Transcript Potential Energy and Conservation of Energy

Physics 111: Mechanics
Lecture 7
Bin Chen
NJIT Physics Department
Chapter 7 Potential Energy and
Energy Conservation





7.1 Gravitational Potential Energy
7.2 Elastic (Spring) Potential Energy
7.3 Conservative and Non-conservative Forces
7.4* Force and Potential Energy
7.5* Energy Diagrams
Definition of Work W

The work, W, done by a constant force on an
object is defined as the product of the component
of the force along the direction of displacement
and the magnitude of the displacement
𝑊 =𝑭∙𝒔
= 𝐹𝑠 cos𝜃



F is the magnitude of the force
s is the magnitude of the
object’s displacement
q is the angle between F and s
andDFx
Work by gravitation force
A 10-kg cannonball was fired from ground
toward a target located in an enemy
compound that is 20 m away and 15 m high.
How much work is done by the gravitational
force on the cannonball as it just hits the
target?
A. 1.5 kJ; B. -1.5 kJ; C. 2.0 kJ;
D. -2.0 KJ; E. 3.2 kJ
Wgrav
mgy1 mgy2
Work done by Gravity and
Gravitational
Potential Energy
U
mgy (gravitational potential energy)
Wgrav
Ugrav
rav
grav
Fs w y1 y2
Wgrav Ugrav,1 Ugrav,2
mgy
mgy1 mgy2
Ugrav,2
Ugrav,1
(
Ugrav
(gravitational potential energy)
K1 Ugrav,1 K2 Ugrav,2
(if only gravity does work)
Ugrav,1 Ugrav,2
Ugrav,2 Ugrav,1
Ugrav
(
1
1
2
Ugrav,1
2
Fs w y1 y2
m
mgy1
2
1
1
2
m
2
2
K2 Ugrav,2
Uel
mgy1
1
2
m
1
2
2
2
1
mgy2 (if only gravity does work)
(if only gravity does work)
kx2
(
(elastic potential energy)
mgy2 (if only gravity does work)
2
(
1
2
(
Potential Energy




Potential energy is associated with the
position of the object
Gravitational Potential Energy is the
energy associated with the relative
position of an object in space near the
Earth’s surface
Shared by both the object and Earth
The gravitational potential energy




m is the mass of an object
g is the acceleration of gravity
y is the vertical position of the mass
relative the surface of the Earth
SI unit: joule (J)
Reference Level

A location where the gravitational potential
energy is zero must be chosen for each
problem


The choice is arbitrary since the change in the
potential energy is the important quantity
Choose a convenient location for the zero
reference height



often the Earth’s surface
may be some other point suggested by the problem
Once the position is chosen, it must remain fixed
for the entire problem
Reference Levels
 Choose
the correct answer. The
gravitational potential energy of an
object
(a) is always positive
(b) is always negative
(c) never equals to zero
(d) can be negative or positive
W
Fscos
(constant force, straight-line d
Work-Kinetic
W FEnergy
s (constantTheorem
force, straight-line disp
When work is done by a net force on an object
2
1
m
(definition
of kinetic ene
and the only changeKin the
object
is its speed,
2
the work done is equal to the change in the
object’s kinetic energy
The work done by the net force on a particle equals the chan
 Speed will increase if work is positive
kinetic energy:


Speed will decrease if work is negative
Wtot
W
x2
x1
K2 K1
K (work energy theorem
1
1
2
𝑊tot = 𝑚𝑣2 − 𝑚𝑣12
2
2
Fx dx (varying x-component of force, straight-
K
y Equations
1
2
m
2
(definition of kinetic energ
Extended Work-Energy Theorem with
Wgrav Fs w y1 y2
mgy1 mgy2
Potential
Energy
The workGravitational
done by the net force
on a particle
equals the chang
 Theenergy:
work-kinetic
energy
to
kinetic
Wgrav
Fs theorem
w y1 can
y2 be extended
mgy1 mgy
2
include
gravitational
potential energy:
Ugrav
mgy (gravitational
potential energy)
Wtot K2 K1
K (work energy theorem)
Wgrav
Ugrav
U grav,1
mgy
Ugrav,2
(gravitational potential energy)
Ugrav,2
Ugrav,1
Ugrav
x2

