Download Physics 106P: Lecture 1 Notes

Work, Energy, and Power
Samar Hathout
KDTH 101
Work
Work is the transfer of energy through motion. In
order for work to take place, a force must be exerted
through a distance. The amount of work done depends
on two things: the amount of force exerted and the
distance over which the force is applied. There are
two factors to keep in mind when deciding when work
is being done: something has to move and the motion
must be in the direction of the applied force. Work
can be calculated by using the following formula:
Work=force x distance
Work
Work is done on the
books when they are
being lifted, but no
work is done on
them when they are
being held or
carried horizontally.
Work can be positive or
negative
•
•
•
•
Man does positive work
lifting box
Man does negative work
lowering box
Gravity does positive
work when box lowers
Gravity does negative
work when box is raised
Work done by a constant Force
• W = F s = |F| |s| cos  = Fs s
|F| : magnitude of force
F

|s| = s : magnitude of displacement
Fs = magnitude of force in
direction of displacement :
Fs = |F| cos 
: angle between displacement and force
vectors
• Kinetic energy : Ekin= 1/2 m v2
• Work-Kinetic Energy Theorem:
Ekin = Wnet
s
Work Done by Gravity

Example 1: Drop ball
Wg = (mg)(S)
Yi = h0
S = h0-hf
Wg = mg(h0-hf)
mg
S
y
= mg(h0-hf)
= Epot,initial – Epot,final
Yf = hf
x
Work Done by Gravity

Example 2: Toss ball up
Wg = (mg)(S)
S = h0-hf
Wg =-mg(h0-hf)
= Epot,initial – Epot,final
Yi = h0
mg
S
y
Yf = hf
x
Work Done by Gravity

Example 3: Slide block down incline
Wg = (mg)(S)cos
h0
Wg = mg(h/cos)cos

h
S = h/cos
hf
mg
Wg = mgh
with h= h0-hf
Work done by gravity is independent of path
taken between h0 and hf
=> The gravitational force is a conservative force.
S
Work done by a Variable Force
The magnitude of the force now depends on the
displacement: Fs(s)
Then the work done by this force is equal to the
area under the graph of Fs versus s, which can be
approximated as follows:

W = S Wi = S Fs(si) s = (Fs(s1)+Fs(s2)+…) s
Concept Question
Imagine that you are comparing three different ways of having a ball
move down through the same height. In which case does the ball reach
the bottom with the highest speed?
1.
2.
3.
4.
Dropping
Slide on ramp (no friction)
Swinging down
All the same
correct
1
2
3
In all three experiments, the balls fall from the same
height and therefore the same amount of their
gravitational potential energy is converted to kinetic
energy. If their kinetic energies are all the same, and their
masses are the same, the balls must all have the same
speed at the end.
Types of Energy
Kinetic Energy
Potential Energy
Forms of Energy
Radiant
Thermal
Electrical
Nuclear
Chemical
Sound
Mechanical
Magnetic
Mechanical Energy
Mechanical energy is the
movement of machine parts.
Mechanical energy is also the
total amount of kinetic and
potential energy in a system.
Wind-up toys, grandfather
clocks, and pogo sticks are
examples of mechanical energy.
Wind power uses mechanical
energy to help create
electricity.
Potential energy + Kinetic energy =
Mechanical energy
Mechanical Energy
Potential energy + Kinetic energy = Mechanical energy
Example of
energy changes
in a swing or
pendulum.
Conservation of Mechanical Energy

or
Total mechanical energy of an object remains constant
provided the net work done by non-conservative forces
is zero:
Etot = Ekin + Epot = constant
Ekin,f+Epot,f = Ekin,0+Epot,0
Otherwise, in the presence of net work done by
non-conservative forces (e.g. friction):
Wnc = Ekin,f – Ekin,0 + Epot,f-Epot,i
Example Problem
Suppose the initial kinetic and potential energies of a system are 75J
and 250J respectively, and that the final kinetic and potential energies
of the same system are 300J and -25J respectively. How much work
was done on the system by non-conservative forces?
1. 0J
2. 50J
correct
3. -50J
4. 225J
5. -225J
Work done by non-conservative forces equals the
difference between final and initial kinetic energies
plus the difference between the final and initial
gravitational potential energies.
W = (300-75) + ((-25) - 250) = 225 - 275 = -50J.
Samar Hathout
Kinetic Energy
1 2
KE = mv
2
Same units as work
Remember the Eq. of motion
v 2f
vi2

= ax
2
2
Multiply both sides by m,
1 2 1 2
mv f  mvi = max
2
2
KE f  KEi = Fx
Samar Hathout
Example
Samar Hathout
Potential Energy

Potential energy exists whenever an
object which has mass has a position
within a force field (gravitational,
magnetic, electrical).
We will focus primarily on
gravitational potential energy
(energy an object has because of
its height above the Earth)
Potential Energy
If force depends on distance,
PE = Fx
For gravity (near Earth’s surface)
PE = mgh
Samar Hathout
Conservation of Energy
PE f  KE f = PEi  KEi
KE = PE
Conservative forces:
• Gravity, electrical, QCD…
Non-conservative forces:
• Friction, air resistance…
Non-conservative forces still conserve energy!
Energy just transfers to thermal energy
Samar Hathout
Example
A diver of mass m drops from
a board 10.0 m above the
water surface, as in the
Figure. Find his speed 5.00 m
above the water surface.
Neglect air resistance.
9.9 m/s
Example
A skier slides down the frictionless slope as shown.
What is the skier’s speed at the bottom?
start
H=40 m
finish
L=250 m
28.0 m/s
Example
Three identical balls are
thrown from the top of a
building with the same initial
speed. Initially,
Ball 1 moves horizontally.
Ball 2 moves upward.
Ball 3 moves downward.
Neglecting air resistance,
which ball has the fastest
speed when it hits the ground?
A)
B)
C)
D)
Ball 1
Ball 2
Ball 3
All have the same speed.
Springs (Hooke’s Law)
F = kx
Proportional to
displacement
from
equilibrium
Potential Energy of Spring
1
 PE = (kx)x
2
PE=-Fx
1 2
PE = kx
2
F
x
Example
A 0.50-kg block rests on a horizontal, frictionless
surface as in the figure; it is pressed against a light
spring having a spring constant of k = 800 N/m, with
an initial compression of 2.0 cm.
x
b) To what height h does the block rise when moving up
the incline?
3.2 cm
Power

Average power is the average rate at which a net force
does work:
Pav = Wnet / t
SI unit: [P] = J/s = watt (W)
Or
Pav = Fnet s /t = Fnet vav
Example
A 1967 Corvette has a weight of 3020 lbs. The 427
cu-in engine was rated at 435 hp at 5400 rpm.
a) If the engine used all 435 hp at 100% efficiency
during acceleration, what speed would the car attain
after 6 seconds?
b) What is the average acceleration? (in “g”s)
a) 120 mph
b) 0.91g
Example
Consider the Corvette (w=3020 lbs) having constant
acceleration of a=0.91g
a) What is the power when v=10 mph?
b) What is the power output when v=100 mph?
a) 73.1 hp
b) 732 hp
(in real world a is larger at low v)