Transcript Lecture11-10

Reading and Review
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
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 =
1
2.
kx
2
The mass moves down to a
lower position, so its gravitational PE decreases,
because PEg = mgh.
Question 8.9 Cart on a Hill
a) 4 m/s
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?
b) 5 m/s
c) 6 m/s
d) 7 m/s
e) 25 m/s
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
When starting from rest, the
cart’s PE is changed into KE:
D PE = D KE 21 = m(4)2
When starting from 3 m/s, the
final KE is:
KEf = KEi + DKE
= 21 m(3)2 + 21 m(4)2
= 21 m(25)
= 21 m(5)2
Speed is not the same as kinetic energy
8-4 Work Done by Nonconservative Forces
In 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:
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
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.
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 I
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?
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
p
p
Going Bowling I
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?
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
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
p
p
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
Momentum 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
Momentum 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
a) greater than
b) less than
c) equal to
rate of change of the pebble’s
momentum?
The rate of change of momentum is, in fact, the force.
Remember that F = D p/Dt. Because the force exerted on
the boulder and the pebble is the same, then the rate of
change of momentum is the same.
Impulse
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 is a vector, in the same direction
as the average force.
Impulse
We can rewrite
as
So we see that
The impulse is equal to the change in momentum.
Why we don’t dive into concrete
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.
Going Bowling II
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?
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
p
p
Going Bowling II
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?
We know:
Dp
Fav =
Dt
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
so D p = Fav Dt
Here, F and Dp are the same for both balls!
It will take the same amount of time to
stop them.
p
p
Going Bowling III
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
a) the bowling ball
b) same distance for both
c) the Ping-Pong ball
d) impossible to say
greater?
p
p
Going Bowling III
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
a) the bowling ball
b) same distance for both
c) the Ping-Pong ball
d) impossible to say
greater?
Use the work-energy theorem: W = DKE. 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.
p
p
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!
With no net force:
•A vector equation
•Works for each coordinate separately
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 system
Internal forces cannot change the
momentum of a system.
However, the momenta of components
of the system may change.
With no net external force:
An example of internal forces moving
components of a system:
Kinetic Energy of a System
Another 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.
p1,i  p2,i = 0 = p1, f  p2, f
BUT:
K1,i  K1,i  K1, f  K1, f = 0
VERY inelastic collision!
Nuclear Fission I
A uranium nucleus (at rest)
undergoes fission and splits
into two fragments, one
heavy and the other light.
Which fragment has the
a) the heavy one
b) the light one
c) both have the same momentum
d) impossible to say
greater momentum?
1
2
Nuclear Fission I
A uranium nucleus (at rest)
undergoes fission and splits
into two fragments, one
heavy and the other light.
Which fragment has the
a) the heavy one
b) the light one
c) both have the same momentum
d) impossible to say
greater momentum?
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.
1
2
Nuclear Fission II
A uranium nucleus (at rest)
undergoes fission and splits
into two fragments, one
heavy and the other light.
Which fragment has the
a) the heavy one
b) the light one
c) both have the same speed
d) impossible to say
greater speed?
1
2
Nuclear Fission II
A uranium nucleus (at rest)
undergoes fission and splits
into two fragments, one
heavy and the other light.
Which fragment has the
a) the heavy one
b) the light one
c) both have the same speed
d) impossible to say
greater speed?
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.
1
2
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 state
and 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.