Conservation of Energy
Energy is Conserved!
Energy is Conserved!
The total energy (in all forms) in a “closed”
system remains constant
 This is one of nature’s “conservation laws”


Conservation applies to:





Energy (includes mass via E = mc2)
Momentum
Angular Momentum
Electric Charge
Conservation laws are fundamental in physics,
and stem from symmetries in our space and time
Conversion of Energy
A Falling object converts gravitational
potential energy into kinetic energy
 Friction converts kinetic energy into
vibrational (thermal) energy
 makes things hot (rub your hands together)
 it is an irretrievable energy
 Doing work on something changes that object’s
energy by amount of work done, transferring
energy from the agent doing the work

Energy Conservation
•Energy can neither be created nor destroyed
•Energy can be converted from one form to another

Need to consider all possible forms of energy in a
system e.g:
 Kinetic energy (1/2 mv2)
 Potential energy (gravitational mgh,
electrostatic)
 Electromagnetic energy
 Work done on the system
 Heat (1st law of thermodynamics of Lord Kelvin)
 Friction  Heat
Energy is measured in Joules [J]
Conservation of Energy
E pf  Ekf  E pi  Eki
0  Ek  Ep
Conservative forces:
• Gravity, electrical …
Non-conservative forces:
• Friction, air resistance…
Non-conservative forces still conserve
energy! Energy just transfers to thermal
energy
A small child slides down the four
frictionless slides A–D. Each has the same
height. Rank in order, from largest to
smallest, her speeds vA to vD at the bottom.
A. vA = vB = vC = vD
B. vD > vA > vB > vC
C. vD > vB > vC > vC
D. vC > vA > vB > vD
E. vC > vB > vA > vD
Conservation of Energy
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. Ball 1
B. Ball 2
C. Ball 3
D. they have the
same speed.
Falling (elastic) ball

a

b

c

d

e

f

g
EPgrav
EPgrav+ EK
EK
EK + EPelastic
EPelastic
EK
EPgrav
Conservation of Energy
Energy Conservation in a Pendulum
Initially the pendulum has Epgrav
Then some Epgrav is changed to Ek but still retains some Epgrav
At the lowest point all the Epgrav is now Ek
Which then converts back to all Epgrav at the top.
Energy Conservation
Demonstrated
Roller coaster car lifted to initial height (energy in)
 Converts gravitational potential energy to motion
 Fastest at bottom of track
 Re-converts kinetic energy back into potential as it
climbs the next hill

Conservation
of Energy
 Total
Energy
= EPgrav + EK
Energy Conservation

The kinetic energy for a mass in motion is
EK = ½mv 2
Book dropped from rest at a height h (EPgrav =
mgh) the book hits the ground with speed v.
Expect ½mv 2 = mgh
 s = h = ut + ½gt 2 where u = 0
 v = u + gt  v 2 = g 2t 2
 mgh = mg  (½gt 2) = ½mg 2t 2 = ½mv 2
sure enough!
 Book has converted its available gravitational
potential energy into kinetic energy: the
energy of motion
Loop-the-Loop

In the loop-the-loop (like in a roller
coaster), the velocity at the top of the loop
must be enough to keep the train on the
track:
v 2/r > g

Works out that train must start ½ r higher
than top of loop to stay on track, ignoring
frictional losses
½r
r
Conservation of Energy: Potential and Kinetic
1.
What is the total energy of the sledder (m = 50 kg) at the top of
the hill?
2. What is the total energy, gravitational potential energy and the
kinetic energy on top of the 15 m bump? Speed?
3. What is the total energy, gravitational potential energy and the
kinetic energy at the bottom of the hill? Speed?
4. How much work was done on the sledder when he was pulled up the
hill by his brother?
Conservation of Energy
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
Conservation of Energy
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
a) After the block is released, find the speed of
the block at the bottom of the incline, position (B).
b) Find the maximum distance d the block travels up
the frictionless incline if the incline angle θ is 25°.
a) 0.8 m/s
b) 7.7 cm
Conservation of Energy
Einitial = Efinal
Example 5.2
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 5.4
Two blocks, A and B (mA=50 kg and
mB=100 kg), are connected by a string as
shown. If the blocks begin
at rest, what will their speeds be after A
has slid
a distance s = 0.25 m? Assume the pulley
and incline are frictionless.
1.51 m/s
s
Example 5.7
A spring-loaded toy gun shoots a 20-g
cork 10 m into the air after the
spring is compressed by a distance of
1.5 cm.
a) What is the spring constant?
b) What is the maximum acceleration
experienced by the cork?
a) 17,440 N/m
b) 13080 m/s2
Mechanical Energy Conservation
W ork = Fd  d =  K E
(W ork - E nergy T heorem )
C onsider the G ravitational Force (a conservative force):
m g(h i  h f )  K E f  K E i
PE i  PE f  KE f  KE i
Re-arranging the equation slightly gives:
 K E =   PE
Re-arranging the same equation slightly differently, we can write:
PE i  KE i  PE f  KE f
We define Mechanical Energy to be the sum of kinetic and
Potential energy:
E = KE + PE
Mechanical Energy Conservation (cont)
PE i  KE i  PE f  KE f
We define Mechanical Energy to be the sum of kinetic and
Potential energy:
E = KE + PE
For the gravitational force (conservative force), we then
have:
Ei  Ef
Example: the tallest roller coaster
The tallest and fastest roller coaster is now the Steel
Dragon in Japan. It has a vertical drop of 93.5
meters. At the top of the drop, cars have a velocity
of 3 m/s. What is the speed of the car at the bottom
of the drop (neglecting friction and air resistance)?
Ei  Ef
m gh i +
vf 
1
2
m v  m gh f 
2
i
2 g(h i  h f )  v
2
i
1
2
2
mvf
 42 .9 m / s
Problem: Roller coaster
m
v0=0
g
h
v=?
y=0
Note: Assume no friction.
Normal force does no work, so irrelevant.
Final velocity is independent of the path taken!!
Example:
Two boxes rest on frictionless ramps. One (the
small box) has less mass than the other. They are
released from rest and allowed to slide. Which
box, if either, has the greater speed at B? Which, if
either, has the greatest kinetic energy?
10 m
8m
Energy
Conversion/Conservation
P.E. = 98 J
Example
K.E. = 0 J
 Drop 1 kg ball from 10 m
P.E. = 73.5 J
K.E. = 24.5 J
6m
P.E. = 49 J
K.E. = 49 J
4m
2m
0m
P.E. = 24.5 J
K.E. = 73.5 J
P.E. = 0 J
K.E. = 98 J
out with mgh = (1 kg)(9.8
m/s2)(10 m) = 98 J of gravitational
potential energy
 halfway down (5 m from floor), has
given up half its potential energy (49
J) to kinetic energy
 starts
 ½mv2
m/s
= 49 J  v2 = 98 m2/s2  v  10
 at
floor (0 m), all potential energy is
given up to kinetic energy
 ½mv2
= 98 J  v2 = 196 m2/s2  v =