What is work?  Work (W) is the energy change of a system due to an external force.

When work is done on a system, W > 0.
When the system does work on the external environment, W < 0.

How do we calculate work?  There are three situations to consider:

1. Constant net force colinear  with the displacement;
2. Constant net force non-colinear  with the displacement;
3. Variable  net force.

1. Constant colinear force:    W = F x    x = distance traveled.

2. Constant non-colinear force:    W = F • Δr = F Δr cos θ = F_x Δx + F_y Δy + F_z Δz
See transparencies for definition of dot product.

Note that:   W > 0 when the force is in the same direction as the displacement;
W < 0 when these are in opposite directions;
W = 0 when F ⊥ Δr;
W is maximized when F || Δr.

3. Variable force:   dW = F(r) • dr → W = $\int$_a^b F(r) • dr

Justification: From Newtons' 2nd law:

F_s = m a_s = m dv_s/dt = m (dv_s/ds) (ds/dt) = m v_s (dv_s/ds).

Multiply by ds:   F_s ds = m v_s dv_s.

Integrating gives:   ∫_a^b F_s ds = ∫_vi^v_f m v_s dv_s= ½ mv_f² - ½ mv_i²

→ W = Δ K