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PHYS 1110
Lecture 3
Professor Stephen Thornton
September 4, 2012
Reading Quiz
Both boys jump off the cliff into the water at the
same time. Boy 2 has an additional horizontal
velocity. Which boy lands in the water first?
A) Boy 1.
B) Boy 2.
C) They land at the
same time.
D) Cannot tell
without further
information.
A
B
Answer: C The horizontal and vertical motion
is independent of each other. Both start with
zero velocity in the y-direction and have the
same acceleration. Their vertical motion is
identical.
Position vector = r
Displacement vector = r  rf  ri
r
Average velocity vector = vav 
t
r
Instantaneous velocity vector = v = lim
t 0 t
v
Average acceleration vector = aav 
t
v
Instantaneous acceleration vector = a  lim
t 0 t
Two dimensional motion
Horizontal and vertical motions are
independent! It is that simple.
We will find later that we only need
to look at the force components along
each direction. It is force that causes
acceleration.
Constant-Acceleration
Equations of Motion
Position as a
function of
time
Velocity as a
function of
time
Velocity as a
function of
position
x = x0 + v0xt
+ ½ axt2
vx = v0x
+ axt
vx2 = v0x2
+ 2axx
y = y0 + v0yt
+ ½ ayt2
vy = v0y
+ ayt
vy2 = v0y2
+ 2ayx
Projectile motion
Assumptions:
 air resistance is ignored for now.
 acceleration of gravity is constant
and has value g = 9.81 m/s2.
 Earth’s rotation is ignored.
Set up solutions
Choose coordinate
system.
y
ay =  g
x
ax = 0 and ay =  g
Now what will the equations in the
table look like?
Projectile motion equations
x  x0  v0 xt
1 2
y  y0  v0 yt  gt
2
vx  v0 x
v y  v0 y  gt
v v
2
x
2
0x
v  v  2 g y
2
y
2
0y
Don’t
memorize these
equations!
Water fountains show projectile
motion
Conceptual Quiz: Look at the demo
(shoot and drop). Two balls are released
at the same time by a spring loaded
mechanism. Ball A falls straight down.
Ball B is propelled out horizontally.
Which ball lands first?
A)
B)
C)
D)
Ball A
Ball B
Impossible to determine.
Balls land at the same time.
Answer
D) Balls land at the same time.
Both balls feel the same vertical
acceleration, -g. Therefore, they drop
at the same rate.
Of course their horizontal motion is
quite different.
Do demo.
Projectile Motion Applet/Physlets
http://physics.bu.edu/~duffy/semester
1/semester1.html
Questions: Three Projectiles
Maximum height
Time of flight
Range
Monkey and the Hunter
Conceptual Quiz
A battleship simultaneously fires two shells at
enemy ships. If the shells follow the parabolic
trajectories shown, which ship gets hit first?
A)
B)
C)
D)
Ship A.
Ship B.
Both at the same time.
Need more information.
Answer: B. Consider the time for
the shell to reach its maximum
height (or fall from its maximum
height). Since shell A goes higher,
it takes a longer time than shell B.
Projectiles with Air Resistance
Newton’s Laws of Motion
We now learn about FORCES
Operational definition: Forces are
pushes and pulls.
Look at spring scale.
Newton’s First Law
A body in motion tends to stay
in motion unless acted upon by
an external (outside) net force.
Do demos
Include air track, air carts, scooter,
tablecloth jerk, hanging masses (quizzes),
bottle and pencil.
Conceptual Quiz.
If we jerk on the bottom string,
what happens?
A) Top string breaks.
B) Bottom string breaks.
C) Can’t possibly tell; it will happen
randomly.
Answer: B. bottom string
breaks.
The bottom string breaks, because the
mass is large and has lots of inertia.
The mass will not respond to a quick
jerk.
Conceptual Quiz.
Now we pull slowly on the
bottom string. What happens?
A) Top string breaks.
B) Bottom string breaks.
C) Can’t possibly tell; it will happen
randomly.
Answer: A) top string breaks.
The top string breaks now, because
we are pulling slowly on it. The
bottom string feels the force from our
hand, but the top string feels the force
from our hand plus the weight of the
mass.
