The test was:
A)  Unfairly difficult
B)  Difficult
C)  Just right
D)  Reasonably easy
E)  Way too easy
©University of Colorado, Boulder
work = force (in the direction of displacement) * distance
units:
[work]=[force]*[distance] = Nm = Joule
x
dx
w = F * cos(θ ) * dx
Clicker question
You push a beer keg up a ramp with constant speed. Suppose you push
parallel to the ramp, with force F. The keg travels a distance d along the
ramp, ending at height h as shown. How much work did YOU do on the
keg?
A)
B)
C)
D)
E)
F*d
F*d*cosθ
zero
F*h (which is equal to F*d*sinΘ)
None of these
A: Your work is simply F*distance*cos (angle between the FORCE and
the DISPLACEMENT). Here, that angle is NOT theta, it's 0 degrees.
You are pushing in the same direction as delta r. So, work is simply
F*d, nothing else.
Clicker question
You push a beer keg up a ramp with constant speed. Suppose you push
parallel to the ramp, with force F. The keg travels a distance d along the
ramp, ending at height h as shown. How much work did GRAVITY do
on the keg?
A) m*g*d
B) m*g*d*cosΘ
C) zero
D) -m*g*h
E) -F*h
D: Gravity is pointing down, so only the vertical component of the
displacement matters, which is pointing up, hence the work done by
gravity is -m*g*h.
Clicker question
Three forces have magnitudes in newtons that are numerically equal to
these quantities:
(A) x ,
(B) x, and
(C) x2,
where x is the position in meters. Each force acts on an object as it moves
from x = 0 to x = 1 m. Notice that all three forces have the same values at
the two endpoints—namely, 0 N and 1 N. Which force does the most
work?
1
2
1/2
∫0 x dx = 3
A:
1
⎡x ⎤
1
n
x
dx
=
=
⎢ n + 1⎥ n + 1
∫0
⎣
⎦0
1
n+1
1
1
∫0 xdx = 2
1
1
∫0 x dx = 3
2
2 or 3 D case:

r2

dr

F

r2

r1
 
F
d
r
=
?
∫

r1
line integral
Work done by a force F between points
A and B
 
W = ∫ Fdr
B
A
Conservative forces
If the work done by a force between two points is
independent of the path, the force is conservative.
Conservative forces conserve mechanical energy.
Non-conservative forces
Any force that is also function of time, and or
velocity
Conservative forces
For a conservative force, the work done in moving between
two points is independent of the path:
Because the work done by a
conservative force is path
independent, the work done
in going around any closed
path is zero:
 
∫ F ⋅ dr = 0
Potential energy
The “stored work” associated with a conservative force is
called potential energy.
The potential is related to the change of internal positions.
The concept of potential is most useful in a closed system.
Closed system: in isolation, no external forces
Not closed system: external forces are present
Potential energy
The “stored work” associated with a conservative force is
called potential energy.
The change in potential energy is the negative of the work
done by a conservative force* if the kinetic energy is
unchanged.
ΔU AB = − ∫
*
B
A
 
F ⋅ dr = −W
For a conservative force the work done is independent of
the path between two points.
Potential-Energy Curves
If there is no external force, the
kinetic energy + potential energy = constant
(Energy conservation)
1 2
mv + mgh = E = constant
2
KE fast
slower
slowest
Potential0
Consider the following possible definitions, and
decide which are correct.
 
i) Work done by a constant force F is W=+ F⋅Δr
ii) The change in potential energy is
ΔPE=+W = −W
ext
field
A) Only1 is true
B) Only 2 is true
C) Both are true
D) Both are false.
©University of Colorado, Boulder
Potential1
Two parallel conducting plates (a capacitor) are
charged as shown. A proton is lifted by an external
agent (tweezers) a distance d as shown. Ignore
gravity in this problem. (There
is a uniform electric

field between the plates) E = −E ŷ
What direction is the force on the proton
due to the E-field in the capacitor?
A: up
B: down
C: zero
y
++++++++++++++++++++++++++++++++
d
x
Proton
-------------------------------------------------------
Potential2
The sign of the work done by the
tweezers is … ?
A: +
B: D: Not enough info
C: zero
++++++++++++++++++++++++++++++++
d
Proton
-------------------------------------------------------
Potential3
The sign of the work done by the E-field
as the proton is moved upwards is…?
A: +
B: D: Not enough info
C: zero
++++++++++++++++++++++++++++++++
d
Proton
-------------------------------------------------------
Potential4
Change in potential energy is :
ΔU = +Wext = −W
field
If we define U(proton) = 0 at the bottom plate,
then U(proton) near the top is...
A: +
B: D: Not enough info
C: zero
++++++++++++++++++++++++++++++++
d
Proton
-------------------------------------------------------