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  xWith 1-dimensional constant force and distance, W = F  x  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.  The work done by the lifter is W = F  x = mgh.  The work done by gravity is W = - F  x = -mgh.  x = h -mg mg

Using the Scalar Product  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) = N m = J F xx

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 xx

Area under a Curve  Separate the total distance into steps  x.  The product within a small step is the area of a rectangle F  x.  The total equals the area between the curve and the x axis. F xx

Work on a Spring  For the spring force the force makes a straight line.  The area under the line is the area of a triangle. F=kx x

Integral Form of Work  The work can be found by taking the area under any force curve.  This technique in calculus is the integral. next F x