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Solved Examples of Work, Power and Energy
The displacement x of a particle moving in one dimension, under the action of a constant force, is related to time t by the equation
t = √x + 3
where x is in meter and t in second.
(a) Plot the variation of displacement with time.
(b) Find the displacement of a particle when its velocity is zero.
(c) Find the work done by the force in the first 6 seconds.
It is given that
t = √x + 3
=> x = (t - 3)2 ...... (A)
So the displacement of particle is varying with time and the variation is parabolic in nature, as shown in Fig. 1.16.
As at t = 0 sec., x = 9 m
And t = 3 sec., x = 0 ,
If we analyze above graph we can say (i) the minimum displacement is zero at t = 3 sec. (ii) the displacement of particle first decreases and then increases. Ans.
Now we have to find the displacement of a particle when its velocity is zero i.e. dx/dt = 0
So, we can differentiate Eq. (A) with respect to time to get dx/dt and then put dx/dt = 0 to get the time at which velocity is zero.
dx/dt = 2(t-3) = 0 => t = 3 sec.
Hence at t = 3 sec., the displacement of particle is zero, using Eq. (A).
We can find the displacement using the plot. Here we have to find displacement of particle when velocity is zero i.e. dx/dt = 0 so that slope of tangent to the curve at that time should be zero. It is clear from the curve that it is at t = 3 sec. At this time the displacement of particle is zero.
To find the work done in the first 6 sec., we can see work energy theorem. According to this theorem,
Work done = Change in K.E.
W = ΔK = 1/2mvf2 - 1/2mvi2
=> W = 1/2m(vf2 - vi2)
Where m is the mass of particle; vf is the velocity when t = 6 sec; and vi is the velocity when t = 0 sec.
As velocity of particle
dx/dt = 2(t - 3)
.·. vf = 2(6 - 3) = 6m/s
vi = -6m/s.
.·. ΔW = ΔK = 1/2 m(36-36) = 0
We can do this part by another method. As work done is defined as W = . .
We can see the final position of particle is same as the initial position. Hence displacement of particle is zero. So the work done by the force acting on particle in the first 6 sec. is zero.