1. for IJSO training course
Pushes and Pulls
for IJSO training course

2. Content What are forces? Measurement of a force
Daily life examples of forces
Useful mathematics: Vectors
Newton’s laws of motion
Free body diagram
Mass, weight and gravity
Density vs. mass
Turning effect of a force

3. A force is the cause of velocity change or deformation.
1. What are forces?
Force, simply put, is a push or pull that an object exerts on another.
We cannot see the force itself but we can observe what it can do:
It can produce a change in the motion of a body. The body may change in speed or direction.
It can change the shape of an object.
A force is the cause of velocity change or deformation.

4. The SI unit of force: N (Newton)
2. Measurement of a force
Force is measured in units called Newton (N). We can measure a force using a spring balance (彈簧秤).
The SI unit of force: N (Newton)
(Wikimedia commons)

5. Many materials including springs extend evenly when stretched by forces, provided that the force is not too large. This is known as Hooke’s law (虎克定律).
A spring balance uses the extension of a spring to measure force. The extension is proportional to the force acting on it as shown below.

6. 3. Daily examples of force
Weight
The weight of an object is the gravitational force acting on it.
Weight

7. Normal: perpendicular to the table surface.
Normal force
A book put on a table does not fall because its weight is balanced by another force, the normal force, from the table.
Normal: perpendicular to the table surface.
normal force
normal force
force by the hand
weight

8. Draw “face-to-face” arrows to represent tension
Tension (張力) in a stretched string tends to shorten it back to the original length.
Once the string breaks or loosens, the tension disappears immediately.
Since tension acts inward to shorten the string, we usually draw two “face-to-face” arrows to represent it.
Draw “face-to-face” arrows to represent tension

9. Example
These two forces counterbalance each other (suppose the weight of the hook is negligible).
“face to face” arrows representing tension
tension 10 N
tension 10 N
The tension balances the weight, therefore the mass does not fall down.
1-kg mass
weight 10 N

10. It always acts in a direction opposite to the motion.
Friction
Friction (摩擦力) arises whenever an object slides or tends to slide over another object.
It always acts in a direction opposite to the motion.
Cause: No surface is perfectly smooth. When two surfaces are in contact, the tiny bumps catch each other.
motion
friction
Friction drags motion.

11. Friction can be useful
We are not able to walk on a road without friction, which pushes us forward.
In rock-climbing, people need to wear shoes with studs. The studs can be firmly pressed against rock to increase the friction so that the climber will not slide easily.
backward push of foot on road
forward push of road on foot

12. The tread patterns on tyres also prevents the car from slipping on slippery roads. Moreover, road surfaces are rough so as to prevents slipping of tyres.
Tread pattern on a car tyre
(Wikimedia commons)
Tread pattern on a mountain bicycle tyre
(Wikimedia commons)

13. Disadvantages of friction
There are some disadvantages of friction. For example, in the movable parts of machines, energy is wasted as sound and heat to overcome friction. Friction will also cause the wear in gears.
Friction can be reduced by the following ways.
bearings
(Wikimedia commons)

14. using lubricating oil using air cushion streamlining
The streamlined shape cuts down the air-friction on the racing car.
1. Propellers 2. Air 3. Fan 4. Flexible skirt
BHC SR-N4 The world's largest car and passenger carrying hovercraft
(All pictures are from Wikimedia commons)

15. 4. Useful mathematics: Vectors
A scalar (標量) is a quantity that can be completely described by a magnitude (size).
Examples: distance, speed, mass, time, volume, temperature, charge, density, energy.
It is not sensible to talk about the direction of a scalar: the temperature is 30oC to the east(?).
A vector (向量) is a quantity that needs both magnitude and direction to describe it.
Examples: displacement, velocity, acceleration, force.
A vector has a direction.

16. Example: displacement
A mouse moves 4 cm northward and then 3 cm eastward.
What is the distance travelled?
Answer = 4 cm + 3 cm = 7 cm
What is the displacement of the mouse?
Answer = 5 cm towards N36.9oE
3 cm
4 cm
5 cm
How to find the angle?