W only have
Fx dx gravitational
(varying x-component
of all
force,
straight-lin
If we
force
and
work
x1
W
Ugrav,2are(ifzero,
Ugrav,2
Ugrav,1
by allU
then
grav,1
Kdone
U grav
Krest
Uforces
only
gravity
does work) U grav
1
grav,1
2
grav,2
𝑊𝑡𝑜𝑡 = 𝑊𝑔𝑟𝑎𝑣
P2
P2
P2
W
F cos 2 dl
F||dl
F d (work done on
1 K
1K
P1
P1 (If only
Pgravity
U
U
(if
only
gravity
does work
1
does
work)
m
mgy
m
mgy
(if
only
gravity
does work)
grav,1
2
grav,2
1
2
2
2
2
2
11
1
m
2U
mgykx
el
1
2
21
m
W
energy)
P potential
(average
mgy
(if only
gravitypower)
does work
2
(elastic
grav
1
Wgrav
2
1
Fs w y1 y2
2
mgy1 mgy2
(gravitational potential
energy)
ConservationU of mgy
Mechanical
Energy
grav




Ugrav
mgy
(gravitational potential energy
We denote the total mechanical energy by
Wgrav U grav,1 Ugrav,2
Ugrav,2
W
U
U
grav,2
K1gravUgrav,1 grav,1
K2 Ugrav,2
Since
So
Ugrav,1
U
Ugrav
U
grav,1
(if grav,2
only gravity
does work
2
2
K112 mU1grav,1
only
gravity
mgy1K2 12 mUgrav,2
mgy2 (if(if
only
gravity
doesdoes
work
2
The total mechanical energy is conserved and remains
the same at all times
2
1
U
kx
2
2 (elastic potential energy)
1
el
21
2
m
1
mgy1
2
m
2
mgy2 (if only gravity does
(If only gravity does
work)
2
1
1
Wel 2 kx1 2 kx22 Uel,1 Uel,2
Uel
1
2
kx
2
Uel
(elastic potential energy)
Problem-Solving Strategy
Define the system
 Select the location of zero gravitational
potential energy



Do not change this location while solving the
problem
Identify two points the object of interest moves
between


One point should be where information is given
The other point should be where you want to find
out something
Platform Diver

A diver of mass m drops
from a board 10.0 m above
the water’s surface. Neglect
air resistance.

Find his speed as he hits the
water
Platform Diver

Find his speed as he hits the water
1 2
1
mvi  mgyi  mv2f  mgy f
2
2
1 2
0  mgyi  mv f  0
2
v f  2 gyi  14m / s
Skateboarding

A boy skateboards from rest down a curved frictionless
ramp. He moves through a quarter-circle with radius R=3m.
The boy and his skateboard have a total mass of 25 kg.

Find his speed at the bottom of the ramp
Skateboarding

Find his speed at the bottom of the ramp
K1  0
U grav,1  mgR
K 2  12 mv22
U grav, 2  0
0  mgR 
1 2
mv2  0
2
1 2
1
mv1  mgy1  mv22  mgy2
2
2
v2  2gR  7.67m / s
Prof. Walter Lewin’s Pendulum
 https://www.youtube.com/watch?v=xXXF
2C-vrQE
Conservation of Mechanical Energy

Three identical balls are thrown from the top of a
building, all with the same initial speed. The first ball
is thrown horizontally, the second at some angle
above the horizontal, and the third at some angle
below the horizontal. Neglecting air resistance, rank
the speeds of the balls as they reach the ground,
from fastest to slowest.
(a) 1, 2, 3
(b) 2, 1, 3
(c) 3, 1 ,2
(d) 3, 2, 1
(e) all three ball strike the ground at the same speed
K
pter 7 Key Equations
2
m
(definition of kinetic
The work
done by
the netGravity
force on a particle
equals
the
When Forces
other
than
Do
Work
W
Fs w y y
mgy mgy
1
2
1
2
kinetic grav
energy:
 The work-kinetic energy theorem can be extended to
include potential
energy:
Wtot K2 potential
K1
K
(work energy the
U
mgy (gravitational
energy)
grav
𝑊𝑡𝑜𝑡 = 𝑊𝑔𝑟𝑎𝑣 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 − 𝐾1


