Download Pushes and Pulls

Pushes and Pulls
for IJSO training course
1
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
2
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.
3
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)
4
• 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.
5
3. Daily examples of force
Weight
• The weight of an object is
the gravitational force
acting on it.
Weight
6
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
weight
force by
the hand
7
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
8
Example
“face to face”
arrows
representing
tension
tension 10 N
These two forces
counterbalance each other
(suppose the weight of the
hook is negligible).
tension 10 N
1-kg mass
The tension balances
the weight, therefore
the mass does not fall
down.
weight 10 N
9
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.
10
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.
11
backward push of foot on road
forward push of road on foot
• 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)
12
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)
13
– using lubricating oil
– using air cushion
– streamlining
The streamlined shape cuts
down the air-friction on the
racing car.
BHC SR-N4 The world's largest car and
passenger carrying hovercraft
1. Propellers
2. Air
3. Fan
4. Flexible skirt
14
(All pictures are from Wikimedia commons)
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.
15
Example: displacement
• A mouse moves 4 cm northward
and then 3 cm eastward.
3 cm
• What is the distance travelled?
– Answer = 4 cm + 3 cm = 7 cm
• What is the displacement of the
mouse?
4 cm
5 cm
– Answer = 5 cm towards N36.9oE
How to find the angle?
16
Example: velocity
• A bird is flying 4 m/s northward.
There suddenly appears a wind of
3 m/s blowing towards the east.
3 m/s
• What is the velocity of the bird?
– Answer = 5 m/s towards N36.9oE
4 m/s
5 m/s
• 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
17
Example: force
• You push a cart with 4 N towards
north. Your friend helps but he
pushes it with 3 N towards the east.
3N
• What is the resultant force?
– Answer = 5 N towards N36.9oE
4N
5N
• What is the magnitude of the force?
– Answer = 5 N
• Note: A magnitude does not have a
direction.
A magnitude does not have a direction.
18
Addition and resolution
• Two usual ways to
denote a vector
F
F
– 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.
c
b
a
Tip-to-tail method
b
c
a
Parallelogram method
19
• 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.
Fx  F cos 
Fy  F sin 
tan  
Fy
Fx
F  Fx  Fy
2
2
20
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.
21
• 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
22
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.
23
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
24
But .. what is acceleration?
• Consider an object moving
from A to B in 2 hours with
a uniform velocity. What is
the velocity?
N
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
Velocity =
Change in displacement
Time required
=
AB
2 hours
25
AB =
2 2  2 2 km  2828.43 m
(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.
Velocity =
Change in displacement
Time required
26
• 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.
N
v2
vc = v2 - v1
E
• Calculate the acceleration.
v1
Change in velocity = vc
Acceleration =
Change in velocity
Time required
=
vc
2s
27
vc =
5  5 m/s  7.07 m/s
2
N
2
(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.
Acceleration =
Change in velocity
Time required
28
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.
29
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
v  u  at
1 2
s  ut  at
2
v 2  u 2  2as
30
Velocity-time graph
Displacement-time graph
v
s
parabola
slope = a
u
0
t
0
t
31
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
32
Force of man accelerates
the cart.
The same force accelerates
two carts half as much.
Twice as much force
produces acceleration twice
as much.
33
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.
34
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 actionand-reaction pair of force
as described by Newton’s
third law?
Normal force = mg (upward)
Weight = mg (downward)
• Answer: No!
35
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)
36
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.
Normal force = mg (upward)
• Both have the same
magnitude mg.
• But they do not cancel
each other because they
are acting on different
bodies.
The force acting on the table
by the block = mg (downward)
37
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.
Example
N
W
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.
38
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.
10 N

Block A
3 kg
Block B
2 kg
39
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
10 N
3 kg
a
f (reaction force
of the pushing
force on Block B)
weight
40
a
10 N
3 kg
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)
41
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)
42
We now have 2 equations in 2 unknowns.
10 - f = 3a
(1)
f = 2a
(2)
Solving them, we have
f=4N
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.
43
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.
44
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
45
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)
46
Consider the free body diagram of Ball B:
T
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.
47
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.
48
• Find the acceleration a of the block in terms
of g, given that
1
o
.
  30 , f  R,  
2 3
Solution
Consider the motion perpendicular
to the motion. The forces are
balanced, therefore we have
R  mg cos 
 mg cos 30 o
3

mg
2
49
Now, consider the motion
parallel to the motion.
Applying Newton’s second
law F = ma, we have
mg sin   f  ma
mg sin 30o  R  ma
1
1
3
mg 
mg  ma
2
2 3 2
1
a g
4
50
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)
51
• 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
52
Weightlessness
• When a girl stands inside a
lift, she cannot feel her own
weight. What she feels is
scale reading = R
the normal force R acting
on her by the lift floor.
weight (W)
reaction (R)
• The scale reading shows
the magnitude of the
reaction force to R, that is,
the force acting on the
scale by her feet.
53
• 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.
weight (W)
reaction (R) • The motion of the lift can
change R, and hence the girl
will feel a different weight.
Apparent weight = R
• If the lift falls freely, R = 0,
the girl will feel weightless.
She is in a state of
weightlessness.
54
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
55
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
56
9. Turning effect of a force
• When we turn on a tap or open a door, the tap or the pivot axis
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
(Wikimedia commons)
Moment = Force  moment arm = Fd
pivot
57
(Wikimedia commons)
• 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.
58