17. What is the velocity of the bird?
Example: velocity
A bird is flying 4 m/s northward. There suddenly appears a wind of 3 m/s blowing towards the east.
What is the velocity of the bird?
Answer = 5 m/s towards N36.9oE
What is the speed of the bird?
Answer = 5 m/s
Note 1: No need to specify the direction.
Note 2: the answer is not simply = 3 m/s + 4 m/s = 7 m/s
3 m/s
4 m/s
5 m/s

18. A magnitude does not have a direction.
Example: force
You push a cart with 4 N towards north. Your friend helps but he pushes it with 3 N towards the east.
What is the resultant force?
Answer = 5 N towards N36.9oE
What is the magnitude of the force?
Answer = 5 N
Note: A magnitude does not have a direction.
3 N
4 N
5 N
A magnitude does not have a direction.

19. Addition and resolution
Two usual ways to denote a vector
Boldface
Adding an arrow
Vectors can be added by using the tip-to-tail or the parallelogram method.
If vectors a and b add up to become c, we can write c = a + b. F F a b c
Tip-to-tail method a b c
Parallelogram method

20. Two vectors can add up to form a single vector, a vector can also be resolved into two vectors.
In physics, we usually resolved a vector into two perpendicular components.
Below, a force F is resolved into two components, Fx and Fy.

21. 5. Newton’s laws of motion
Isaac Newton developed three laws of motion, which give accurate description on the motion of cars, aircraft, planet, etc.
The laws are important but simple. They are just the answers to three simple questions.
Consider a cue and a ball.

22. Newton’s 3 laws of motion answer 3 questions:
If the cue does not hit the ball, what will happen to the ball?
Newton’s first law
If the cue hits the ball, what will happen to the ball?
Newton’s second law
If the cue hits the ball, what will happen to the cue?
Newton’s third law

23. The first law
Also called “The law of inertia” (慣性定律)
A body continues in a state of rest or uniform motion in a straight line unless acted upon by some net force.
Galileo discovered this.
If the cue does not hit the ball, the ball will remain at rest.

24. In the form of equation, the second law can be written as F = ma
The acceleration of an object is directly proportional to, and in the same direction as, the unbalanced force acting on it, and inversely proportional to the mass of the object.
In the form of equation, the second law can be written as F = ma
F is the acting force
m is the mass of the object
a is the acceleration (a vector) of the object
If the cue hits the ball, the ball will accelerate.
Second law: F = ma

25. But .. what is acceleration?
Consider an object moving from A to B in 2 hours with a uniform velocity. What is the velocity?
B (1 km, 3 km)
A (3 km, 1 km)
Final displacement from O = OB O E
Initial displacement from O = OA
Change in displacement = OB – OA = AB
Change in displacement AB
Velocity = =
Time required
2 hours

26. (Note: speed is also a scalar.)
AB =
(Note: This AB does not have an arrow. It indicates a length, which is a scalar.) N
B (1 km, 3 km)
Speed = AB / 7200 s = 0.39 m/s
(Note: speed is also a scalar.)
A (3 km, 1 km) O E
Velocity = 0.39 m/s towards NW.
Change in displacement
Velocity =
Time required

27. Calculate the acceleration.
Consider a bird. At time t = 0 s, it was moving 5 m/s towards SE. Its velocity gradually changed such that at t = 2 s, its velocity became 5 m/s towards NE.
Calculate the acceleration. N v2
vc = v2 - v1 E v1
Change in velocity = vc
Change in velocity vc
Acceleration = =
Time required
2 s

28. Acceleration = 3.54 m/s2 towards N.
vc =
(Note: This vc does not have an arrow. It indicates a magnitude.) v2
vc = v2 - v1
Magnitdue of acceleration
= vc / 2 s = 3.54 m/s2 E v1
Acceleration = 3.54 m/s2 towards N.
Change in velocity
Acceleration =
Time required

29. Equations of motion in 1D
In the 1D, there are only two directions, left and right, up and down, back and forth, etc.
For these simple cases, once we have chosen a positive direction, we can use + and - signs to indicate direction. We can also use a symbol without boldface to denote a vector.
Example: If we choose downward positive, the velocity v = -5 m/s describes an upward motion of speed 5 m/s.