Since
Then
Wgrav
x2
U grav,1W Ugrav,2Fx dx U(varying
Ugrav,1
Ugravof force, stra
x-component
grav,2
x1
𝑊𝑜𝑡ℎ𝑒𝑟 + 𝑈𝑔𝑟𝑎𝑣,1 − 𝑈𝑔𝑟𝑎𝑣,2 = 𝐾2 − 𝐾1
K1 Ugrav,1
K2 Ugrav,2P2
(if only gravity
does
P2
P2 work)
Fdo
coswork
dl
F||dl
F d (work do
 If forces other than W
gravity
P1
P1
P1
1
2
𝐾1 2+ 𝑈
𝑊2 𝑜𝑡ℎ𝑒𝑟
=(if𝐾only
𝑔𝑟𝑎𝑣,11 +
2+𝑈
𝑔𝑟𝑎𝑣,2does work)
m
mgy
m
mgy
gravity
1
1
2
2
2
W
1
1
𝑚𝑣12 + 𝑚𝑔𝑦11 +2 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝑚𝑣22 P
+av 𝑚𝑔𝑦t2
Uel 2 kx
(elastic 2potential energy)
2
Wel
1
2
kx12
1
2
kx22
UPel,1 lim
Uel,2
W
dW
Uel
(average power
(instantaneous
Lifting bucket from well

A person lifted a 10.0-kg bucket from the
bottom of a 10-m deep well. The bucket started
from rest and had a 1.00 m/s speed as it
reached the top. How much work, in J, is done
by this person? (g=9.8 m/s2. Neglect friction.)
A.
B.
C.
D.
E.
975
985
980
-985
-975
Spring Force: An Elastic Force

Hooke’s Law gives the force
𝐹𝑥 = 𝑘𝑥




Where 𝐹𝑥 is the force exerted on
the spring in the same direction
of x
The force exerted by the spring
is 𝐹𝑠 = −𝐹𝑥 = −𝑘𝑥
k is the spring constant. Unit: N/m.
Work done on the spring from x1
to x2
x2
x2
x1
x1
W   Fx dx   kxdx 
1 2 1 2
kx2  kx1
2
2
Potential Energy in a Spring

Work done by the spring
𝑥2
𝑊𝑒𝑙 =
𝑥1

1 2 1 2
−𝑘𝑥 𝑑𝑥 = 𝑘𝑥1 − 𝑘𝑥2
2
2
Elastic Potential Energy:


SI unit: Joule (J)
related to the work required to compress
a spring from its equilibrium position to
some final, arbitrary, position
m
1
2
1
2
m
mgy1
2
1
1
2
K1 Ugrav,1
m
2
2
mgy2 (if only gravity does
K2 UKgrav,212 m 2 (definition
only gravity
doesenergy)
work)
of kinetic
Work-Energy(ifTheorem
with
Extended
m 12 mgy1 12 m 22 mgy2 (if only gravity does work)
Elastic
Potential
Energy
2
1
U
kx
(elastic potential energy)
2
2
1
1 2 force
The work
done
by the
net
on a particle
change
el
m
mgy
m
mgy
(if only equals
gravitythe
does
work)in the p
1
1
2
2
2
2
 kinetic
The work-kinetic
energy theorem can be extended to
energy:
2
1
Uel elastic
(elastic
potential energy)
include
energy:
2 kx potential
K22 2K1(elastic
(work energy
theorem)
1 K 2potential
UelWtot 12 1kx
energy)
Wel

1
2
kx12
1
2
x2
2
kx1
2
kx2
kx22 Uel,1 Uel,2
Uel,1 Uel,2
Uel
Uel
2
2
1
1
W have
F
dx
(varying
x
-component
of
force, straight-line
displa
W
kx
kx
Uand
U
U
If we only
spring
force
all
done
x
1
2
el,1
el,2work
el
2
2
x1 el
by all
forces
zero, then 𝑊the=elastic
Krest
Kare
force does
𝑊
1 Uel,1
2 Uel,2 (if only
K1 Uel,1
2
1
Wel
𝑒𝑙
K2 Uel,2 (if only the elastic force𝑡𝑜𝑡
does work)
P2 only the
P2 elastic force does work)
K1 Uel,1P2 K2 Uel,2 (if
(If
the elastic
W
F cos dl
F||dl
F only
d (work
done on a curved
P1
P1
P1
force does work)
2
2
1 1 2 2 1 12
1
m
kx
11 kx 2 2 mv
kx
22 1 1 2 2 2 2 12 2 22
1
2
m
1
K1 U1
kx22 (if only the elastic force does work)
W
Pav
(average power)
Wother K2 U2
(validt in general)
K
in general)
K1 UU1 WWotherK K
U2 U2(valid in(valid
general)
1
2
kx1
elastic
force the
doeselastic
work) force
mv2(if only
kxthe
(if only
2
2
1
2
mv22
1
2
Hitting a Spring

