Work and Energy
A calculus-based perspective
AP Physics C
The old “special case”

W=F d is a special case



F must be in the direction
of motion
F is a constant force
What if F is not parallel to
d?



The Dot or Scalar Product.
One vector times another
to “make” a scalar
W= Fcosθd
If the graph below shows the force exerted by the Death
Star’s Tractor beam. How could you calculate the work
done on a ship being pulled from position a to b?
A.
B.
C.
Find the slope of the
line between a and b
Find the Area under
the curve from point
a to b
Multiply force x (b-a)
Area Under the Curve



For Hooke’s Law, the force is linear.
For a force vs. distance graph, the work
is just the area under the curve. The
shape is a triangle.
What if the area under the curve is not
geometric?
The Reality
•Most Forces aren’t
•Finding an integral is a
constant.
way of accurately
•Springs
finding
the area under a
curve.
•Magnetic fields
•My looking at
•Pushes/pulls
smaller and smaller
•Gravity
“pieces” over
of x,
large
distances
multiplying
them by
the force, and
summing them we
can find total area.
Integration Rules!!
Example: Work done to
stretch a spring.


F(x) = kx
Hooke’s Law
How much work do you do if you
stretch a spring from an initial
position of 0m to the 1m position if
the spring constant is 49 N/m
A.
B.
C.
D.
49
25
98
10
Nm
Nm
Nm
Nm
Work for a varying force
x2
Work   F ( x) dx
x1
b
b
a
a
Wab   F  ds   F cos  ds
A force F(x) acts on a particle. The force is related to the
position of the particle by the formula F(x) = Cx3, where C is a
constant. Find the work done by this force on the particle
when the particle moves from x = 1.5 m to x = 3 m.
A.
B.
C.
D.
1J
34 J
0J
19C J
Solution
3
W

3
F ( x) dx 
1.5
Cx
dx

3
1.5
3
 x 
C 4
4
W  C    3  1.5   19C J
 4 1.5 4
4
Energy
Work-Energy Theorem

The change in the kinetic energy of an
object is equal to the net work done on
the object.
Types of Force

Conservative


Obeys conservation of
energy
Examples



Spring force
Gravity
Non-conservative


Energy is transferred
into non-mechanical
forms
Examples


Friction
Air drag
Equilibrium


Occurs when net force = 0
If force = F(x), then equilibrium exists at points
where F(x) = 0.
U    F ( x)dx

Equilibrium exists where dU/dx = 0
Power
Power takes many forms
W Fd mad
P


 Fv
t
t
t