30. Uniform acceleration Let We can prove that
t = the time for which the body accelerates
a = acceleration (which is assumed constant)
u = the velocity at time t = 0, the initial velocity
v = the velocity after time t, the final velocity
s = the displacement travelled in time t
We can prove that

31. v s u t t Velocity-time graph Displacement-time graph slope = a
parabola u t t

32. Back … to the second law: F = ma
Mass is a measure of the inertia, the tendency of an object to maintain its state of motion. The SI unit of mass is kg (kilogram).
1 Newton (N) is defined as the net force that gives an acceleration of 1 m/s2 to a mass of 1 kg.
The same formula can be applied to the weight of a body of mass m such that W = mg.
W: the weight of the body. It is a force, in units of N.
g: gravitational acceleration = 9.8 m/s2 downward, irrespective of m.
W = mg

33. Force of man accelerates the cart.
The same force accelerates two carts half as much.
Twice as much force produces acceleration twice as much.

34. For every action, there is an equal and opposite reaction.
The third law
For every action, there is an equal and opposite reaction.
When the cue hits the ball, the ball also “hits” the cue.
Action: the man pushes on the wall.
Reaction: the wall pushes on the man.
Action: Earth pulls on the falling man.
Reaction: The man pulls on Earth.

35. Normal force = mg (upward)
Example
The block does not fall because its weight is balanced by a normal force from the table surface.
Are the weight and the normal force an action-and-reaction pair of force as described by Newton’s third law?
Answer: No!
Normal force = mg (upward)
Weight = mg (downward)

36. Explanation
Action and reaction act on different bodies. They cannot cancel each other.
The “partner” of the weight is the gravitational attraction of the block on the Earth.
Weight = mg (downward)
Gravitational attraction of the block on the Earth = mg (upward)

37. Normal force = mg (upward)
Explanation
The “partner” of the normal force acting on the block by the table surface is the force acting on the table by the block surface.
Both have the same magnitude mg.
But they do not cancel each other because they are acting on different bodies.
Normal force = mg (upward)
The force acting on the table by the block = mg (downward)

38. 6. Free body diagram
To study the motion of a single object in a system of several bodies, one must isolate the object and draw a simple diagram to indicate all the external forces acting on it. This diagram is called a free body diagram. N
Example
For an object of mass m at rest on a table surface, there are two external forces acting on it:
1. Its weight W
2. Normal force from the table surface N.
Obviously, W = -N, and W = N = mg. W

39.  Consider two blocks, A and B, on a smooth surface. Find
Worked Example 1
Consider two blocks, A and B, on a smooth surface.
Find
(a) the pushing force on Block B by Block A.
(b) the acceleration of the blocks. 
Block A
3 kg
Block B
2 kg
10 N

40. Take rightward positive. Let a be the acceleration of the blocks.
Solution: Method 1
Take rightward positive.
Let a be the acceleration of the blocks.
Let f be the pushing force on Block B by Block A.
Consider the free body diagram of Block A
normal force from the table surface a
3 kg
10 N
f (reaction force of the pushing force on Block B)
weight

41. a
3 kg
10 N f
Vertical direction: No motion. The weight and the normal force from the table balance each other.
Horizontal direction: Applying Newton’s second law (F = ma), we have (with units neglected)
10 - f = 3a
(1)

42. Then consider the free body diagram of Block B
normal force from the table surface a
2 kg f
weight
We ignore the vertical direction because the forces are balanced. Consider the horizontal direction. Applying the second law again, we have
f = 2a
(2)

43. We now have 2 equations in 2 unknowns.
10 - f = 3a
(1)
f = 2a
(2)
Solving them, we have
f = 4 N
a = 2 m/s2
(a) The pushing force on Block B by Block A = 4 N towards the right.
(b) The acceleration of the blocks = 2 m/s2.

