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Calculating Work
The Joule

Work is force acting over a distance, and has units
equal to units of force times units of distance.
• With 1-dimensional constant force and distance, W = F Dx

The unit of work is the joule (J) = 1 N m.
Constant Force

Lifting an object requires a
constant force equal to
gravity, F = mg.
mg
Dx = h

The work done by the lifter is
W = F Dx = mgh.
-mg
mg

The work done by gravity is
W = - F Dx = -mgh.
-mg
Using the Scalar Product

Dx
F
A man is letting a 300 kg
piano slide 4 m at constant
velocity down a 30° incline
while exerting a 400 N force
on the horizontal. What work
does he do?
• The component of the force
is (400 N)(cos 30) = -350 N
• Negative since it is opposite
the displacement
• The work is (-350 N)(4 m) =
-1400 N m = -1400 J
Variable Force

The force applied to a spring
increases as the distance
increases.

The work must be calculated
over each separate interval.

The work increases over a
small interval as the force
increases.
F
Dx
Area under a Curve

Separate the total distance
into steps Dx.

The product within a small
step is the area of a
rectangle F Dx.

The total equals the area
between the curve and the x
axis.
F
Dx
Work on a Spring
F=kx

For the spring force the force
makes a straight line.

The area under the line is
the area of a triangle.
1
Fs x
2
1
Ws  (kx) x
2
1
Ws  kx2
2
Ws 
x
Integral Form of Work

The work can be found by
taking the area under any
force curve.
F

This technique in calculus is
the integral.
W   F||dx
 
W   F  dr
x
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