Conservation of Energy
Forms of Energy
Mechanical Energy
Ug UsK
Thermal Energy
Eth
Other forms include
Echem Enuclear
Law of Conservation of Energy
What you put in is what you get out
Total energy is conserved
Practical Applications
Gasoline converts to energy which moves the car
A battery converts stored chemical energy to electrical energy
Dams convert the kinetic energy of falling water into electrical energy
Example : step 1
Example : step 2
Example : step 3
Example : last step
Conservation of Mechanical Energy
Emghmv 2
2
1
Total Energy
Potential Energy
Kinetic Energy
m = mass
v = velocity
g = gravitational acceleration
h = height
ILYA, did you know that
even though it was a
bumpy ride, our energy remained constant!
Conservation of Mechanical Energy We denote the total mechanical energy by
Since
The total mechanical energy is conserved and remains the same at all times
PEKEE
iiff KEPEPEKE
ffii mghmvmghmv 22
2
1
2
1
A block projected up a incline Point A (initial state): Point B (final state):
mcmxyv iii 1.010,0,0
m
smkg
mmN
mg
kxd i
28.1
30sin)/8.9)(5.0(
)1.0)(/625(5.0
sin
2
2
221
2222
2
1
2
1
2
1
2
1fffiii kxmgymvkxmgymv
0,sin,0 fff xdhyv
sin2
1 2 mgdmgykx fi
Trucks with the noted masses moving at the noted speeds crash into barriers that bring them to rest with a constant force. Which truck compresses the barrier by the largest distance?
Questions :
2 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 …
The work done by a conservative force is independent of the path, and depends only on the starting and ending points.
Closed path, W=0. Pick any starting and ending points.
A
B
A
B
W1
W2
W1 = WAB
W2 = WBA
W1 + W2 = 0W3
W1 + W3 = 0
So, all paths from B to A take the same amount of work.
W1
Path doesn’t matter!
gHv
gHv
MvMgH
2
2
02
10
2
2
Initial Final
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
fifig mgymgyPEPEW
22
2
1
2
1fisfsis kxkxPEPEW
Ex n.1: Block-Spring Collision A block having a mass of 0.8 kg is given an initial velocity vA = 1.2
m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Assuming the surface to be frictionless, calculate the maximum compression of the spring after the collision.
msmmN
kgvk
mx A 15.0)/2.1(
/50
8.0max
002
100
2
1 22max Amvmv
2222
2
1
2
1
2
1
2
1iiifff kxmghmvkxmghmv
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
sotherforceknc WdfdFW
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
2222
2
1
2
1
2
1
2
1iiifff kxmghmvkxmghmv
)()( iiffnc PEKEPEKEW
)2
1
2
1()
2
1
2
1( 2222
iiifffsotherforce kxmghmvkxmghmvWfd
Ex n 2:Block-Spring Collision A block having a mass of 0.8 kg is given an initial velocity vA = 1.2
m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Suppose a constant force of kinetic friction acts between the block and the surface, with µk = 0.5, what is the maximum compression xc in the spring.
)002
1()
2
100(0 22 Ack mvkxNd
)2
1
2
1()
2
1
2
1( 2222
iiifffsotherforce kxmgymvkxmgymvWfd
ckAc mgxmvkx 22
2
1
2
1
cxdmgN and
058.09.325 2 cc xx mxc 093.0
Connected Blocks in Motion Two blocks are connected by a light string that passes over a
frictionless pulley. The block of mass m1 lies on a horizontal surface and is connected to a spring of force constant k. The system is released from rest when the spring is unstretched. If the hanging block of mass m2 falls a distance h before coming to rest, calculate the coefficient of kinetic friction between the block of mass m1 and the surface.
22 2
10 kxghmNxk
PEKEWfd sotherforce
hxmgN and
)02
1()0( 2
2 kxghmPEPEPE sg
221 2
1khghmghmk gm
khgm
k1
2 21
gRv min
jmggmW ˆ
jR
vmFc ˆ
2
To stay on loop, the normal force, N, must be greater than zero.
mgR
vm
mgR
vmN
jmgNjR
vm
jmgjNWNjR
vmFc
2min
2
2
2
0
ˆ)(ˆ
ˆˆˆ
jNN ˆ
gRv min
Ex n. 3 :Block on wheel of death
RRRh
mghRmgmgR
mghRmgmv
2
52
2
1
)2(2
1
)2(2
1 2min
Mass must start higher than top of loop
h
gRRgghv
mghmgmv
52
522
)0(2
1 2
gRRgghv
mgRRmgmv
mghRmgmv
32
322
2
5)(
2
1
)(2
1
2
2
h
h = 5/2 R
Vmax = gR5
gRv min
Vmax
Vfin
gR3Vfin =
Potential Energy vs. Force
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