PHYS 1110
Lecture 4
Professor Stephen Thornton
September 6, 2012
Reading Quiz
A) yes
Is it possible to do
work on an object
that remains at rest?
B) no
C) depends on mass
Reading Quiz
A) yes
Is it possible to do
work on an object
that remains at rest?
B) no
C) depends on mass
Work requires that a force acts
over a distance. If an object does
not move at all, there is no
displacement, and therefore no
work done.
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
A
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!
Drag Forces
1
2
FD = r ACD v
2
Here,
FD is the drag force;
ρ the density of the
medium;
A the cross-sectional
area of the object;
CD the drag coefficient.
Velocity-Dependent Forces:
Drag and Terminal Velocity
When an object moves through a fluid at low
speed, it experiences a drag force that
depends on the velocity of the object.
FD  bv
For small velocities, the force is approximately
proportional to the velocity; for higher speeds,
the force is approximately proportional to the
square of the velocity.
2
FD
v
If the drag force on a falling
object is proportional to its
velocity, the object gradually
slows until the drag force and
the gravitational force are equal.
Then it falls with constant
velocity, called the terminal
velocity (~120 mph for humans).
mg  bv when equal, no net force
mg
vT 
terminal velocity
b
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.
B
DO DEMO!
E
D
A
·
C
Quiz: 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.
Let the force and displacement be
in the same direction (for now).
W = Fd
called Work
unit: N m, newton meter
Work is so important that we give it its own unit: joule
1J=1Nm
How much work is a joule?
• Lifting an apple about 1 meter is a joule.
• Total annual U.S. energy use is ~1020 J.
• Lifting a text book 1 m.
F = (2.6 kg)(9.81 m/s2) = 26 N
W = (26 N)(1 m) = 26 J
Curiously there is no work
done if there is no displacement in the
direction of the force.
Example: hold a heavy object in front
with your arm extended. No work is
done, because there is no displacement in
the direction of gravitational
force. This is true even if you
walk while holding it.
We use component of
force in direction of
displacement.
What happens when force is not in
same direction as displacement?
W  ( F cos )d  Fd cos
W = Fd cos. Work is a scalar.
We can determine work two ways: Work is the
1) component of force in the direction of
displacement times the magnitude of displacement.
2) component of displacement in the direction of the
force times the magnitude of the force.
F cos
W  F  d = Fd cos scalar or dot product
Work can be negative. See case on right.
Gravity does work on the apple as it falls.
The apple accelerates. Gravity does negative
work on the apple if we throw it up.
Notes about work
•Work is a scalar, not a vector.
•Work can be positive, negative, or zero.
•The angle  is always the angle between F
and d . Be careful about this.
•There is often more than one force acting
on an object. The total work is the sum of
the work done by all the forces.
We had a kinematic equation that stated:
v  v  2ad
2
f
2
i
or rearranging,
2ad  v  v
2
f
2
i
 Fnet 
2
2
2
 d  v f  vi ,
 m 
Fnet d 
1
1
multiply by m / 2
mv  mv
2
2
1 2 1 2
Wnet  Wtotal  mv f  mvi
2
2
2
f
2
i
Definition of kinetic energy
1 2
K  mv
2
Unit of kinetic energy is the joule, J.
Work-Energy Theorem
The total work done on an object is
equal to the change in its kinetic
energy:
Wtotal  K  K f  Ki 
1
1
mv  mv
2
2
2
f
This is a general result, even for a
force not constant in magnitude and
direction.
2
i
Conceptual Quiz:
Two marbles, one twice as heavy as the other, are
dropped to the ground from the roof of a building.
Just before hitting the ground, the heavier marble
has
A) the same kinetic energy as the lighter one.
B) half as much kinetic energy as the lighter one.
C) twice as much kinetic energy as the lighter one.
D) four times as much kinetic energy as the
lighter one.
Answer: C
The velocities will be the same in this
case, so the only difference in the
kinetic energy is due to the mass.
Because the mass is twice as much,
the kinetic energy is twice as much.
Conceptual Quiz:
A force F pushes a block along a horizontal
surface against the force of friction f. If the
block undergoes a displacement d at
constant velocity, the work done by the net
force on the block is (hint on next slide)
A)
B)
C)
D)
zero.
equal to the work done by friction.
increases the kinetic energy of the block.
decreases the kinetic energy of the block.
Conceptual Quiz:
A force F pushes a block along a horizontal
surface against the force of friction f. If the
block undergoes a displacement d at
constant velocity, the work done by the net
F
force on the block is
f
A)
B)
C)
D)
zero.
equal to the work done by friction.
increases the kinetic energy of the block.
decreases the kinetic energy of the block.
Answer: A
The key here is that the velocity is
constant, so there is no change in the
kinetic energy. The total work done
by the net force is zero, but the work
done by the force F is positive, while
the friction force is negative.
Graphical Representation of the
Work Done by a Constant Force
Work Done by a Nonconstant Force
Work Done by a
Continuously Varying Force
We approximate the curve along
various parts of it. We add up
each area to obtain the total.
The spring force
varies with
position. It is a
good example of
the kind of force
we are
considering.
Work Needed to Stretch a
Spring a Distance x
1 2
W  kx
2
The Work Done by a Spring
Can Be Positive or Negative
Work done
by spring is
positive in
this case.
The Work Done by a Spring
Can Be Positive or Negative
Work done
by spring is
negative in
this case.
Power
Power measures how fast work is done.
