The equation W = F^{->}.S^{->} = FS cos θ is applicable when remains constant but when the force is variable work is obtained by integrating F^{->}.ds^{->}

Thus, W = ∫F^{->}.ds^{->}

An example of a variable force is the spring force in which force depends on the extension x,

i.e., F^{->}= F^{->}(x)

When the force is time dependent, we have

W = ∫F^{->}.dX^{->} = ∫F^{->}.v^{->}dt

where F^{->} and v^{->} are the instantaneous force and velocity vectors.

**Example :** A body acted upon by a force F^{->} , given by, F^{->} = – k [(cos ωt) i^{^} + (sin ωt )j^{^} ] undergoes displacement, where the position vector r^{->} of the body is given by r^{->} = a[cos (ωt + α) +sin (ωt + α) ]. Find the work done by the force from time t = 0 to time t = 2π /ω

**Solution :** The position of body is given by

r^{->} = a[cos (ωt + α)i^{^} + sin (ωt + α)j^{^} ]

Its velocity is given by,

v^{->} = dr^{->}/dt = d/dt [ a cos (ωt + α) + a sin (ωt + α) ]

= – aω sin (ωt + α)i^{^} + a ωcos (ωt + α)j^{^}

The power developed by this force is,

= (aωk) cos (ωt) sin (ωt + α) – (aωk) sin (ωt) cos (ωt + α)

= aωk [ sin (ωt + α) cos ωt – cos (ωt + α) sin ωt]

= aαk sin (ωt + α – ωt)

= aαk sin α

W = ∫ dW = aωk sin α _{o}∫ ^{2π/ω}dt

= aωk sin α x 2π/ω = 2π a k sin α

**Exercise 3:** A particle is acted upon by a force given by = A cos ωt +B and its position vector is given by r^{->} = a[(cos (ωt)i^{^} + sin (ωt)j^{^} ] + (1/2) bt^{2}k^{^}

Find the work done on the particle by the force F^{->} from time t = π/ 2ω to time t = π /ω