that if you are told that a certain object is stationary, OR is moving at a constant speed, then you can declare with great confidence that the it is in a state of equilibrium and so no matter how many forces are acting on it, they are BALANCED and have a Resultant Force of .....ZERO.

For example: A child slides down a slide due to a downward force of 200N. He slides at a constant speed. What can you say about any other forces acting on the child?

Well, you can say that there is an UPWARD force acting on the child and that it is BALANCING out the downward force, because the child is moving at constant speed (or is in a state of EQUILIBRIUM), and that its size is exactly 200N. Easy!

Notice - you can say all of this only because the child (the object) is in a state of Equilibrium, moving at constant speed; the condition for Equilibrium is to remain at Constant Speed or to be stationary.

#### Resultant Forces And Free Body Diagrams

Luckily for most KS4 students it will NEVER be necessary to have to draw fancy force diagrams such as these:

All you NEED to do is to draw basic rectangle shapes with arrows to represent forces, as we did above.
When we do this, we are drawing "free body diagrams".

Free body diagrams are just a simple way of illustrating the forces that act upon an object. The only condition is that the arrows used are drawn so that their lengths represent the sizes of the forces and their directions represent the directions of the forces involved.

The car diagram above could have been drawn as a "free body diagram" by simply replacing the car shape for a basic rectangle.

OK, since this little section is on "free body diagrams" we have to draw at least one, but as stated, we have already been drawing quite a lot of them, so we need to choose one that is a bit different from those we have done; how about a diagram involving vertical forces rather than horizontal forces? We haven't done one like that yet!

Let's draw a free body diagram for a sky diver falling in "free-fall" (ie falling at constant speed).

Actually, I'm not going to draw this one, you are! I will give you instructions:

1. Draw a simple rectangle and write beside it "sky-diver in free fall".
2. Draw one force arrow from the centre of the rectangle pointing downwards below the rectangle. Label this Weight.
3. Draw one force arrow from the centre of the rectange pointing upwards above the rectangle, but make it exactly the same length as the first arrow. Label this Air Resistance.

That's it! You have drawn a free body diagram for a sky diver in free-fall. Simple, isn't it.

But you should have practice drawing more free body diagrams, so have a go at the following.

#### Finding the Resultant Force when the forces are at strange angles!

OK, back to our Force Arrows.

First, a reminder of something we said earlier: the Force Arrows that are being used to illustrate the forces in the above simple diagrams are also known as Vector Arrows since Forces are Vector Quantities and so they have size (or magnitude) and direction; the size of the force is indicated by the length of the arrow and the direction of the force is indicated by the direction of the arrow.

In all of the above examples the forces happen to be acting along the same straight line, but what if the forces are not acting in such a simple way. For example, what if one force is acting at 45 degrees to another force as shown in the diagram below; how then would you be able to work out the Resultant Force or the Resultant Vector?

So, how do you find the Resultant of these two vectors (or forces) ?

It is no longer so easy... or at least it doesn't look so easy, but I hope to show you that it isn't too difficult.

This is what you do:

You “pick up” one of the force arrows (or vector arrows) and whilst maintaining its angle you move it to the arrow end of the other one. See the next diagram

Finally, the Resultant is a vector that goes from the start of these two vectors to the end of these two vectors. See the next diagram.

The arrow in black is the Resultant Force; the single vector that replaces the other two, now shown in grey.

So now we can see the effect that the two forces will have on the object; it will feel a force mostly to the right but also a little bit upwards, so it is likely to move to the right and slightly upwards.

Notice; it doesn’t matter which of the original two vectors you “pick up” and move; as long as you move it correctly, keeping its angle fixed and move it to the end of the other one, then you will get the same answer; try it with this example.

Finally, when we add vectors like this it is known as "Vector Addition" and until very recently it was not taught until at least AS Level ! So, you are doing very well if you have grasped it.

#### Resolving Vectors

If two Vectors can be added together to make one Resultant Vector, then one vector can be “split apart” into two vectors!

In other words, "resolving vectors" is just the reverse of what we have just been doing; it is taking one vector and making it into two.

