1. Teach A Level Maths
Force Diagrams
and
Newton’s 1st Law

2. “Teach A Level Maths” Yr1/AS Mechanics
Force Diagrams
and
Newton’s 1st Law
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3. We have already seen how to find the resultant of a set of forces acting at a point and how to deal with forces in equilibrium.
In this presentation we look at the nature of the forces and how to identify them in simple practical situations.

4. Weight is just one of a set of forces that may act on an object.
In every day language mass and weight are confused.
Mass is a scalar quantity and is measured in kilograms.
Weight has both magnitude and direction so it is a vector. The unit of weight is a newton.
Weight is just one of a set of forces that may act on an object.
In mechanics we usually call an object a “body”.

5. The principal forces we meet in 1st year Mechanics are:
The weight of a body.
The tension in a string.
The contact force between 2 surfaces. This force is made up of 2 components:
the frictional component, along the surface,
and the normal reaction, perpendicular to the surface.
We need a different letter to label unknown values of each force and here we have to make a decision.

6. ( The r must be written as a subscript because Fr looks like F  r. )
Weight
We will use W
Tension
We will use T
The component of friction gives us a problem.
An important formula in mechanics is F = ma.
In this formula, F stands for the resultant of the forces acting on a body.
Using F for friction and for the total force leads to terrible confusion.
I will always use Fr for the frictional component.
( The r must be written as a subscript because Fr looks like F  r. )

7. Friction
We will use Fr
For the Normal Reaction N or R are sensible letters.
Most books use R for one normal reaction and N for a 2nd one. I’ll do the same.
Normal Reaction
We will use N or R
Some questions also contain a pulling or pushing force and if it isn’t in a string we often use the letter P.
Occasionally a force that is described as a resisting force may be a mixture of air resistance and friction. We can choose any letter that seems suitable.

8. As well as letters to label unknown forces, we need lines with arrows to show the directions of the forces. Fr
e.g.
The length of the lines in these diagrams isn’t proportional to the size of the forces.
But we also need lines, arrows and letters to show velocity and acceleration.
I will use the following: v a
Velocity
Acceleration
We must use different arrows for the three quantities.

9. In the 1600’s, Newton wrote in his 1st law of motion that . . .
A body stays at rest, or continues to move with constant velocity, unless a resultant force acts on it.
(I’ve simplified Newton’s words)
So, if a parcel resting on a table has no resultant force acting on it, it stays where it is.
But, if you are in a moving car which brakes suddenly, you will continue at the same speed ( although the car doesn’t ). . .
unless you are wearing a seat belt ( to provide the resultant force to
slow you down ).

10. However, having no resultant force doesn’t mean there are no forces.
Can you and your partner decide what forces are acting on a parcel lying on the table ?
Ans: The weight of the parcel acting downwards . . . R
table
The weight of a body acts on itself and that’s why, if the table were not there, the parcel would fall to the floor.
The normal reaction from the table acting upwards. W
We will model the parcel as a particle so the forces all act at one point.

11. Fr always acts against the motion or the tendency to move.
The parcel is now a particle but we still need to be able to see it ! Fr T
table
If we attach a string to the parcel . . .
and try to pull it along the table, the force diagram changes.
We have a tension in the string . . .
The parcel now has a “tendency to move”, so the contact force has a frictional component . . .
Fr always acts against the motion or the tendency to move.

12. R Fr T W If we attach a string to the parcel . . .
table T W R Fr
If we attach a string to the parcel . . .
and try to pull it along the table, the force diagram changes.
We have a tension in the string . . .
The parcel now has a “tendency to move”, so the contact force has a frictional component . . .
We could replace R and Fr by the total contact force but we don’t know its exact direction.
There would also be air resistance but the models at this stage either ignore this or include it with the frictional component as a “resisting force”.

13. R Fr T W Do you think the parcel starts to move or stays at rest ?
table T W R Fr
Do you think the parcel starts to move or stays at rest ?
Newton’s 1st law says that it depends on whether there is a resultant force.
If there is no resultant force, the forces balance and the parcel stays where it is.
If there is no resultant force, we say the body is in equilibrium.

14. Fr is the component of the contact force trying to prevent motion.
table T W R Fr
Fr is the component of the contact force trying to prevent motion.
As we pull harder and increase the size of T,
Fr gets larger to try to prevent the parcel moving.
At some stage, depending on the slipperiness of the surfaces, the frictional force cannot get any larger and the parcel starts to slide.
(As it is clumsy to say frictional component all the time, we usually just say frictional force.)

