Lesson 13
Work
I.
Work
A.
Definition

The infinitesimal amount of work done by a force, F , upon a body when acting

over an infinitesimal displacement, ds , is given by
 
dW  F  d s
Thus, from Calculus we know that the total work done by a force upon an object
over the path from point A to point B is given by
B

W   F  ds
A

F

ds
Note: When calculating work, you must first identify the work done by which
FORCE!!

F
B.
Special Cases:
If we look at the integrand
we see that if the ___________________ between the ____________________
and _______________________ is ___________________ and if the
__________________________ of the _______________________ is
________________________ then the definition of work is simply
C.
Important Facts About Work
1.
The work done by a force is _________________ if any of these four cases hold:
a)
The force is __________________________.
b)
The displacement is _______________________.
c)
The force is __________________________ to the displacement.
d)
The integral (AREA under a force-distance graph) is ________________
Example:
The work done by Earth's gravity on the moon assuming that the moon
travels in a circular orbit.

v
Earth
The force is NOT
Zero
The displacement is NOT
Zero
The work is __________________________________
2.
Work is a _____________________________ .
Thus, the net work on an object (ie the work by the net external force) is just
3.
The unit of work is ____________________________ . We don't use
____________ because another quantity, TORQUE, also has these units.
Example 1: Calculate the work done by the 5.0 N force as the block is moved
5 m up an inclined plane with 40 degree inclination as shown below:
F=5N
40 
Solution:
Example 2: What is the work done by gravity on a ball as it falls a vertical
distance h?
Example 3: What is the work done by gravity on a ball as it rises vertically by a
distance h?
Let Us Solve Example 3 Using Calculus:
Evaluate Integral Analytically
OR
Graphically
II.
Hooke Springs
A.
Hooke's Law
The magnitude of the restoring force on an ideal spring is ___________________
_____________________ to the ____________________ the spring is
___________________ from its un-stretched position.
The direction of the spring force opposes the direction of ___________________
or ______________________.
x=0
Un-stretched Position
Magnitude:
Direction:
x=0
x
Stretched Position
B.
Spring Constant
The spring constant contains ALL ___________________________
about how the spring was ___________________ .
Big k
small k
C.
means a ________________________ spring.
means a ___________________________________ spring.
Are real springs actually Hooke Springs?
______________ If the __________________ or ____________________ is
_________________!
Math Reason: Any analytical function f (x), ie smooth, finite function, can be
represented as a power series expansion about some point xo. This is known as a Taylor
series expansion and the formula is
f(x  x 0 )  A 0  A1 (x  x 0 )  A 2 (x  x 0 ) 2  A 3 (x  x 0 ) 3  .....
where the coefficients are found by the formula
An
dnf
dx n
x x0
If the difference (x-x0) is very small then only the first few terms are needed to
approximate the function. Expanding the force of the spring around the un-stretched
position x0 = 0, A0 is zero so the first non-zero term is A1x. The coefficient A1 is our
spring constant k!
The higher order terms are called anharmicity terms for springs and are related to thermal
expansion properties when using spring forces to represent the binding of atoms in a
solid.
Homework Problems:
The Taylor expansion is very useful for making calculations with a computer as well as
for modeling real systems in physics and engineering.
A)
Derive the Taylor series expansions for Cos(x) and Sin(x).
B)
What are the small angle approximations for these functions?
C)
Why are these small angle approximations only valid if the angle is measured in
radians and not degrees? Hint: Perform dimensional analysis on the Taylor series
and then look up the definitions of degrees and radians.
D.
Work Done By A Hooke Spring (Example of Calculating Work For NonConstant Forces)
Let us use Calculus to determine the work by a spring upon the mass as it is
stretched a distance X from its un-stretched position.
Graphically
III.
Types of Work
We divide work into two basic types based upon whether the amount of work
done depends upon the path that the object takes or only upon its starting and
ending point!!
A.
If the __________________ done by a force upon an object only depends on
the objects ___________________ and __________________ location and
_________________ upon the ___________________ then the force is
___________________________ .
B.
All other forces are __________________________________________ .
C.
The work done over a ________________________ path by a
_________________________ force is always ______________________.
D.
If the curl of a force is ______________________ then the force is
_________________________ .
Problem:
Prove that gravity is a conservative force.
IV.
Work VS Force Calculations
Since Newton's Laws tell us everything about mechanics, why do we want to
develop new concepts? First, the concept of work will simplify the math
necessary to perform calculations for more complex problems. Second, the
concept of energy and work will work on problems where Newton's Laws do not
apply (Quantum Mechanics, etc).
EXAMPLE:
Let us assume that we want to determine the speed of an object at some
instant of time.
Math Difficulty #1:
We know that from Newton II that a Force causes a particle to accelerate. If we
then want to find the speed of the particle, we would need to integrate the
acceleration with respect to time. Only in the special case of constant acceleration
can we use the kinematic equations to avoid integration. Work will often enable
us to go directly to speed without doing an integral.
Math Difficulty #2:
Forces are vectors while work is a scalar!! Scalar math is much easier than
vector math!
Math Difficulty #3:
We may not know the form of a particular force. However, if we can determine
that the force does no work then we don't care!
Final Solution: Bring together our concepts on work with the concepts of
mechanical energy, kinetic energy, and potential energy in next lesson so that we
can find the speed without integrals or vector math.