But "why?" you would ask, would anybody want to take one vector and make it into two?

Well, if you were presented with a vector like the one we ended up with at the end of our "red rectangle" example all you can really say is that the red rectangle will feel a force that is a certain amount to the right and another amount upwards. What "resolving" this vector would give you would be:
1. One vector that is perfectly vertical, so you would know exactly the size of the vertical vector. We call this, the vertical COMPONENT.
2. One vector that is perfectly horizontal, so you would know exactly the size of the horizontal vector. We call this, the horizontal COMPONENT.

For example: Imagine that a cannonball is fired from a cannon at an angle of 30 degrees and it exits the cannon with a certain velocity vector. How can you work out how high it will reach and how far it will travel before hitting the ground? Well, you can't get any of the answers until you find its VERTICAL velocity COMPONENT and its HORIZONTAL velocity COMPONENT.

Once you have "Resolved" the single vector into its two COMPONENTS then you can use them seperately to answer the two questions.

Now, don't panic, nobody is going to ask you such hard questions (until you get to A level), but you need to know, for KS4, how to "resolve a vector into its two components". And its not difficult! So let's look at how we do it.

#### How to resolve a vector (eg a force) into its two components

Consider a single vector acting on an object at some angle to the horizontal. There's a perfect illustration of such an object below.

Your task is to "resolve" this single vector into its 2 components.

The first thing we do is to draw a rectangle (use dashed lines) such that the single vector is the diagonal of the rectangle; see the next diagram.

Finally, the two components are the horizontal and vertical sides of this rectangle, as shown on the third diagram.

The two components are simply the vertical and horizontal sides of the rectangle.

That didn't take long, did it? By the way, we could call this method of solution, "the rectangle method" if that helps you to remember it.

Hopefully you can see that it all makes sense; the grey arrow is effectively the addition of the two components, hence the two black arrows are the "components" of the grey.
In other words, resolving vectors is simply the reverse of the addition of vectors (which is what we were doing in the previous section).

Before we leave this, it is worth noting that the 3rd diagram can also be drawn as shown in the next diagram.

Take a moment to agree that it is the same; the vertical arrow is still the same vector because it has the same length and the same direction whether you draw it on the left or right side of the rectangle. Its your choice.

#### Resolving Vectors Questions

Now its your turn to have a go at "resolving vectors".

Copy out the 6 vectors shown below, keeping their angles as they are shown on the screen, to the best of your ability. Then by using the rectangle method, find the horizontal and vertical components for each of the six vectors.

When you are ready (but not before!), press the button to reveal the answers. The answer diagram will appear below the question diagram.

### The Effect of A Resultant Force

(A brief excursion into section 4.5.6 Forces and motion.)

A resultant force ALWAYS causes some change.

We saw earlier that when the Resultant Force on an object is zero, the object is in a state of Equilibrium, which basically means that the forces acting on it are cancelling each other out and so causing no change to the motion of the object. We called such forces, Balanced Forces.

But, when the Resultant Force is NOT zero:

the object will NOT be in Equilibrium.
So, change to its motion WILL occur.

But what sort of change?

The object might start to move IF it had been stationary.
The object might stop IF it had been moving.
The object might speed up.
The object might slow down.
The object might change direction.
The object might start to rotate.
These are all Accelerations!

The object might also be stretched, squashed or twisted, and later we will have to look at some of these in more detail. In other words, all kinds of changes are possible.

The essential point is :

#### A Resultant Force always brings about some change to the motion of an object or to its shape.

OK. I wonder are you thinking – is there a link between the AMOUNT of change and the size and direction of the resultant force?

What a good question! What a perceptive student you are.

There is a strong proportional link. (Do you remember this word from earlier?)

The amount of change is proportional to the size of the resultant force and it acts in the direction of the resultant force.

Anyway, we are going to leave this part of Forces for now; we will return to it later on when we delve into the world of Newton’s Laws of Motion in the AQA KS4 section called 4.5.6 Forces and motion.
I recommend that you go to that section now, but if you follow the order in the AQA section listing then the next section is 4.5.2 Work done and Energy transfer.