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Work
Work = Force
×
distance
(the force must be parallel to movement)
OR
Work = (Force)(cos
θ
)(distance)
When you are determining the force parallel to the movement you can do this manually and keep
track of the direction of movement or you can use the general formula that W = Fcos
θ
d. In this
formula Fcos
θ
is equal to the force parallel to the movement. The drawback to using this generic
formula is that you must pick the correct
θ
or the answer will be wrong. Alternatively, you can
manually determine the vector that is parallel to the movement, decide if the movement is in the
same direction as the force (positive force and therefore positive work) or in the opposite
direction as the force (negative force and therefore negative work). Then use W = Fd with the
appropriate sign for the force. Either method is acceptable, you will need to decide which works
better for you.
In addition to net work being equal to the net force times the distance moved, net work is equal
to the change in kinetic energy of an object. The work done by gravity is equal to the change in
the gravitational potential energy. Work done by a spring is equal to the change in the spring’s
potential energy. Each of these statements can be expressed as an equation:
W
net
= KE
f
–KE
i
(KE = ½ mv
2
)
Kinetic Energy
W
gravity
= PE
f
– PE
i
(PE
gravity
= mgh)
Potential Energy due to gravity
W
spring
= PE
f
– PE
i
(PE
spring
= ½ kx
2
)
Potential Energy due to a spring
Since energy is conserved, we can combine all of these into:
Overall Work Equation
(also called work energy theorem)
½
mv
1
2
+ mgy
1
+ ½ kx
1
2
−
F
f
d
±
W
ae
= ½ mv
2
2
+ mgy
2
+ ½ kx
2
2
KE
i
PE
i
spring
i
friction
KE
f
PE
f
spring
f
(W
ae
= work from anything else)
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 Spring '10
 Carter
 mechanics, Force, Work

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