1. Employ conservative and non-conservative forces
Lecture 16
Chapter 11 (Work)
Employ conservative and non-conservative forces
Relate force to potential energy
Use the concept of power (i.e., energy per time)
Chapter 12
Define rotational inertia
Define rotational kinetic energy
Assignment:
HW7 due Tuesday, Nov. 1st
For Monday: Read Chapter 12, (skip angular momentum and explicit integration for center of mass, rotational inertia, etc.)
Exam 2 7:15 PM Thursday, Nov. 3th 1

2. Definition of Work, The basics
Ingredients: Force ( F ), displacement (  r ) 
 r
displacement F
Work, W, of a constant force F
acts through a displacement  r :
W = F · r (Work is a scalar)
Work tells you something about what happened on the path!
Did something do work on you?
Did you do work on something?

3. Net Work: 1-D Example (constant force)
A force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m.
Start
Finish F
q = 0° x
Net Work is F x = 10 x 5 N m = 50 J
1 N m ≡ 1 Joule (energy)
Work reflects positive energy transfer

4. Net Work: 1-D 2nd Example (constant force)
A force F = 10 N is opposite the motion of a box across a frictionless floor for a distance x = 5 m.
Start
Finish
q = 180° F x
Net Work is F x = -10 x 5 N m = -50 J
Work again reflects negative energy transfer

5. Work: “2-D” Example (constant force)
An angled force, F = 10 N, pushes a box across a frictionless floor for a distance x = 5 m and y = 0 m
Start
Finish F Fx
q = -45° x
(Net) Work is Fx x = F cos(-45°) x = 50 x 0.71 Nm = 35 J

6. Exercise Work in the presence of friction and non-contact forces
A box is pulled up a rough (m > 0) incline by a rope-pulley-weight arrangement as shown below.
How many forces (including non-contact ones) are doing work on the box ?
Of these which are positive and which are negative?
State the system (here, just the box)
Use a Free Body Diagram
Compare force and path v 2 3 4 5

7. Work and Varying Forces (1D)
Area = Fx Dx
F is increasing
Here W = F · r
becomes dW = F dx
Consider a varying force F(x) Fx x Dx

8. Example: Work Kinetic-Energy Theorem with variable force
How much will the spring compress (i.e. x = xf - xi) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vo) on frictionless surface as shown below ? to vo m
spring at an equilibrium position x F
V=0 t m
spring compressed

9. Compare work with changes in potential energy
Consider the ball moving up to height h
(from time 1 to time 2)
How does this relate to the potential energy?
Work done by the Earth’s gravity on the ball)
W = F  Dx = (-mg) -h = mgh
DU = Uf – Ui = mg 0 - mg h = -mg h
DU = -W mg h mg

10. Conservative Forces & Potential Energy
If a conservative force F we can define a potential energy function U:
The work done by a conservative force is equal and opposite to the change in the potential energy function.
(independent of path!)
W = F ·dr = - U ò ri rf Uf Ui

11. A Non-Conservative Force, Friction
Looking down on an air-hockey table with no air flowing (m > 0).
Now compare two paths in which the puck starts out with the same speed (Ki path 1 = Ki path 2) .
Path 2
Path 1

12. A Non-Conservative Force
Path 2
Path 1
Since path2 distance >path1 distance the puck will be traveling slower at the end of path 2.
Work done by a non-conservative force irreversibly removes energy out of the “system”.
Here WNC = Efinal - Einitial < 0  and reflects Ethermal

13. There is no transformation because energy is conserved.
A child slides down a playground slide at constant speed. The energy transformation is
U  K
U  ETh
K  U
K ETh
There is no transformation because energy is conserved.
STT11.1
Answer: B

14. Conservative Forces and Potential Energy
So we can also describe work and changes in potential energy (for conservative forces)
DU = - W
Recalling (if 1D)
W = Fx Dx
Combining these two,
DU = - Fx Dx
Letting small quantities go to infinitesimals,
dU = - Fx dx
Or,
Fx = -dU / dx

15. Spring: Fx = 0 => dU / dx = 0 for x=xeq
Equilibrium
Example
Spring: Fx = 0 => dU / dx = 0 for x=xeq
The spring is in equilibrium position
In general: dU / dx = 0  for ANY function establishes equilibrium U U
stable equilibrium
unstable equilibrium

16. Work & Power:
Two cars go up a hill, a Corvette and a ordinary Chevy Malibu. Both cars have the same mass.
Assuming identical friction, both engines do the same amount of work to get up the hill.
Are the cars essentially the same ?
NO. The Corvette can get up the hill quicker
It has a more powerful engine.

17. Work & Power: Power is the rate, J/s, at which work is done.
Average Power is,
Instantaneous Power is,
If force constant, W= F Dx = F (v0 Dt + ½ aDt2)
and P = W / Dt = F (v0 + aDt)

18. Work & Power: Example
Average
Power:
1 W = 1 J / 1s
Example:
A person, mass 80.0 kg, runs up 2 floors (8.0 m) at constant speed. If they climb it in 5.0 sec, what is the average power used?
Pavg = F h / Dt = mgh / Dt = 80.0 x 9.80 x 8.0 / 5.0 W
P = 1250 W

19. Top Middle Bottom Exercise Work & Power
Starting from rest, a car drives up a hill at constant acceleration and then suddenly stops at the top.
The instantaneous power delivered by the engine during this drive looks like which of the following, Z3
time
Power
Top
Middle
Bottom

20. Exercise Work & Power
P = dW / dt and W = F d = (m mg cos q - mg sin q) d
and d = ½ a t2 (constant accelation)
So W = F ½ a t2  P = F a t = F v
(A)
(B)
(C)
Power
time
Power Z3
time
Power
time

21. Chap. 12: Rotational Dynamics
Up until now rotation has been only in terms of circular motion with ac = v2 / R and | aT | = d| v | / dt
Rotation is common in the world around us.
Many ideas developed for translational motion are transferable.

22. Rotational motion has consequences
Katrina
How does one describe rotation (magnitude and direction)?

23. Rotational Variables Rotation about a fixed axis:
Consider a disk rotating about an axis through its center:
Recall :
(Analogous to the linear case )  

24. System of Particles (Distributed Mass):
Until now, we have considered the behavior of very simple systems (one or two masses).
But real objects have distributed mass !
For example, consider a simple rotating disk and 2 equal mass m plugs at distances r and 2r.
Compare the velocities and kinetic energies at these two points. w 1 2

25. For these two particles
1 K= ½ m v2
Twice the radius, four times the kinetic energy w
2 K= ½ m (2v)2 = 4 ½ m v2

26. For these two particles
1 K= ½ m v2 w
2 K= ½ m (2v)2 = 4 ½ m v2

27. For these two particles

28. Rotation & Kinetic Energy...
The kinetic energy of a rotating system looks similar to that of a point particle:
Point Particle Rotating System
v is “linear” velocity
m is the mass.
 is angular velocity
I is the moment of inertia
about the rotation axis.

29. Calculating Moment of Inertia
where r is the distance from the mass to the axis of rotation.
Example: Calculate the moment of inertia of four point masses
(m) on the corners of a square whose sides have length L,
about a perpendicular axis through the center of the square: m m L m m

30. Calculating Moment of Inertia...
For a single object, I depends on the rotation axis!
Example: I1 = 4 m R2 = 4 m (21/2 L / 2)2
I1 = 2mL2
I2 = mL2
I = 2mL2 m m L m m

31. Lecture 16
Assignment:
HW7 due Tuesday Nov. 1st 1