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Physics C
Energy and Momentum
When does a force do positive work?
Name:___________________
b) The work done by the rope on the sled
When does a force do zero work?
c) The work done by friction on the sled.
When does a force do negative work?
Units of Work
SI System:
Dot product equation for work:
British System:
cgs System:
How to evaluate the dot product for work if
given magnitudes and directions of vectors:
Atomic Level:
Problem: A droplet of water of mass 50 mg falls
at constant speed under the influence of gravity
and air resistance. After the drop has fallen 1.0
km, what is the work done by a) gravity and b)
air resistance?
How to evaluate the dot product for work if
given unit vector representation of vectors:
Applications of the dot product:
Problem: A sled loaded with bricks has a mass
of 20.0 kg. It is pulled at constant speed by a
rope inclined at 25o above the horizontal, and it
moves a distance of 100 m on a horizontal
surface. If the coefficient of kinetic friction
between the sled and the ground is 0.40,
calculate:
a) The tension in the rope.
When is the dot product biggest?
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• Problem: Determine the work done by the force as the
particle moves from x = 2 m to x = 8 m.
Problem: Vector A has a magnitude of 8.0 and
vector B has a magnitude of 12.0. The two
vectors make an angle of 40o with each other.
Find A•B.
F (N)
40
20
0
2
4
6
8
10
12
x (m)
-20
-40
Problem: A force F = (5.0i + 6.0j – 2.0k)N acts
on an object that undergoes a displacement of r
= (4.0i – 9.0j + 3.0k)m. How much work was
done on the object by the force?
Problem: A force acting on a particle is Fx = (4x
– x2)N. Find the work done by the force on the
particle when the particle moves along the x-axis
from x= 0 to x = 2.0 m.
Problem: A force F = (5.0i – 3.0j) N acts upon a
body which undergoes a displacement d = (2.0i –
j) m. How much work is performed, and what is
the angle between the vectors?
Problem: Derive an expression for the work
done by a spring as it is stretched from its
equilibrium position
Formula for calculating work for infinitesimal
displacement:
Problem: How much work does an applied force
do when it stretches a nonlinear spring where the
force varies according to the expressions F =
(300 N/m) x – (25 N/m2) x2 from its equilibrium
length to 20 cm?
Integral expression for work when force is
variable with position:
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Define what is meant by “net work”
What is the power expended by the probe’s
engines? The acceleration due to gravity of
Zombie is ½ that of earth’s.
State the work-energy theorem in equation form:
Problem: Develop an expression for the power
output of an airplane cruising at constant speed v
in level flight. Assume that the aerodynamic drag
force is given by FD = bv2.
Give an equation for kinetic energy:
Problem: A force of F1 = (4.0 i + j) N and
another of F2 = -4.0 j N act upon a 1 kg object at
rest at the origin. What is the speed of the object
after it has moved a distance of 3.0 m?
By what factor must the power be increased to
increase airspeed by 25%?
Problem: Using what you know about units,
calculate how many Joules is in a kilowatt-hour.
Define power:
Give several equations for calculating power:
Give the SI unit of power:
Conservative forces:
Work in moving an object is path
independent.
Work in moving an object along a closed
path is zero.
Work is directly related to a negative change
in potential energy
Ex: gravity, electrostatic, magnetostatic,
springs
Problem: A 1000-kg space probe lifts straight
upward off the planet Zombie, which is without
an atmosphere, at a constant speed of 3.0 m/s.
Non-conservative forces:
Work is path dependent.
Work along a closed path is NOT zero.
Work may be related to a change in
mechanical energy, or thermal energy
Ex: friction, drag, magnetodynamic
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Define potential energy:
Problem: Gravitational potential energy for a
body a large distance r from the center of the
earth is defined as shown below. Derive this
equation from the Universal Law of Gravity.
Give examples of potential energy:
How does potential energy relate to work?
Law of Conservation of Mechanical Energy
Gravitational potential energy:
Problem: A single conservative force of F = (3i
+ 5j) N acts on a 4.0 kg particle. Calculate the
work done if the particle if the moves from the
origin to r = (2i - 3j) m. Does the result depend
on path?
Spring potential energy:
What is the speed of the particle at r if the speed
at the origin was 4.0 m/s?
