1. Potential Energy and Energy Conservation
PHYSICS 220
Lecture 10
Potential Energy and Energy Conservation
Lecture 10
Purdue University, Physics 220

2. Purdue University, Physics 220
Potential Energy
Work done by gravity is independent of path
Wg = -mg (yf - yi) = - PEg
Define PEg = mgy
Only the difference in potential energy is physically meaningful, i.e., you have the freedom to choose the reference (or zero potential energy) point.
Works for any CONSERVATIVE force
Careful that independent of path really but does depend on initial and final points.
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There is a fact, or if you wish, a law, governing natural phenomena that are known to date. There is no known exception to this law; it is exact, so far we know. The law is called conservation of energy; it states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity, which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same.
Richard Feynman
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Energy
Energy is “conserved” meaning it can not be created nor destroyed
Can change form
Can be transferred
Total Energy of an isolated system does not change with time
Types
Kinetic Energy
Potential Energy
Rest Energy (E=mc2)
Thermal Energy
… …
Units: Joule (J) = N m = kg m2 / s2
Maybe do example of conservation (like cookies) and counter example like acceleration
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Work by Variable Force
W = Fx Dx
Work is the area under the F vs x plot
Spring: F spring= -k x
Potential Energy:
-W=PEs = 1/2 k x2
Lecture 10
Purdue University, Physics 220

6. Work-Energy with Conservative Forces
Work-Energy Theorem
Move work by conservative forces to other side
If there are NO non-conservative forces
Conservation of mechanical energy
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7. Purdue University, Physics 220
iClicker
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?
A) Dropping B) Slide on ramp (no friction) C) Swinging down D) All the same A B C
correct
Conservation of Energy (Wnc=0)
KEinitial + PEinitial = KEfinal + PEfinal
0 + mgh = ½ m v2final + 0
vfinal = sqrt(2 g h)
Lecture 10
Purdue University, Physics 220

8. Skiing Example (no Friction)
A skier goes down a 78 meter high hill with a variety of slopes. What is the maximum speed the skier can obtain starting from rest at the top?
Conservation of energy:
KEi + PEi = KEf + PEf
½ m vi2 + m g yi = ½ m vf2 + m g yf
0 + g yi = ½ vf2 + g yf
vf2 = 2 g (yi-yf)
vf = sqrt( 2 g (yi-yf))
vf = sqrt( 2 x 9.8 x 78) = 39 m/s
Lecture 10
Purdue University, Physics 220

9. Purdue University, Physics 220
Skiing with Friction
A 50 kg skier goes down a 78 meter high hill with a variety of slopes. She is observed to be going 30 m/s at the bottom of the hill. How much work was done by friction?
Work Energy Theorem:
Wnc = (KEf + PEf) - (KEi + PEi)
= (½ m vf2 + m g yf) - (½ m vi2 + m g yi)
= ½ (vf2 - g yi )m
= (½ (30)2 – 9.8 x 78) 50
= (450 – 764) 50 Joules
= Joules
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10. Purdue University, Physics 220
Power (Rate of Work)
Pav = W / Dt
Units: Joules/Second = Watt
W = F r cosq
= F (v t) cosq
P = F v cosq
How much power does it take for a (70 kg) student to run up the stairs (5 meters) in 7 seconds?
Pav = W / t
= m g h / t
= (70 kg) (9.8 m/s2) (5 m) / 7 s
= 490 J/s or 490 Watts
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11. Purdue University, Physics 220
Pendulum Exercise
As the pendulum falls, the work done by the string is
A) Positive B) Zero C) Negative
How fast is the ball moving at the bottom of the path?
W = F d cos q. But q = 90 degrees so Work is zero.
Conservation of Energy (Wnc=0)
SWnc = DKE + DPE
0 = KEfinal - KEinitial + PEfinal - PEinitial
KEinitial + PEinitial = KEfinal + PEfinal
0 + mgh = ½ m v2final + 0
vfinal = sqrt(2 g h) h
Lecture 10
Purdue University, Physics 220

12. Gravitational Potential Energy
If the gravitational force is not constant or nearly constant, we have to start from Newton’s gravitational force law
The gravitational potential energy is:
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Problem: How High?
A projectile of mass m is launched straight up from the surface of the earth with initial speed v0. What is the maximum distance from the center of the earth RMAX it reaches before falling back down.
RMAX m RE v0 M
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Problem: How High...
All forces are conservative:
WNC = 0
KE = -PE
And we know:
RMAX m RE v0
hMAX M
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15. Purdue University, Physics 220
Problem: How High...
RMAX m RE v0
hMAX M
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16. Purdue University, Physics 220
Escape Velocity
If we want the projectile to escape to infinity we need to make the denominator in the above equation zero:
We call this value of v0 the escape velocity, vesc
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17. Purdue University, Physics 220
Exercise
A box sliding on a horizontal frictionless surface runs into a fixed spring, compressing it a distance x1 from its relaxed position while momentarily coming to rest.
If the initial speed of the box were doubled and its mass were halved, how far x2 would the spring compress ?
A) B) C) x
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18. Purdue University, Physics 220
Exercise
Use the fact that Ei = Ef
In this case, Ef = 0 + 1/2 kx2
and Ei = 1/2 mv2 + 0
so kx2 = mv2
In the case of x1
So if v2 = 2v1 and m2 = m1/2 x1 v1 m1
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19. Purdue University, Physics 220
iClicker
How high will the pendulum swing on the other side now?
A) h1 > h B) h1 = h2 C) h1 < h2
Conservation of Energy (Wnc=0)
SWnc = DKE + DPE
KEinitial + PEinitial = KEfinal + PEfinal
0 + mgh1 = 0 + mgh2
h1 = h2 m h1 h2
Lecture 10
Purdue University, Physics 220