Question

a force is acting on a mass of 6kg displacement x of the mass is related to time t as x=t2/4 meter.work done by the force in 2 sec is

Answer 1

Given,
Mass of body , m = 6 kg
displacement is the function of time , x = t²/4
∵ Workdone = force × displacement , but here no any external force act on body.
∴ we have to use ,
Workdone = Change in kinetic energy

Now, x = t²/4
differentiate x with respect to time ,
dx/dt = 2t/2 = t/2
We know, dx/dt = velocity of body
At t = 2sec, velocity of body , v = 2/2 = 1 m/s
So, change in kinetic energy = final kinetic energy - intial kinetic energy of body
= 1/2mv² - 0 [because initially body is in rest because velocity of body at t = 0, is 0 ]
∴ change in kinetic energy = 1/2 × 6 × (1)²
= 1/2 × 6 = 3 J

Hence , Workdone = 3J

Answer 2

We know that:
            W = Integration of ( F dx)
       
       We are given: x = t²/4
        Now replace x in terms of t as shown in the attachment to get to the answer 3 Joules