A 1 kg mass is moved part way around a vertical square loop as
shown. The square is 1 meter on a side and the final position of the
mass is 0.5 m below its original position. Assume that g = 10 m/s2.
What is the work done by the force of gravity during this journey? 0J
A) 10 J B
10 J
A
-5 J
C
B)  5 J C)  0 J D
Wg = (10N)(0.5m) = 5 J
D)  –10 J E)  –5 J 0J

 
 
 

WBB ++W
WCC++W
WDD=
FA ⋅ ΔrA + FB ⋅ ΔrB + FC ⋅ ΔrC + FD ⋅ ΔrD
Wgg =
= WAA ++ W
= -(10N)(0.5m) + 0 + (10N)(1m) + 0 = -5 J + 0 + 10 J + 0 = 5J L20 F 10/10/14 a*er lecture 1 

Definition:
 Conservative Force F. WF depends only on ri
and rf , and is therefore independent of path between the
endpoints.
e.g., gravity, springs (elastic forces)
We can define a potential energy associated with each conservative force.
e.g., gravitational potential energy, elastic potential energy
This is important when we consider conservation of energy.
L20 F 10/10/14 a*er lecture 2 Assignments
For this week and next:
•  You should have read Ch. 6 of Wolfson and Prof. Dubson’s notes. Now read Ch. 7.
•  Begin CAPA 7 and HW 7 over the weekend.
Today:
•  Work, energy, and move toward conservation of energy.
L20 F 10/10/14 a*er lecture 3 Last Time: Work, Kinetic Energy
Definition: Energy is the scalar conserved quantity that obeys the
first law of thermodynamics:
W = ΔU
(Unit = joule or J)

Definition: Work done by a force F on an object that undergoes a

displacement Δr = Δx = d :

WF = component of F along
 the
 displacement times the displacement =
F ⋅ Δr
 Dot Product
 
A
A ⋅ B = AB cosθ
θ

A cosθ
B
L20 F 10/10/14 a*er lecture 4 Last Time: Work, Kinetic Energy
Definition: Energy is the scalar conserved quantity that obeys the
first law of thermodynamics:
W = ΔU
(Unit = joule or J)

Definition: Work done by a force F on an object that undergoes a

displacement Δr = Δx = d :

WF = component of F along
 the
 displacement times the displacement =
F ⋅ Δr
B

F
Assumes:

1.  F constant

2.  Δr points in a straight line
L20 F 10/10/14 a*er lecture 
F
More generally:
 
WF = ∫ F ⋅dr
B
A
A

F
5 Last Time: Work, Kinetic Energy
Definition: Energy is the scalar conserved quantity that obeys the
first law of thermodynamics:
W = ΔU
(Unit = joule or J)

Definition: Work done by a force F on an object that undergoes a

displacement Δr = Δx = d :

WF = component of F along
 the
 displacement times the displacement =
F ⋅ Δr
B
 
WF = ∫ F ⋅dr
A
Definition: Kinetic energy is the energy of motion. An object of mass
m moving with speed v has kinetic energy equal to:
L20 F 10/10/14 a*er lecture 1 2
KE = mv
2
6 Work-Kinetic Energy Principle
The work done by the net force on a single object is equal to the
change in kinetic energy of that object:
Wnet = WFnet = ΔKE = KEf - KEi
(Applies only for the net work – much consider the effect of all forces.)
L20 F 10/10/14 a*er lecture 7 Work-KE Principle
Wnet = WFnet = ΔKE = KEf - KEi
Example 1: (Motivate the W-E Prinl.) Push a book across a frictionless
table with constant force Fext.
(1)  Fnet = Fext because N and mg cancel.
(2)  Thus, a = Fext /m.
(3)  Wnet = Wext = +Fext Δx
⇒ Fext =
Wext
Δx
(4)  vf2 = vi2 + 2aΔx = vi2 + 2(Fext /m)Δx
= vi2 + 2(Wext /Δx m)Δx = vi2 + 2(Wext /m)
Wext
(5) v − v = 2
m
2
f
2
i
L20 F 10/10/14 a*er lecture 1 2 1 2
⇒
mv f − mvi = Wext
2
2
⇒ ΔKE = Wext
8 A projectile is fired upward with an initial speed v on an airless
world. A short time later, it comes back down and has a final
speed v (just before it hits the ground). What was the sign of the
total work done by the force of gravity during the flight? Up
A) +
B) –

Δr
W = -mgh

< 0
Fg
Down
W = mgh
> 0


Δr
Fg
C) 0
Two ways to look at this:
1.  The negative work on the way up cancels the positive work on the way down.
2.  By the Work-KE Principle, Wgrav = Wnet = ΔKE = 0.
L20 F 10/10/14 a*er lecture 9 A projectile is fired upward through air with an initial speed v0. A
short time later, it comes back down and has a final speed v < v0 .
Air resistance is NOT negligible. What was the sign of the work
done by friction during the flight?
Up
A) +
B) –
W < 0

Δr

Ff
C) 0
Down

Ff
 Δr
F
W < 0
Fnet = Fgrav + Ffriction
Wnet = Wgrav + W friction = 0 + (−) = ΔKE < 0
L20 F 10/10/14 a*er lecture 10 
More general definition of Work due to a force F that varies along a

rf
particle’s path:
N

rf
i=1

ri

F
 
 
WF = lim ∑ Fi ⋅ Δri = ∫ F ⋅ dr
N→∞

F
Example of a variable force: spring

ri
Hooke’s “Law”:
Force exerted by
the spring on the
hand
L20 F 10/10/14 a*er lecture 
F
Fspring = −kx
Force opposes
displacement x
spring
constant
stiff spring: k large
floppy spring: k small
11 One end of a spring is fixed in place. I stretch the spring by pulling
on the other end. What is the sign of the work done by the force from
my hand? A) positive B) negative C) zero
D) answer depends on the direction in
which I pull
E) answer depends on the coordinate
system I choose No matter which way I pull, the force I exert is in the same direction
as the displacement. When force and displacement are in the same
direction, the work done is positive.
L20 F 10/10/14 a*er lecture 12 A force is applied in the x-direction
while an object moves a distance x along
the x-axis. The force varies with position
according to this graph:
The work done by this force is 1
A) F0 x
2
B) F0 x
Wnet
3
C) F0 x
2
D) 2F0 x
5
E) F0 x
2
⎛ x⎞
⎛ x⎞ 3
= F0 ⎜ ⎟ + 2F0 ⎜ ⎟ = F0 x
⎝ 2⎠
⎝ 2⎠ 2
= area under the curve (or the integral)
L20 F 10/10/14 a*er lecture 13