44. Method 1 is a long method, below is a shorter one.
Solution: Method 2
Method 1 is a long method, below is a shorter one.
The whole system is a mass of 5 kg.
We take rightward positive and define the same f and a as those in Method 1.
Applying the second law (F = ma), we have 10 = 5a, hence a = 2 m/s2.
Consider only Block B. The only force acting on it is f. Hence f = 2a = 4 N.

45. Worked Example 2
Consider a pulley and two balls, A and B. For convenience, take g = 10 m/s2.
Find
(a) the acceleration of Ball A.
(b) the tension in the string.
A: 4 kg
B: 1 kg

46. Solution
Take downward positive.
Let tension = T and acceleration of Ball A = a.
Consider the free body diagram of Ball A: T
A: 4 kg a
Weight = 4g
We can apply F = ma and get
4g - T = 4a
(1)

47. Consider the free body diagram of Ball B:
B: 1 kg a
Weight = g
We apply F = ma and get
g - T = -a
(2)
Solving Equations (1) and (2), we get a = 6 m/s2 and T = 16 N.
(a) The acceleration of Ball A = 6 m/s2 downward.
(b) The tension in the string = 16 N.

48. Consider a block on an inclined plane.
Worked Example 3
Consider a block on an inclined plane.
Label all forces acting on the block and resolve them into components parallel and perpendicular to the plane.

49. Find the acceleration a of the block in terms of g, given that
Solution
Consider the motion perpendicular to the motion. The forces are balanced, therefore we have

50. Now, consider the motion parallel to the motion
Now, consider the motion parallel to the motion. Applying Newton’s second law F = ma, we have

51. 7. Mass, weight and gravity
In everyday life, people often confuse mass with weight.
A piece of meat does not weigh 500 g, but its mass is 500 g and it weighs about 5 N on the Earth.
(Wikimedia commons)

52. g varies slightly with positions on the Earth.
The mass of an object is a measure of its inertia. It is always the same wherever the object is.
On the other hand, the weight W of an object is the pull of the gravity acting on it. It depends on its mass m and the gravitational acceleration g.
W = mg
g varies slightly with positions on the Earth.
g is different on different celestial objects:
Earth
Moon
Venus
Jupiter
m/s2
1.622 m/s2
8.87 m/s2
24.79 m/s2

53. Weightlessness
When a girl stands inside a lift, she cannot feel her own weight. What she feels is the normal force R acting on her by the lift floor.
The scale reading shows the magnitude of the reaction force to R, that is, the force acting on the scale by her feet.
scale reading = R
weight (W)
reaction (R)

54. Only two forces are acting on the girl,
Weight of the girl = W
Normal force acting on her = R
The scale reading (= R) is the girl’s apparent weight.
The motion of the lift can change R, and hence the girl will feel a different weight.
If the lift falls freely, R = 0, the girl will feel weightless. She is in a state of weightlessness.
weight (W)
reaction (R)
Apparent weight = R

55. 8. Density vs. mass
Density (密度) is a commonly-used concept in daily life. We say, for example, a plastic foam board is less dense than a piece of metal.
Intuition tells us that more mass packed into a small volume will give a higher density.
In fact, the density of an object is defined as
mass
Density =
volume

56. Measurement of density
To find the density of an object, one must know both the mass and volume.
Mass: can be measured by a balance.
Volume: How to measure?
Answer:
From the rise in level, we can measure the volume.
Measuring the density of an irregular solid
Measuring the density of a liquid

57. 9. Turning effect of a force
axis
When we turn on a tap or open a door, the tap or the door handle will rotate about an axis or a fixed point called the pivot.(支點). The perpendicular distance between the force and the pivot is called the moment arm (力臂).
The moment of a force is a measure of this turning effect. Moment is a vector quantity and its direction is indicated by either clockwise or anticlockwise. Its definition is
pivot
(Wikimedia commons)
Moment = Force  moment arm = Fd
pivot
(Wikimedia commons)

58. Principle of moments (力矩原理)
When a body is in balance, the total clockwise moment about any point is equal to the total anticlockwise moment about the same point.