Newton’s first law is also known as the law of
inertia.
Inertia means the body wants to keep its present
motion, whether at rest or not.
If a body is moving at constant velocity, it wants
to remain moving at constant velocity.
If at rest, it wants to remain at rest.
It keeps its inertia unless a net force acts on it!
Newton’s second law of motion
Let’s do some experiments on the air
track with a constant force (fan car).
We will use the fan to push various
masses and observe the acceleration
of the masses.
Do experiments.
We learn that the acceleration is
proportional to F/m.
F
a
m
Because there may be several
forces on the object, we have to
take the net force.
Newton’s Second Law
Determine the net force.
Fnet   Fi
i
Now Newton’s Second Law appears as
Fnet  ma
Unit is newton. 1 N = 1 kg·m/s2
Forces
Lots of things to learn about forces.
Find net force – free-body diagrams
are very helpful.
Different kinds of forces:
W weight
N normal force, also FN
T tension, for example, a rope
f friction
several others
Forces are vectors
In some cases we will need to use the
vector notation:
W , N, T, f
Newton’s Third Law
When an object 1 exerts a force on
object 2, then object 2 will exert an
equal, but opposite, force on object 1.
Forces always come in pairs and are
equal and opposite.
Math form of Newton’s
rd
3
Law
F12   F21
Force on body 1 due to 2, F12 , is
equal and opposite to the force on 2
due to 1, F21 .
We often say “for every action there
is an equal and opposite reaction”.
Law of action and reaction.
Examples of ActionReaction Force Pairs
Do demos
Air track reaction cars
Two carts with students
Important Points
Action-reaction force pairs always act on
different objects!
When dealing with forces, we want the Fnet
on a particular object. In a 3rd law force
pair, one force acts on our object and the
other force acts on another object. This is
a big source of confusion!
Example – which object is
the force acting upon?
Quiz: Which of Newton’s laws
refers to an action and a reaction force?
A) First law.
B) Second law.
C) Third law.
D) This is a trick question. There is only one
Newton’s law.
Answer: C
Newton’s third law is more commonly
known by its abbreviated form, “for every
action there is an equal and opposite
reaction”.
Weight
Weight is a force
W = mg called gravitational force
Note that weight is not a mass!
W  mg
Weight and Mass
The Normal Force
May Equal the Weight
Quiz: Is the normal force on a body
always equal to its weight?
A) Yes.
B) No: it depends on the net external
force due to other forces acting on the
body.
C) No: it depends on the shape of the body.
D) No: it depends on the area of contact
between the body and the surface.
Answer: B
There can be
other external
forces that reduce
the normal force
on a body.
An Object on an Inclined Surface
W
Force pushing downhill
Rubbing surfaces cause friction
Facts about friction
• Leonardo da Vinci performed
experiments on friction (~1500).
He found it was the normal force
that was important, not the surface
area.
• Show block on table.
• Friction is bad. $1B of research/yr
• Friction is good. Many examples;
we walk, drive cars, etc.
Kinetic Friction and the
Normal Force
Constant velocity.
F   fk
2 F  2 f k
More experimental results
 Friction is proportional to the
magnitude of the normal force.
 Friction is independent of the
relative speed of the surfaces.
 Friction is (to first order)
independent of the area of contact
between surfaces.
We are examining kinetic friction.
Static friction
In kinetic or sliding friction the two objects
are moving with respect to each other.
In static friction the object is at rest, but we
want it to move. We have to overcome
friction to have it move. We find there is a
certain amount of force needed to do this.
Example of static friction
Experimental result for static friction
There is an upper limit of static
friction that we have to overcome in
order to move an object.
Static friction can have any value up
to this maximum amount.
0  fs  fs,max
fs, max  s N
These are concepts!
Facts about static friction
Static friction can have any value
from zero to its maximum amount.
Value:
0  fs  fs,max
Static friction is independent of area
of contact between surfaces.
Static friction is parallel to surface of
contact, and in the direction that
opposes relative motion.