A 2 kg block slides with no friction and with an
initial speed of 4 m/s. It hits a spring with
spring constant k=1400 N/m. The block
compresses the spring in a straight line for a
distance 0.1 m. What is the block’s speed, in
m/s, at that point?
A.
B.
C.
D.
E.
1.0
2.0
3.0
4.0
5.0
K1 Ugrav,1
K2 Ugrav,2
(if only gravity does work)
K
1
2
m
2
(
(definition of kinetic energy)
Extended Work-Energy Theorem
2
2
1
1
m
mgy
m
mgy2 (ifand
only Elastic
gravity does
work) Energy (
with
BOTH
Gravitational
Potential
1
1
2
2
2
The work done by the net force on a particle equals the change in the
 The
work-kinetic
energy theorem can be extended to
kinetic
energy:
include both
of potential energy:
2
1 types
Uel 2 kx
(elastic potential energy)
Wtot
K2
K1
K (work energy theorem)
(
𝑊𝑔𝑟𝑎𝑣 = 𝑚𝑔𝑦1 − 𝑚𝑔𝑦2 = 𝑈𝑔𝑟𝑎𝑣,1 − 𝑈𝑔𝑟𝑎𝑣,2 = −∆𝑈𝑔𝑟𝑎𝑣
Wel 12 kx1x2 12 kx22 Uel,1 Uel,2
Uel
(7
W
Fx dx (varying x-component of force, straight-line displ
x
 If the system only involves gravitational force and
spring force or all work done by all rest forces are zero,
Kthen
K2 U
(if
only
the+Pelastic
force
does work)
(7
1 Uel,1
P
𝑊Pel,2
=
𝑊
𝑊
𝑡𝑜𝑡
𝑔𝑟𝑎𝑣
𝑒𝑙
2
1
W
1
2
m
2
P1
F cos dl
2
P1
2
F||dl
P1
F d
(work done on a curved
𝐾1 + 𝑈𝑔𝑟𝑎𝑣,1 + 𝑈𝑒𝑙,1 = 𝐾2 + 𝑈𝑔𝑟𝑎𝑣,2 + 𝑈𝑒𝑙,2
2
1
1
2
kx12 1 12 mv22
1
2
kx22 1 (if only
1 the elastic force
1 does work)
𝑚𝑣12 + 𝑚𝑔𝑦1 + 𝑘𝑥12 = 𝑚𝑣22 +W𝑚𝑔𝑦2 + 𝑘𝑥22
(average
power)
2
2
2Pav
2
K1 U1 Wother
K2 U2
(7
t
(valid in general)
W
dW
(7
Mechanical Energy Conservation with BOTH
Gravitational and Elastic Potential Energy

We denote the total mechanical energy by

Since 𝐸2 = 𝐸1

The total mechanical energy is conserved and remains
the same at all times
1
1 2 1
1 2
2
2
𝑚𝑣1 + 𝑚𝑔𝑦1 + 𝑘𝑥1 = 𝑚𝑣2 + 𝑚𝑔𝑦2 + 𝑘𝑥2
2
2
2
2
A block projected up an incline



A 0.5-kg block rests on a horizontal, frictionless surface.
The block is pressed back against a spring having a
constant of k = 625 N/m, compressing the spring by
10.0 cm to point A. Then the block is released.
(a) Find the maximum distance d the block travels up
the frictionless incline if θ = 30°.
(b) How fast is the block going when halfway to its
maximum height?
A block projected up a incline
Point A (initial state):
 Point C (final state):

A block projected up a incline
Point A (initial state):
 Point B (final state):

