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Reading and Review
A mass attached to a vertical
spring causes the spring to
stretch and the mass to move
downwards. What can you
say about the spring’s
potential energy (PEs) and the
gravitational potential energy
(PEg) of the mass?
a) both PEs and PEg decrease
b) PEs increases and PEg decreases
c) both PEs and PEg increase
d) PEs decreases and PEg increases
e) PEs increases and PEg is constant
Question 8.5 Springs and Gravity
A mass attached to a vertical
spring causes the spring to
stretch and the mass to move
downwards. What can you
say about the spring’s
potential energy (PEs) and the
gravitational potential energy
(PEg) of the mass?
a) both PEs and PEg decrease
b) PEs increases and PEg decreases
c) both PEs and PEg increase
d) PEs decreases and PEg increases
e) PEs increases and PEg is constant
The spring is stretched, so its elastic PE increases,
because PEs = kx2. The mass moves down to a
lower position, so its gravitational PE decreases,
because PEg = mgh.
Question 8.5 Springs and Gravity
12
Question 8.9 Cart on a Hill
A cart starting from rest rolls down a hill
and at the bottom has a speed of 4 m/s. If
the cart were given an initial push, so its
initial speed at the top of the hill was 3 m/s,
what would be its speed at the bottom?
a) 4 m/s
b) 5 m/s
c) 6 m/s
d) 7 m/s
e) 25 m/s
Question 8.9 Cart on a Hill
When starting from rest, thecart’s PE is changed into KE:
PE = KE = m(4)2
A cart starting from rest rolls down a hill
and at the bottom has a speed of 4 m/s. If
the cart were given an initial push, so its
initial speed at the top of the hill was 3 m/s,
what would be its speed at the bottom?
a) 4 m/s
b) 5 m/s
c) 6 m/s
d) 7 m/s
e) 25 m/s
When starting from 3 m/s, thefinal KE is:
KEf = KEi + KE
= m(3)2 + m(4)2
= m(25)
= m(5)2
Speed is not the same as kinetic energy
12
12
12
12
12
8-4 Work Done by Nonconservative ForcesIn the presence of nonconservative forces, the total mechanical energy is not conserved:
Solving,
(8-9)
8-4 Work Done by Nonconservative Forces
In this example, the nonconservative force is water resistance:
You see a leaf falling to the ground
with constant speed. When you
first notice it, the leaf has initial
total energy PEi + KEi. You watch
the leaf until just before it hits the
ground, at which point it has final
total energy PEf + KEf. How do
these total energies compare?
a) PEi + KEi > PEf + KEf
b) PEi + KEi = PEf + KEf
c) PEi + KEi < PEf + KEf
d) impossible to tell from
the information provided
Question 8.10a Falling Leaves
You see a leaf falling to the ground
with constant speed. When you
first notice it, the leaf has initial
total energy PEi + KEi. You watch
the leaf until just before it hits the
ground, at which point it has final
total energy PEf + KEf. How do
these total energies compare?
a) PEi + KEi > PEf + KEf
b) PEi + KEi = PEf + KEf
c) PEi + KEi < PEf + KEf
d) impossible to tell from
the information provided
As the leaf falls, air resistance exerts a force on it opposite to its direction of motion. This force does negative work, which prevents the leaf from accelerating. This frictional force is a nonconservative force, so the leaf loses energy as it falls, and its final total energy is less than its initial total energy.
Question 8.10a Falling Leaves
Follow-up: What happens to leaf’s KE as it falls? What net work is done?
8-5 Potential Energy Curves and Equipotentials
The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:
8-5 Potential Energy Curves and Equipotentials
The potential energy curve for a spring:
8-5 Potential Energy Curves and Equipotentials
Contour maps are also a form of potential energy curve:
Lecture 11
Linear Momentum
Linear Momentum
Momentum is a vector; its direction is the same as the direction of the velocity.
Going Bowling IGoing Bowling I
p
p
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy?
Going Bowling IGoing Bowling I
p
p
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy?
Momentum is p = mv
so the ping-pong ball must have a much greater velocity
Kinetic Energy is KE = 1/2 mv2
so (for a single object): KE = p2 / 2m
Momentum and Newton’s Second Law
Newton’s second law, as we wrote it before:
is only valid for objects that have constant mass. Here is a more general form, also useful when the mass is changing:
Change in Momentum
Change in momentum:
(a) mv
(b) 2mv
A net force of 200 N acts on a 100-
kg boulder, and a force of the same
magnitude acts on a 130-g pebble.
How does the rate of change of the
boulder’s momentum compare to
the rate of change of the pebble’s
momentum?
a) greater than
b) less than
c) equal to
Momentum and ForceMomentum and Force
A net force of 200 N acts on a 100-
kg boulder, and a force of the same
magnitude acts on a 130-g pebble.