Typical Coefficients of Friction
Materials
Kinetic, k
Static, s
Rubber on concrete (dry)
0.80
0.90
Steel on steel
0.57
0.74
Glass on glass
0.40
0.94
Wood on leather
0.40
0.50
Copper on steel
0.36
0.53
Rubber on concrete (wet)
0.25
0.30
Steel on ice
0.06
0.10
Waxed ski on snow
0.05
0.10
Teflon on Teflon
0.04
0.04
Note that s  k
When we walk,
is it static friction
or kinetic friction
when our feet are
in contact with
the ground?
Answer: static
Same with car
tires.
Conceptual Quiz
An object is held in place by friction on an
inclined surface. The angle of inclination
is slowly increased until the object starts
to move. If the surface is kept at this
angle, the object
A) speeds up.
B) moves at uniform speed.
C) slows down.
Answer: A
Remember that the coefficient of static
friction is greater than that of kinetic
friction. So when static friction is
overcome and the object starts sliding, the
frictional force becomes smaller, and the
net force (due to gravity) is down the plane.
Tension in a Heavy Rope
Heavy rope:
T T T
3
2
1
Light rope:
T T T
3
= mg
mass m
g
2
1
We usually consider
light ropes.
A Pulley Changes
the Direction of a
Tension
Tension in a String
Spring Forces
Spring Forces
F  kx
Notice signs of
the force in both
cases.
Spring Forces
Equation F = -kx is known as
Hooke’s Law.
The force is always in the direction to
restore the spring to equilibrium.
The minus sign simply indicates that
the force is a restoring force.
Concepts!
Quiz: A person hoists a bucket from a well
using a rope. Let the bucket be at rest. She
then ties the other end of the rope to the handle.
In which case is the tension in the rope the
greatest? 1
2
A) Case 1
B) Case 2
C) They are the same
Answer: A) The tension is
greatest in the first case.
See the next slide.
Circular motion
Do demo with
string and ball.
Note that the
direction of the
velocity is
changing. The
ball is
accelerating!
v  v f  vi
Notice that v tends to
point towards the center of
the circle. As  becomes
smaller and smaller, v
points directly to center.
Therefore the acceleration
points towards the center
of the circle.
Centripetal acceleration
Centripetal means “center seeking”.
v v2  v1
aav 

t
t
The derivation is straightforward, but
we will not do it. The result is that the
magnitude of the centripetal
a
acceleration acp is
2
v
acp 
r
where r is the radius and v is the speed.
v
Circular motion
Results for circular motion:
 Consider an object moving in a
circle of radius r with a constant
speed v.
 A centripetal acceleration of
magnitude v2/r must cause it.
 There must be a centripetal force
Fcp of value
mv 2
Fcp  macp 
r
Centripetal force
Where in the world did this centripetal
force come from?
There has to be a force to keep the object
moving in a circle. In the case of the ball
and string, it was the tension in the string.
The tension always pointed towards the
center!
The direction of the centripetal force must
also be towards the center!
The moon rotates around the Earth in a circle.
What is the centripetal force that causes this?
If you drive around in a circle with a bicycle
or even with a car, what is the centripetal
force?
In a simple atomic model of the hydrogen
atom, the electron rotates around the proton in
a circle. What is the centripetal force?
Conceptual Quiz
A ball is attached to a string and swung
in a horizontal circle of constant radius.
Immediately after the string is released
the ball will move in what direction?
D
A
B
·
E
C
Answer: B
Remember that the velocity is always
tangent when we have circular motion.
This is the instantaneous velocity. So
right when the string is released, it has
to go in the direction of the velocity at
that instant. Therefore it must go in its
tangential direction.
DO DEMO!
What other forces are
exerted on the ball
besides mg?
A) Friction
B) Tension
C) A normal force
perpendicular to mg.
D) A normal force
perpendicular to the
surface of the cone at
the ball.
Answer: D
The only other
possible force is the
normal force, and it
must be
perpendicular to the
surface that the ball
is rolling upon.
Quiz: What is the
direction of the net force?
A) towards the center of
the dashed circle at the
ball (radially).
B) away from the center
of the circle at the ball.
C) up at the ball.
D) down at the ball.
E) cannot tell with
information given.
Answer: A
Because the ball is
moving at constant
speed in a circle, the
net force must be
along the radial
direction, towards the
center of the circle.
This is the centripetal
force.