15. In ice hockey, at one point, a puck is sliding across the ice with speed v.
Decide with your partner what forces are acting on the puck. Fr W
Ans: We still have W, R and Fr .
The force from the stick that started the puck moving only acted when the puck was hit. That force has now disappeared.

16. In ice hockey, at one point, a puck is sliding across the ice with speed v.
Decide with your partner what forces are acting on the puck. Fr W
Ans: We still have W, R and Fr .
Can you see from the diagram that there is definitely a resultant as there is no force to balance Fr ?
Newton’s 1st law tells us that the puck will not continue to move with constant velocity. The frictional force will slow it down.

17. Now, tell your partner what happens if we assume the frictional force is so small that we can ignore it.
Ans: Newton’s 1st law tells us that the puck will continue to move with constant velocity ( that is, constant speed in a straight line ).
ice v R W

18. SUMMARY
Force is a vector.
Weight is a force. Mass is not.
Arrow used in diagrams for force:
Notation for forces:
Weight: W
Tension: T
Components of the contact force:
Normal Reaction: N or R
Friction: Fr
Newton’s 1st Law: A body stays at rest, or continues to move with constant velocity, unless a resultant force acts on it.

19. You will meet some words in mechanics which give you information about the model you need to use. Here are some of the words and the meaning you need to attach to them.
“particle”
the forces act at a point
“smooth”
friction should be ignored
“rough”
friction must be included
“inextensible”
the string or cord or rope doesn’t stretch
“light”
the weight should be ignored.

20. When we solve problems involving forces acting on a body we always draw a diagram showing the forces.
e.g.1 Show the forces acting on a particle attached to a light inextensible string hanging freely from a fixed peg.
peg
particle
The weight is always present and always acts vertically down. T
There is a tension in the string.
The arrows need to be clear and not too close to the particle. However, they mustn’t be too far away either.
Tension always acts along a string, away from the body on which it acts. W
The tension acting on the peg is downwards so it would be wrong if the arrow for the tension acting on the particle was more than halfway up the string.

21. When we solve problems involving forces acting on a body we always draw a diagram showing the forces.
e.g.1 Show the forces acting on a particle attached to a light inextensible string hanging freely from a fixed peg.
peg
particle
The weight is always present and always acts vertically down. T
There is a tension in the string.
The particle is not in contact with a surface so there is no contact force. W

22. e.g.2 Ignoring air resistance, show the forces acting on the body which is sliding with constant velocity down a rough slope.
We start with the weight.
There is no string so no tension.
The only other force is the contact force ( or its components ). W
30

23. e.g.2 Ignoring air resistance, show the forces acting on the body which is sliding with constant velocity down a rough slope.
We start with the weight. C
Tell your partner what Newton tells us about the forces in this question. W
30
Ans: There is no resultant.
So, the total contact force balances the weight.
We usually solve problems using the components of C but we don’t need to study that until next year.

24. Here are some common errors found in force diagrams.
Diagrams are too small to clearly see forces, velocities, accelerations, lengths and angles.
Some forces are missed out.
A force and its components are shown.
Friction is put in the wrong direction or missed out.
Mass is shown instead of weight.

26. The summary page follows in a form suitable for photocopying.

27. FORCE DIAGRAMS AND NEWTON’S 1ST LAW
Summary
FORCE DIAGRAMS AND NEWTON’S 1ST LAW
TEACH A LEVEL MATHS – MECHANICS 1
Force is a vector.
Arrow used in diagrams for force:
Notation for forces:
Weight: W
Tension: T
Normal Reaction: N or R
Friction: Fr
Components of the contact force:
Newton’s 1st Law: A body stays at rest, or continues to move with constant velocity, unless a resultant force acts on it.
Weight is a force. Mass is not.
CONTINUED

28. FORCE DIAGRAMS AND NEWTON’S 1ST LAW
Summary continued:
FORCE DIAGRAMS AND NEWTON’S 1ST LAW
TEACH A LEVEL MATHS – MECHANICS 1
Meanings of words used in modelling
“particle”
the forces act at a point
“smooth”
friction should be ignored
“inextensible”
the string or cord or rope doesn’t stretch
“light”
the weight should be ignored.
“rough”
friction must be included
Some common errors found in force diagrams:
Diagrams are too small to clearly see forces, velocities, accelerations, lengths and angles.
Some forces are missed out.
A force and its components are shown.
Friction is put in the wrong direction or missed out.
Mass is shown instead of weight.