Download Physics 201 - University of Virginia

PHYS 1110
Lecture 5
Professor Stephen Thornton
September 11, 2012
Reading Quiz
D) same speed
for all balls
A
B
C
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?
D) same speed
for all balls
A
B
C
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?
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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 and 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
Ui
U  U i  U f  Wc  mgy
U i  mgy  U f
Ui  U f
y
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
Example 3-4 The water flowing through Hoover
Dam’s turbines is about 1.1 x 1010 m3 each year.
The water falls on average 160 m from the water
intake system down to the turbine. How much work
does gravity do each year when the water drops?
What is the potential energy loss?
Strategy If we know how much water mass passes
over Hoover Dam each year, and we know the
height of the water drop, we can find the work done
by using Equation (3-19), (3-20), or (3-21). We can
convert the 4.2 billion kWh into joules to determine
the efficiency.
is B.
Springs
The work required to compress a
1
spring is kx 2 .
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
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.
L
Case 1
Case 2
Coordinate system origin
Powerhouse (y = 0)
Upper reservoir (y = 0)
Potential energy zero
y=0
y=0
Potential energy at top
U i = mgL
Potential energy at bottom
Uf = 0
Kinetic energy at top
Ki = 0
Kinetic energy at bottom
Kf =
Energy at top
E = mgL
Energy at bottom
E=
1 2
mv
2
1 2
mv
2
Ui = 0
U f = - mgL
Ki = 0
Kf =
1 2
mv
2
E= 0
E = - mgL +
1 2
mv
2
In both cases the energy E has to be conserved, and in both cases we must have
1
mgL = mv 2
2
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
In a baseball game, the
catcher stops a 90-mph
A) catcher has done positive work
pitch. What can you say
B) catcher has done negative work
about the work done by
C) catcher has done zero work
the catcher on the ball?
Conceptual Quiz
In a baseball game, the
catcher stops a 90-mph
A) catcher has done positive work
pitch. What can you say
B) catcher has done negative work
about the work done by
C) catcher has done zero work
the catcher on the ball?
The force exerted by the catcher is opposite in direction to the
displacement of the ball, so the work is negative. Or using the
definition of work (W = F d cos q ), because q = 180º, then W < 0.
Note that because the work done on the ball is negative, its
speed decreases.
Follow-up: What about the work done by the ball on the catcher?
Conceptual Quiz
A child on a skateboard is
moving at a speed of 2 m/s.
After a force acts on the
child, her speed is 3 m/s.
What can you say about
the work done by the
external force on the child?
A) positive work was
done
B) negative work was
done
C) zero work was done
Conceptual Quiz
A child on a skateboard is
moving at a speed of 2 m/s.
After a force acts on the
child, her speed is 3 m/s.
What can you say about
the work done by the
external force on the child?
A) positive work was
done
B) negative work was
done
C) zero work was done
The kinetic energy of the child increased because her
speed increased. This increase in KE was the result of
positive work being done. Or, from the definition of
work, because W = KE = KEf – KEi and we know that
KEf > KEi in this case, then the work W must be
positive.
Follow-up: What does it mean for negative work to be done on the child?
Conceptual Quiz
If a car traveling 60 km/hr
can brake to a stop within
20 m, what is its stopping
distance if it is traveling
120 km/hr? Assume that
the braking force is the
same in both cases.
A)
B)
C)
D)
E)
20 m
30 m
40 m
60 m
80 m
Conceptual Quiz
If a car traveling 60 km/hr
can brake to a stop within
20 m, what is its stopping
distance if it is traveling
120 km/hr? Assume that
the braking force is the
same in both cases.
1
2
F d = Wnet = KE 1= 0 – mv2,
and thus, |F| d = 2 mv2.
Therefore, if the speed doubles,
the stopping distance gets four
times larger.
A)
B)
C)
D)
E)
20 m
30 m
40 m
60 m
80 m
Conceptual Quiz
By what factor does
A) no change at all
the kinetic energy of
B) factor of 3
a car change when
its speed is tripled?
C) factor of 6
D) factor of 9
E) factor of 12
Conceptual Quiz
By what factor does the
A) no change at all
kinetic energy of a car
B) factor of 3
change when its speed
C) factor of 6
is tripled?
D) factor of 9
E) factor of 12
Because the kinetic energy is
1
2
mv2, if the speed increases
by a factor of 3, then the KE will increase by a factor of 9.
Follow-up: How would you achieve a KE increase of a factor of 2?
