Question

The displacement x of a particle of mass m moving in a straight line varies with time t as x=kt^3/2 under the action of a force F where k is constant. Calculate the work done by the force

Answer 1

Calculate the work done by the force:

Given: Displacement (x) is given as a function of time(t)

             $x = kt^{\frac{3}{2} }$

             F = force

             m = mass

             k = constant

Solution:

Work done by Force = Force x Displacement

                                    = F x $kt^{\frac{3}{2} }$

Force = mass x acceleration

Acceleration = $\frac{d^{2}x }{dt^{2} }$ = k · $\frac{d^{2}(t^{\frac{3}{2} }) }{dt^{2} }$

                               = k ·  $\frac{d }{dt }(\frac{3}{2} t^{\frac{1}{2} })$

                               = $\frac{3k}{4} t^{\frac{-1}{2} }$

                               = $\frac{3k}{4\sqrt{t} }$

Force = mass x acceleration

       F   = m x $\frac{3k}{4\sqrt{t} }$

      F     =  $\frac{3km}{4\sqrt{t} }$

Work done by Force = Force x Displacement          

                                W = $\frac{3km}{4\sqrt{t} }$ x $kt^{\frac{3}{2} }$

                                W = $\frac{3tk^{2} m}{4} }$

Therefore, Work done by Force F is $\frac{3tk^{2} m}{4} }$.