Problem: Three identical springs (X, Y, and Z)
are hung as shown. When a 5.0-kg mass is hung
on X, the mass descends 4.0 cm from its initial
point. When a 7.0-kg mass is hung on Z, how far
does the mass descend?
What is the change in potential energy of the
system?
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Sample Problem: A bead slides on the loop-theloop shown. If it is released from height h = 3.5
R, what is the speed at point A? How great is the
normal force at A if the mass is 5.0 g?
What relationship exists between force and
potential energy?
Stable Equilibrium:
General statement of the Law of Conservation of
Energy (including dissipative forces).
Unstable Equilibrium
Sample Problem: A 2,000 kg car starts from
rest and coasts down from the top of a 5.00 m
long driveway that is sloped at an angel of 20o
with the horizontal. If an average friction force
of 4,000 N impedes the motion of the car, find
the speed of the car at the bottom of the
driveway.
Neutral Equilibrium
Equation that relates force and potential energy:
Problem: A parachutist of mass 50 kg jumps out
of a hot air balloon 1,000 meters above the
ground and lands on the ground with a speed of
5.00 m/s. How much energy was lost to friction
during the descent?
Stable equilibrium equations:
Unstable equilibrium equations:
Neutral equilibrium equations:
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Problem: The potential energy of a two-particle
system separated by a distance r is given by U(r)
= A/r, where A is a constant. Find the radial
force F that each particle exerts on the other.
What is the momentum of the system of football
players?
Define Impulse:
Problem: A potential energy function for a twodimensional force is of the form U = 3x3y – 7x.
Find the force acting at a point (x,y).
Equations for impulse:
Problem: Restate Newton’s 2nd Law in terms of
impulse.
Define linear momentum:
Problem: A 150-g baseball moving at 40 m/s
15o below the horizontal is struck by a bat. It
leaves the bat at 55 m/s 35o above the horizontal.
What is the impulse exerted by the bat on the
ball?
Equations for linear momentum:
Problem: How fast must an electron move to
have the same momentum as a proton moving at
300 m/s?
If the collision took 2.3 ms, what was the
average force?
Problem: A 90-kg tackle runs north at 5.0 m/s
and a 75-kg quarterback runs east at 8.0 m/s.
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State the Law of Conservation of Linear
Momentum
What kinds of energy and momentum changes
characterize each type of collision?
Elastic:
Inelastic:
Problem: An 85-kg lumberjack stands at one
end of a floating 400-kg log that is at rest relative
to the shore of a lake. If the lumberjack jogs to
the other end of the log at 2.5 m/s relative to the
shore, what happens to the log while he is
moving?
Explosion:
Problem: A 1.5 kg cart traveling at 1.5 m/s
collides with a stationary 0.5 kg cart and sticks to
it. At what speed are the carts moving after the
collision?
Problem: Two blocks of mass 0.5 kg and 1.5 kg
are placed on a horizontal, frictionless surface. A
light spring is compressed between them. A cord
initially holding the blocks together is burned;
after this, the block of mass 1.5 kg moves to the
right with a speed of 2.0 m/s. A) What is the
speed and direction of the other block? B) What
was the original elastic energy in the spring?
Problem: A 1.5 kg cart traveling at 1.5 m/s
collides elastically with a stationary 0.5 kg cart.
At what speed are each of the carts moving after
the collision?
Describe an elastic collision:
Problem: What is the recoil velocity of a 120-kg
cannon that fires a 30-kg cannonball at 320 m/s?
Describe an inelastic collision:
Describe an explosion:
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Sample problem: A pool player hits a cue ball
in the x-direction at 0.80 m/s. The cue ball
knocks into the 8-ball, which moves at a 35o
angle above the x-axis. Assuming the collision is
elastic, determine the angle of deflection of the
cue ball.
Center of Mass Problem
y
2R
x
How do you calculate the center of mass for a
system of point masses?

Sample Problem
y
Problem: Find the x-coordinate of the center of
mass of a rod of length L whose mass per unit
length varies according to the expression  = x.
Determine the
Center of Mass.
2 kg
Problem: A thin strip of material of mass M is
bent into a semicircle of radius R. Find its center
of mass.
x
0
1 kg
-2
-2
0
Determine the
center of mass
of this shape.
For more complicated problems, how do you
determine the center of mass?
Define center of mass
2

3 kg
2
4
Equation of motion of a System of Particles
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