Wel
1
2
kx12
1
2
kx22 Uel,1 Uel,2
Uel
If other
forces
do
work
K U
K U (if only the elastic force does wo
1

el,1
2
el,2
If work done by other forces that cannot be described
in terms
of 2potential
2 energy
2
2
1
1
1
1
2
m
1
2
kx1
2
mv2
2
kx2
(if only the elastic force do

General case

If Wother is positive, E = K + U increases
K U E =UKint+ U0 decreases
(law of conservation of energy)
If Wother is negative,
If Wother is zero,
E = K + U keeps constant


Fx x
K1 U1 Wother
dU x
dx
K2 U2
(valid in general)
(force from potential energy, one dime
Types of Forces

Conservative forces



Work and energy associated
with the force can be recovered
Examples: Gravity, Spring Force,
EM forces
Nonconservative forces


The forces are generally
dissipative and work done
against it cannot easily be
recovered
Examples: Kinetic friction, air
drag forces, normal forces,
tension forces, applied forces …
March 4, 2015
Conservative Forces

A force is conservative if the work it does on an
object moving between two points is
independent of the path the objects take
between the points




The work depends only upon the initial and final
positions of the object
Any conservative force can have a potential energy
function associated with it
Work done by gravity
Work done by spring force
Nonconservative Forces

A force is nonconservative if the work it does
on an object depends on the path taken by the
object between its final and starting points.




The work depends upon the movement path
For a non-conservative force, potential energy can
NOT be defined
Work done by a nonconservative force
It is generally dissipative. The dispersal
of energy takes the form of heat or sound
Conservation of Energy in General

Any work done by conservative forces can be accounted
for by changes in potential energy
𝑊𝑐 = 𝑈1 − 𝑈2 = − 𝑈2 − 𝑈1 = −∆𝑈

Law of conservation of energy

Energy is never created or destroyed. It only changes
form.
Problem-Solving Strategy

Define the system to see if it includes non-conservative
forces (especially friction, drag force …)
Without non-conservative forces

With non-conservative forces

Select the location of zero potential energy



Do not change this location while solving the problem
Identify two points the object of interest moves between


One point should be where information is given
The other point should be where you want to find out something
Homework #7 review I:
Conservative Forces
A child's toy consists of a block that attaches to a table with a suction cup, a spring
connected to that block, a ball, and a launching ramp. The spring has a spring
constant k, the ball has a mass m, and the ramp rises a height y above the table, the
surface of which is a height H above the floor.
Initially, the spring rests at its equilibrium length. The spring then is compressed a
distance s, where the ball is held at rest. The ball is then released, launching it up the
ramp. When the ball leaves the launching ramp its velocity vector makes an
angle θ with respect to the horizontal. ignore friction and air resistance.


Calculate vr, the speed of the ball when it leaves the
launching ramp. Express the speed of the ball in terms
of k, s, m, g, y, and/or H.
With what speed will the ball hit the floor? Express the speed
in terms of k, s, m, g, y, and/or H.
Work-Energy Example for NonConservative Forces

A 5-N force pushes an 2-kg block across a
surface a distance 2 m. The block starts with a
speed of 4 m/s and ends with 3 m/s. How much
work, in J, does the friction do?
A. -10
B. -3
C. -17
D. 3
E. 9
Another Example:
Sliding down a slope

A crate slides down a slope with an angle of
inclination of 30 degrees. The coefficient of
moving friction between the crate and the slope
is 0.15. What is the speed of the crate after it
N
moves 1 m along the slope?
A.
B.
C.
D.
E.
2.7 m/s
3.1 m/s
3.5 m/s
1.2 m/s
Can’t tell, need mass
fk
Changes in Mechanical Energy for Non-conservative forces
A 3-kg crate slides down a ramp. The ramp is 1 m in length and
inclined at an angle of 30° as shown. The crate starts from rest at the
top. The surface in contact have a coefficient of kinetic friction of 0.15.
Use energy methods to determine the speed of the crate at the bottom
of the ramp.
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N
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Energy Review
 Kinetic

Energy
Associated with movement of members of a
system
 Potential
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
Determined by the configuration of the system
Gravitational and Elastic
 Internal

Energy
Energy
Related to the temperature of the system
Energy is conserved
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Energy cannot be created nor destroyed
It can be transferred from one object to another or
change in form
If the total amount of energy in a system changes,
it can only be due to the fact that energy has
crossed the boundary of the system by some
method of energy transfer