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Pushes and PullsPushes and PullsPushes and PullsPushes and Pulls

for IJSO training coursefor IJSO training course

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Content

1. What are forces?

2. Measurement of a force

3. Daily life examples of forces

4. Useful mathematics: Vectors

5. Newton’s laws of motion

6. Free body diagram

7. Mass, weight and gravity

8. Density vs. mass

9. Turning effect of a force

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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.

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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 ).

(Wikimedia commons)

The SI unit of force: N (Newton)

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• 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.

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3. Daily examples of force

• The weight of an object is

the gravitational force

acting on it.

Weight

Weight

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• 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

weight

force by the hand

normal force

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• 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.

Tension

Draw “face-to-face” arrows to represent tension

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Example

“face to face” arrows representing tension

These two forces counterbalance each other (suppose the weight of the hook is negligible).

The tension balances the weight, therefore the mass does not fall down.

tension 10 N

tension 10 N

weight 10 N

1-kg mass

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• 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.

Friction

Friction drags motion.

motion

friction

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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

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• 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

Tread pattern on a mountain bicycle tyre

(Wikimedia commons)

(Wikimedia commons)

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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)

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– using lubricating oil

– using air cushion

– streamlining

(All pictures are from Wikimedia commons)

BHC SR-N4 The world's largest car and passenger carrying hovercraft

1. Propellers2. Air3. Fan4. Flexible skirt

The streamlined shape cuts down the air-friction on the racing car.

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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.

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• 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

Example: displacement

3 cm

4 cm5 cm

How to find the angle?

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• 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

Example: velocity

3 m/s

4 m/s5 m/s

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• 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.

Example: force

3 N

4 N5 N

A magnitude does not have a direction.

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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

bc

a

bc

Tip-to-tail method

Parallelogram method

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• 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.

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tan

sin

cos

yx

x

y

y

x

FFF

F

F

FF

FF

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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.

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• 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

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• 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.

The first law

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The second law

• 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

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But .. what is acceleration?

• Consider an object moving

from A to B in 2 hours with

a uniform velocity. What is

the velocity?

E

N

A (3 km, 1 km)

B (1 km, 3 km)

OFinal displacement from O = OB

Initial displacement from O = OA

Change in displacement = OB – OA = AB

Velocity = Change in displacement

Time required=

AB

2 hours

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E

N

A (3 km, 1 km)

B (1 km, 3 km)

O

AB =

Velocity = 0.39 m/s towards NW.

m 43.2828 km 22 22

Speed = AB / 7200 s = 0.39 m/s

(Note: This AB does not have an arrow. It indicates a length, which is a scalar.)

(Note: speed is also a scalar.)

Velocity = Change in displacement

Time required

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• 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.E

N

Change in velocity = vc

Acceleration = Change in velocity

Time required=

vc

2 s

v1

v2

vc = v2 - v1

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E

N

Acceleration = Change in velocity

Time required

v1

v2

vc = v2 - v1

vc =

Acceleration = 3.54 m/s2 towards N.

m/s 07.7 m/s 55 22

Magnitdue of acceleration= vc / 2 s = 3.54 m/s2

(Note: This vc does not have an arrow. It indicates a magnitude.)

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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.

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Uniform acceleration

• Let• 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

asuv

atuts

atuv

2

2

1

22

2

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v

t0

uslope = a

Velocity-time graph Displacement-time graph

t0

s

parabola

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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

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Force of man accelerates the cart.

The same force accelerates two carts half as much.

Twice as much force produces acceleration twice as much.

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• For every action, there is an equal and opposite reaction.

• When the cue hits the ball, the ball also “hits” the cue.

The third law

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.

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• 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!

Example

Normal force = mg (upward)

Weight = mg (downward)

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• 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.

Explanation

Weight = mg (downward)

Gravitational attraction of the block on the Earth = mg (upward)

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• 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.

Explanation

Normal force = mg (upward)

The force acting on the table by the block = mg (downward)

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• 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.

6. Free body diagram

N

W

ExampleFor 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.

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• 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 A3 kg Block B

2 kg10 N

Worked Example 1

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3 kg10 N

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

f (reaction force of the pushing force on Block B)

a

weight

normal force from the table surface

Solution: Method 1

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3 kg10 N f

a

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)

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2 kg

Then consider the free body diagram of Block B

weight

normal force from the table surface

f

a

We ignore the vertical direction because the forces are

balanced. Consider the horizontal direction. Applying the

second law again, we have

f = 2a (2)

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We now have 2 equations in 2 unknowns.

f = 2a (2)

10 - f = 3a (1)

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.

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• 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.

Solution: Method 2

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• 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.

Worked Example 2

A: 4 kgB: 1 kg

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Solution

Consider the free body diagram of Ball A:

A: 4 kg

Take downward positive.

Let tension = T and acceleration of Ball A = a.

T

Weight = 4g

We can apply F = ma and get

4g - T = 4a (1)

a

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Consider the free body diagram of Ball B:

B: 1 kg

T

Weight = g

We apply F = ma and get

g - T = -a (2)

a

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.

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• 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.

Worked Example 3

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• Find the acceleration a of the block in terms

of g, given that

,30o .32

1,Rf

Solution

Consider the motion perpendicular

to the motion. The forces are

balanced, therefore we have

mg

mg

mgR

2

3

30cos

coso

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Now, consider the motion

parallel to the motion.

Applying Newton’s second

law F = ma, we have

ga

mamgmg

maRmg

mafmg

4

1

2

3

32

1

2

1

30sin

sino

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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)

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• 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

9.80665 m/s2 1.622 m/s2 8.87 m/s2 24.79 m/s2

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• 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.

Weightlessness

weight (W)reaction (R)

scale reading = R

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• 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.Apparent weight = R

weight (W)reaction (R)

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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

Density = mass

volume

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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:

Measuring the density of an irregular solid

From the rise in level, we can measure the volume.

Measuring the density of a liquid

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9. Turning effect of a force

• 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

Moment = Force moment arm = Fd

axispivot

(Wikimedia commons)

pivot(Wikimedia commons)

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• 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.