Conceptual Quiz
Two stones, one twice the
mass of the other, are dropped
from a cliff. Just before hitting
the ground, what is the kinetic
energy of the heavy stone
compared to the light one?
A) quarter as much
B) half as much
C) the same
D) twice as much
E) four times as much
Conceptual Quiz
Two stones, one twice the
mass of the other, are dropped
from a cliff. Just before hitting
the ground, what is the kinetic
energy of the heavy stone
compared to the light one?
A) quarter as much
B) half as much
C) the same
D) twice as much
E) four times as much
Consider the work done by gravity to make the stone
fall distance d:
KE = Wnet = F d cosq
KE = mg d
Thus, the stone with the greater mass has the greater
KE, which is twice as big for the heavy stone.
Follow-up: How do the initial values of gravitational PE compare?
Conceptual Quiz
A car starts from rest and accelerates to 30
mph. Later, it gets on a highway and
A) 0  30 mph
accelerates to 60 mph. Which takes more
B) 30  60 mph
energy, the 0  30 mph, or the 30  60
C) both the same
mph?
Conceptual Quiz
A car starts from rest and accelerates to 30
mph. Later, it gets on a highway and
A) 0  30 mph
accelerates to 60 mph. Which takes more
B) 30  60 mph
energy, the 0  30 mph, or the 30  60
C) both the same
mph?
1
The change in KE ( 2 mv2 ) involves the velocity squared.
So in the first case, we have:
In the second case, we have:
1
2
1
2
1
m (302 − 02) = 2 m (900)
m (602 − 302) = 21 m (2700)
Thus, the bigger energy change occurs in the second case.
Follow-up: How much energy is required to stop the 60-mph car?
Conceptual Quiz
The work W0 accelerates a car from
A) 2 W0
0 to 50 km/hr. How much work is
B) 3 W0
needed to accelerate the car from
C) 6 W0
50 km/hr to 150 km/hr?
D) 8 W0
E) 9 W0
Conceptual Quiz
The work W0 accelerates a car from
A) 2 W0
0 to 50 km/hr. How much work is
B) 3 W0
needed to accelerate the car from
C) 6 W0
50 km/hr to 150 km/hr?
D) 8 W0
E) 9 W0
Let’s call the two speeds v and 3v, for simplicity.
We know that the work is given by W = KE = KEf – Kei.
1
2
Case #1: W0 =
Case #2: W =
1
2
m (v2 – 02) =
1
2
m ((3v)2 – v2) =
m (v2)
1
m
2
(9v2 – v2) =
1
2
m (8v2) = 8 W0
Follow-up: How much work is required to stop the 150-km/hr car?
Conceptual Quiz
Two blocks of mass m1 and m2 (m1 > m2)
A) m1
slide on a frictionless floor and have the
B) m2
same kinetic energy when they hit a long
C) they will go the
rough stretch (m > 0), which slows them
same distance
down to a stop. Which one goes farther?
m1
m2
Conceptual Quiz
Two blocks of mass m1 and m2 (m1 > m2)
A) m1
slide on a frictionless floor and have the
B) m2
same kinetic energy when they hit a long
C) they will go the
rough stretch (m > 0), which slows them
same distance
down to a stop. Which one goes farther?
With the same KE, both blocks
m1
must have the same work done
to them by friction. The friction
force is less for m2 so stopping
m2
distance must be greater.
Follow-up: Which block has the greater magnitude of acceleration?
Conceptual Quiz
A golfer making a putt gives the ball an initial
velocity of v0, but he has badly misjudged the
putt, and the ball only travels one-quarter of
the distance to the hole. If the resistance force
due to the grass is constant, what speed
should he have given the ball (from its original
position) in order to make it into the hole?
A) 2 v0
B) 3 v0
C) 4 v0
D) 8 v0
E) 16 v0
Conceptual Quiz
A golfer making a putt gives the ball an initial
velocity of v0, but he has badly misjudged the
putt, and the ball only travels one-quarter of
the distance to the hole. If the resistance force
due to the grass is constant, what speed
should he have given the ball (from its original
position) in order to make it into the hole?
A) 2 v0
B) 3 v0
C) 4 v0
D) 8 v0
E) 16 v0
In traveling four times the distance, the resistive force
will do four times the work. Thus, the ball’s initial KE
must be four times greater in order to just reach the
hole—this requires an increase in the initial speed by a
1
factor of 2, because KE = 2 mv2.