Average power = P = W/t
Power is so important that it also has its
own unit. SI unit: watt
1 watt = 1 W = 1 J/s = 1 joule/sec
1 horsepower = 1 hp = 746 watt ***
Table 3-1 Typical Power Generator Stations by Units (megawatt)
Type__________
Nuclear reactor
Per unit
1000/reactor
Total
1,000
Tennessee Valley Authority
245/ generator 14,500
(11 coal-fired plants, 59 generators)
Hoover Dam
(hydroelectric) (17 turbines)
130/turbine
2,100
Three Georges Dam
700/turbine
(hydroelectric, China) (32 turbines)
22,500
Typical wind farm
(150 wind turbines)
1.5/turbine
Typical solar farm
0.0003/panel
(250,000 photovoltaic modules)
225
70
To find work, we have to be sure about
what force is exerting the effort. Here we
might ask about the work done by friction,
gravity, air resistance, or the engine.
We need to make sure we can find the
work done by every force.
Wengine  Fd
Wfriction  Ffriction  d   Ffriction d
Wgravity  Fgravity  d   mgd sin 
Wair resis  Fair resis  d   Fair resis d
Conservative and
Nonconservative Forces
• A conservative force does zero total work on
any closed path.
•
• The work done by a conservative force in
going from an arbitrary point A to an
arbitrary point B is independent of the path
from A to B.
•B
A•
Doing Work Against Gravity
Energy is reclaimed in this case.
Doing Work Against Friction
Energy is not reclaimed in this case.
Work Done by Gravity on
a Closed Path is Zero.
Work Done by Friction on a
Closed Path is Not Zero.
Conservative Forces
Gravity
Springs
Nonconservative Forces
Friction
Tension
Potential Energy
When we do work, say to lift a
box off the floor, then we give the
box energy. We call that energy
potential energy. Potential
energy, in a sense, has potential to
do work. It is like stored energy.
However, it only works for
conservative forces.
Do potential energy demo. Burn
string and let large mass drop.
Notes on potential energy
Potential energy is part of the workenergy theorem. Potential energy
can be changed into kinetic energy.
Think about gravity for a good
example to use.
There is no single “equation” to use
for potential energy.
Remember that it is only useful for
conservative forces.
Definition of potential energy
We will use a subscript on Wc to
remind us about conservative forces.
This doesn’t work for friction.
Wc U U (U U )U
i
f
f
i
SI unit is the joule (still energy).
Remember gravity
The work done by a conservative force
is equal to the negative of the change in
potential energy.
Hold a box up. It has potential energy.
Drop the box. Gravity does positive
work on the box. The change in the
gravitational potential energy is
negative. The box has less potential
energy when it is on the floor.
Gravity Is a Conservative Force:
Kinetic energy, potential energy, and
speed are the same at points A to D.
Gravitational Potential Energy
Boy does +mgy work
W  F d  mgy
to climb up to y.
(Gravity does
negative work, -mgy).
He has potential
energy mgy. Gravity
does work on boy to
bring him down. The
potential energy is
converted into kinetic
energy.
More potential energy (PE) notes
Gravitational potential energy = mgh
Only change in potential energy U is
important.
There is no absolute value of PE.
We choose the zero of PE to be at the most
convenient position to solve problem.
Gravitational potential energy
Wc  mgy
U  U i  U f  Wc  mgy
Ui
U i  mgy  U f
Ui  U f
Uf
Because we can choose the “zero” of
potential energy anywhere we want, it
might be convenient to place it at y = 0
(but not always!).
Where might we choose the zero of
potential energy to be here?
Do demos
Loop the loop
Bowling ball
is B.
Springs
The work required to compress a
spring is 1 2 .
kx
2
The potential energy
of springs is
1 2
U  kx
2
1 2
W  kx
2
Conservation of mechanical energy
Mechanical energy E is defined to be
the sum of K + U.
E=K+U
Mechanical energy is conserved.
Only happens for conservative forces.
Solving a Kinematics Problem
Using Conservation of Energy
E = mgh
E=0
15)
Ball rolling on a frictionless track
Gravitational potential energy vs
position for the previous track.
See also kinetic and total energy.
A Mass on a Spring
E
K
U
Bath County, Virginia, pumped storage facility
electrical power plant.
Day – water flows down from upper reservoir
producing electricity.
Night – use power from other plants to pump water
back up.
Contour Map
Conceptual Quiz:
Two unequal masses are hung from a string
that pass over an ideal pulley. What is true
about the gravitational potential energy U and
the kinetic energy K of the system after the
masses are released from rest?
A)
B)
C)
D)
E)
U > 0 and K < 0.
U > 0 and K > 0.
U > 0 and K = 0.
U = 0 and K = 0.
U < 0 and K > 0.
Answer: E
Initially the system is at rest. Let the
potential energy be zero at this point.
Therefore the total mechanical energy
is zero. If the system starts moving,
then K > 0. Since E = 0, then U <
0.
Conceptual Quiz
4) same speed
for all balls
1
2
3
Three balls of equal mass start from rest
and roll down different ramps. All ramps
have the same height. Which ball has the
greater speed at the bottom of its ramp?
Three balls of equal mass start from rest
and roll down different ramps. All ramps
have the same height. Which ball has the
greater speed at the bottom of its ramp?
4) same speed
for all balls
1
2
3
All of the balls have the same initial
gravitational PE, since they are all at the
same height (PE = mgh). Thus, when they
get to the bottom, they all have the same final
KE, and hence the same speed (KE = 1/2 mv2).
Follow-up: Which ball takes longer to get down the ramp?