How does the rate of change of the
boulder’s momentum compare to
the rate of change of the pebble’s
momentum?
a) greater than
b) less than
c) equal to
The rate of change of momentum is, in fact, the
force. Remember that F = p/t. Because the force
exerted on the boulder and the pebble is the same,
then the rate of change of momentum is the same.
Momentum and ForceMomentum and Force
Impulse
Impulse is a vector, in the same direction as the average force.
The same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time.
Impulse quantifies the overall change in momentum
Impulse
We can rewrite
as
So we see that
The impulse is equal to the change in momentum.
Why we don’t dive into concreteThe same change in momentum may be
produced by a large force acting for a short time, or by a smaller force acting for a longer time.
Going Bowling IIGoing Bowling II
p
p
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest?
Going Bowling IIGoing Bowling II
We know:
Here, F and p are the same for both balls!
It will take the same amount of time to stop them. p
p so p = Fav t
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest?
av tp
F
Going Bowling IIIGoing Bowling III
p
p
A bowling ball and a Ping-Pong
ball are rolling toward you with
the same momentum. If you
exert the same force to stop each
one, for which is the stopping
distance greater?
a) the bowling ball
b) same distance for both
c) the Ping-Pong ball
d) impossible to say
Going Bowling IIIGoing Bowling III
p
p
Use the work-energy theorem: W = KE. The ball with less mass has the greater speed, and thus the greater KE. In order to remove that KE, work must be done, where W = Fd. Because the force is the same in both cases, the distance needed to stop the less massive ball must be bigger.
A bowling ball and a Ping-Pong
ball are rolling toward you with
the same momentum. If you
exert the same force to stop each
one, for which is the stopping
distance greater?
a) the bowling ball
b) same distance for both
c) the Ping-Pong ball
d) impossible to say
Conservation of Linear Momentum
The net force acting on an object is the rate of change of its momentum:
If the net force is zero, the momentum does not change!
•A vector equation•Works for each coordinate separately
With no net force:
Internal Versus External Forces
Internal forces act between objects within the system.
As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero:
Therefore, the net force acting on a system is the sum of the external forces acting on it.
Momentum of components of a systemInternal forces cannot change the momentum of a system.
However, the momenta of components of the system may change.
An example of internal forces moving components of a system:
With no net external force:
Kinetic Energy of a SystemAnother example of internal forces moving components of a system:
The initial momentum equals the final (total) momentum.
But the final Kinetic Energy is very large
Opposite case:
Two identical cars travelling at identical
speeds in opposite directions collide head on.
BUT:
VERY inelastic collision!
1, 2, 1, 2,0i i f fp p p p
1, 1, 1, 1, 0i i f fK K K K
Nuclear Fission INuclear Fission I
A uranium nucleus (at rest) A uranium nucleus (at rest)
undergoes fission and splits undergoes fission and splits
into two fragments, one into two fragments, one
heavy and the other light. heavy and the other light.
Which fragment has the Which fragment has the
greater momentum?greater momentum?
a) the heavy one a) the heavy one
b) the light oneb) the light one
c) both have the same c) both have the same
momentummomentum
d) impossible to sayd) impossible to say
11 22
Nuclear Fission INuclear Fission I
A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the
greater momentum?
a) the heavy one
b) the light one
c) both have the same
momentum
d) impossible to say
1 2
The initial momentum of the uranium
was zero, so the final total momentum
of the two fragments must also be
zero. Thus the individual momenta
are equal in magnitude and opposite
in direction.
Nuclear Fission IINuclear Fission II
a) the heavy one
b) the light one
c) both have the same speed
d) impossible to say
1 2
A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the
greater speed?
Nuclear Fission IINuclear Fission II
We have already seen that the
individual momenta are equal and
opposite. In order to keep the
magnitude of momentum mv the
same, the heavy fragment has the
lower speed and the light fragment
has the greater speed.
a) the heavy one
b) the light one
c) both have the same speed
d) impossible to say
1 2
A uranium nucleus (at rest) undergoes fission and splits into two fragments, one heavy and the other light. Which fragment has the
greater speed?
Systems with Changing Mass: Rocket Propulsion
If a mass of fuel Δm is ejected from a rocket with speed v, the change in momentum of the rocket is:
The force, or thrust, is
A plate drops onto a smooth floor and shatters into three pieces of equal mass. Two of the pieces go off with equal speeds v along the floor, but at right angles to one another. Find the speed and direction of the third piece.
We know that px=0, py = 0 in initial stateand no external forces act in the horizontal
An 85-kg lumberjack stands at one end of a 380-kg floating log, as shown in the figure. Both the log and the lumberjack are at rest initially. (a) If the lumberjack now trots toward the other end of the log with a speed of 2.7 m/s relative to the log, what is the lumberjack’s speed relative to the shore? Ignore friction between the log and the water. (b) If the mass of the log had been greater, would the lumberjack’s speed relative to the shore be greater than, less than, or the same as in part (a)? Explain.