Pushes and Pulls

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Pushes and Pulls

• for 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. Newtons 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 (???).

The SI unit of force N (Newton)
(Wikimedia commons)
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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 Hookes law
  (????).
• 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
Weight

• The weight of an object is the gravitational
  force acting on it.

Weight
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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
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Tension

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

• Friction (???) 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.
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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
(Wikimedia commons)
Tread pattern on a mountain bicycle tyre
(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

The streamlined shape cuts down the air-friction
on the racing car.
1. Propellers2. Air3. Fan4. Flexible skirt
BHC SR-N4 The world's largest car and passenger
carrying hovercraft
(All pictures are from Wikimedia commons)
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4. Useful mathematics Vectors

• A scalar (??) 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 (??) 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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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?
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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
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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.
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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
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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5. Newtons 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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• Newtons 3 laws of motion answer 3 questions
• If the cue does not hit the ball, what will
  happen to the ball?
• Newtons first law
• If the cue hits the ball, what will happen to the
  ball?
• Newtons second law
• If the cue hits the ball, what will happen to the
  cue?
• Newtons third law

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The first law

• Also called The law of inertia (????)
• 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.

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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?
N

• 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
E
O
Initial displacement from O OA
Change in displacement OB OA AB
Change in displacement
AB

Velocity
Time required
2 hours
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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)
E
O
Velocity 0.39 m/s towards NW.
Change in displacement
Velocity
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.

N
v2
vc v2 - v1
E
v1
Change in velocity vc
Change in velocity
vc
Acceleration

Time required
2 s
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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
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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

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Velocity-time graph
Displacement-time graph
v
s
parabola
slope a
u
0
0
t
t
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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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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.
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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
  Newtons third law?
• Answer No!

Normal force mg (upward)
Weight mg (downward)
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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)
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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)
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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
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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
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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
f (reaction force of the pushing force on Block B)
10 N
weight
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a
3 kg
f
10 N
Vertical direction No motion. The weight and the
normal force from the table balance each other.
Horizontal direction Applying Newtons second
law (F ma), we have (with units neglected)
(1)
10 - f 3a
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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
(2)
f 2a
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We now have 2 equations in 2 unknowns.
(1)
10 - f 3a
(2)
f 2a
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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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.

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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
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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
(1)
4g - T 4a
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Consider the free body diagram of Ball B
T
B 1 kg
a
Weight g
We apply F ma and get
(2)
g - T -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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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.

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• 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
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Now, consider the motion parallel to the motion.
Applying Newtons second law F ma, we have
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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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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)
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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 girls 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
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8. Density vs. mass

• Density (??) 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
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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

From the rise in level, we can measure the volume.
Measuring the density of an irregular solid
Measuring the density of a liquid
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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.(??). The
  perpendicular distance between the force and the
  pivot is called the moment arm (??).
• 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)
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• Principle of moments (????)
• When a body is in balance, the total clockwise
  moment about any point is equal to the total
  anticlockwise moment about the same point.