Title: Pushes and Pulls
1Pushes and Pulls
2Content
- What are forces?
- Measurement of a force
- Daily life examples of forces
- Useful mathematics Vectors
- Newtons laws of motion
- Free body diagram
- Mass, weight and gravity
- Density vs. mass
- Turning effect of a force
31. 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.
42. 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)
5- 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.
63. Daily examples of force
Weight
- The weight of an object is the gravitational
force acting on it.
Weight
7Normal 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
8Tension
- 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
9Example
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
10Friction
- 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.
11Friction 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)
13Disadvantages 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. Propellers2. Air3. Fan4. Flexible skirt
BHC SR-N4 The world's largest car and passenger
carrying hovercraft
(All pictures are from Wikimedia commons)
154. 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.
16Example 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?
17Example 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
18Example 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.
19Addition 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
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.
215. 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.
22- 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
23The 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.
24The 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
25But .. 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
26AB
(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
27- 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
28N
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
29Equations 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.
30Uniform 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
31Velocity-time graph
Displacement-time graph
v
s
parabola
slope a
u
0
0
t
t
32Back 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
33Force of man accelerates the cart.
The same force accelerates two carts half as much.
Twice as much force produces acceleration twice
as much.
34The 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.
35Example
- 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)
36Explanation
- 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)
37Explanation
- 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)
386. 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
39Worked 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
40Solution 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
41a
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
42Then 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
43We 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.
44Solution 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.
45Worked 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
46Solution
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
47Consider 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.
48Worked 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
50Now, consider the motion parallel to the motion.
Applying Newtons second law F ma, we have
517. 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- 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
53Weightlessness
- 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 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
558. 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
56Measurement 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
579. 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)
58- 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.