1. PHYSICS 231 Lecture 11+12: How much energy goes into problems?
Remco Zegers
PHY 231

2. Previously NEXT: Work: W=Fcos()x Energy transfer
Power: P=W/t Rate of energy transfer
Potential energy (PE) Energy associated with position.
Gravitational PE: mgh Energy associated with
position in grav. field.
Kinetic energy KE: ½mv2 Energy associated with
motion
NEXT:
Conservative force: Work done does not depend on path
Non-conservative force: Work done does depend on path
Mechanical energy ME: ME=KE+PE
Conserved if only conservative forces are present KEi+PEi=KEf+PEf
Not conserved in the presence of non-conservative forces (KEi+PEi)-(KEf+PEf)=Wnc
PHY 231

3. Work and energy
WORK
POTENTIAL
ENERGY
KINETIC
ENERGY
PHY 231

4. Mechanical Energy Mechanical energy Gravitational Potential
Energy (mgh)
Kinetic
Energy
½mv2
PHY 231

5. Conservative forces
A force is conservative if the work done by the force when
Moving an object from A to B does not depend on the path
taken from A to B.
Example: gravitational force
Using the stairs:
Wg=mghf-mghi=mg(hf-hi)
Using the elevator:
The path taken (longer or shorter)
does not matter: only the displacement does!
h=10m
PHY 231

6. Non conservative forces
A force is non-conservative if the work done by the force when moving an object from A to B depends on the path taken from A to B.
Example: Friction
You have to perform more work
Against friction if you take the
long path, compared to the short
path. The friction force changes
kinetic energy into heat.
Heat, chemical energy (e.g battery or fuel in an engine)
Are sources or sinks of internal energy.
PHY 231

7. Conservation of mechanical energy only holds if the system is closed AND all forces are conservative
MEi-MEf=(PE+KE)i-(PE+KE)f=0 if all forces are conservative
Example: throwing a snowball from a building neglecting air resistance
MEi-MEf=(PE+KE)i-(PE+KE)f=Wnc if some forces are nonconservative.
Wnc=work done by non-conservative forces.
Example: throwing a snowball from a building taking into account air resistance
PHY 231

8. Overview Conservation of mechanical energy Newton’s second Law F=ma
Wnc=0
Closed system
Work
W=(Fcos)x
Work-energy Theorem
Wnc=Ef-Ei
Equations of kinematics
X(t)=X(0)+V(0)t+½at2
V(t)=V(0)+at
PHY 231

9. Conservation of mechanical energy
A) what is the speed of m1 and m2 when they pass each other?
PHY 231

10. work How much work is done by the gravitational force when the masses
pass each other?
PHY 231

11. Friction (non-conservative)
The pulley is not completely frictionless.
The friction force equals 5 N. What is the
speed of the objects when they pass?
PHY 231

12. A spring k: spring constant (N/m) Fs=-kx +x Fs(x=0)=0 N Fs(x=-a)=ka
Fs=(0+ka)/2=ka/2
Ws=Fsx=(ka/2)*(a)=ka2/2
The energy stored in a spring depends on the location of the endpoint: elastic potential energy.
PHY 231

13. PINBALL The ball-launcher spring has a constant k=120 N/m. A player
pulls the handle 0.05 m. The
mass of the ball is 0.1 kg. What
is the launching speed?
PHY 231

14. Ball on a track A h end B h end
In which case has the ball the highest velocity at the end?
A) Case A B) Case B C) Same speed
In which case does it take the longest time to get to the end?
A) Case A B) Case B C) Same time
PHY 231

15. Race track
KE PE TME NC
With friction
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16. PHY 231

17. A swing If relieved from rest, what is the velocity of the ball at the
lowest point?
30o
L=5m h
PHY 231

18. PHY 231

19. A running person While running, a person dissipates about
0.60 J of mechanical energy per step per
kg of body mass. If a 60 kg person develops
a power of 70 Watt during a race, how
fast is she running (1 step=1.5 m long)
What is the force the person exerts on the
road?
W=Fx P=W/t=Fv
PHY 231

20. Parabolic motion Where is the kinetic energy… highest? lowest ?
Assume height of catapult is
negligible to the maximum height
of the stone.
Parabolic motion
And what about potential energy? 
t=0
t=1
t=2
t=3 E A C D B
t=5
PHY 231

21. PHY 231

22. question An object is lowered into a deep hole in the surface of
the earth. What happens to its potential energy?
increase
decrease
remain constant
cannot tell from information given
don’t know
PHY 231

23. question An outfielder throws a baseball of 0.15 kg at a speed of
40 m/s and angle of 30 degrees with the field. What is the
kinetic energy of the baseball at the highest point,
ignoring friction?
0 J
30 J
90 J
120 J
don’t know
PHY 231

24. question A worker pushes a sled with a force of 50 N over
a distance of 10 m. A frictional force acts on the
wheelbarrow in the opposite direction, with a
magnitude of 30 N. What net work is done on the
wheelbarrow?
don’t know
100 J
200 J
300 J
500 J
PHY 231

25. question Old faithful geyser in Yellowstone park shoots water hourly
to a height of 40 m. With what velocity does the water
leave the ground?
7.0 m/s
14 m/s
20 m/s
28 m/s
don’t know
PHY 231

26. quiz (for credit)
A ball of 1 kg rolls up a ramp, with initial velocity of 6 m/s.
It reaches a maximum height of 1 m (I.e. velocity 0 at
at that point). How much work is done by friction? 0.
8.2 J
9.8 J
18 J
27.8 J
kinetic energy: 0.5mv2
potential energy: mgh
PHY 231