1. Purdue University, Physics 220
Unimportable #8279AE
Lecture 10
Purdue University, Physics 220

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

3. Purdue University, Physics 220
Work Done by Gravity
Ball is tossed up with velocity vA.
What is the work done by gravitational force?
How far up it will go? B
Only force/work done be gravity
Wg = - mgh = KEB – KEA = ½m vB2 - ½m vA2
-mgh = -½m vA2
h = vA2 / 2g h A
Lecture 9
Purdue University, Physics 220

4. Purdue University, Physics 220
Work Done by Gravity
Slide block down incline
Wg = (mg)(s)cos
s = h/cos
Wg = mg(h/cos)cos
Wg = mgh h  mg S
Lecture 9
Purdue University, Physics 220

5. Purdue University, Physics 220
Work Done by Gravity
Depends only on initial and final height!
Wg = -mg(yf - yi)
Independent of path
If you end up where you began, Wg = 0
Whenever the work that is done by
a force is independent of its path and
it's only determined by the starting point and the end point that force is called a "conservative force“.
Lecture 9
Purdue University, Physics 220

6. 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.
Lecture 10
Purdue University, Physics 220

7. 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
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
Lecture 10
Purdue University, Physics 220

10. 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

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. Purdue University, Physics 220
Pendulum Exercise
With no regard for his own personal safety your physics professor will risk being smashed by a bowling ball pendulum! If released from a height h, how far will the bowling ball reach when it returns?
Conservation of Energy (Wnc=0)
SWnc = DKE + DPE
0 = KEfinal - KEinitial + PEfinal- PEinitial
KEinitial + PEinitial = KEfinal + PEfinal
0 + mghinitial = 0 + mghfinal
hinitial = hfinal h
Lecture 10
Purdue University, Physics 220

13. 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
Lecture 10
Purdue University, Physics 220

14. 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:
Lecture 10
Purdue University, Physics 220

15. Purdue University, Physics 220
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
Lecture 10
Purdue University, Physics 220 6

16. Purdue University, Physics 220
Problem: How High...
All forces are conservative:
WNC = 0
KE = -PE
And we know:
RMAX m RE v0
hMAX M
Lecture 10
Purdue University, Physics 220 7

17. Purdue University, Physics 220
Problem: How High...
RMAX m RE v0
hMAX M
Lecture 10
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18. 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
Lecture 10
Purdue University, Physics 220 9

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