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Question

A particle of mass 102 kg is moving along the positive x-axis under the influence of a force F(x)=K2x2 where K=102 Nm2. At time t=0, particle is at x=1 m and its velocity is v=0. The time at which it reaches x=0.25 m is

A
1.7 s
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B
1.2 s
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C
1.9 s
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D
1.5 s
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Solution

The correct option is D 1.5 s
Given,
mass of the particle, m=102 kg

Force acting on the particle,

F=K2x2...(1)

We know that force acting on the moving particle,

F=ma...(2)

Equating (1) and (2)

ma=K2x2

Substituting the values of m and K in the above equation,

102×a=1022x2

a=12x2

vdvdx=12x2 [a=vdvdx]

Rearranging and integrating the equation, as x=1 m,v=0 m\s,

v0vdv=12x1x2dx

12[v2]v0=+12[x1]x1

v2=[1x1]=1xx

v=±1xx

Initially the particle was moving in +x-direction at x=1. When the particle is at x=0.5, obviously its velocity will be in -x-direction. The force acting in -x-direction first decreases the speed of the particle, bring it momentarily at rest and then changes the direction of motion of the particle.
Thus, we should consider the negative velocity.

v=dxdt=1xx

0.251x1xdx=t0dt

On taking x=sin2θ, we get dx=2sinθcosθdθ.

And limits as when x=1,θ=90
and when x=0.25,θ=30.

3090sin2θ1sin2θ×2sinθcosθdθ=t0dt

30902sin2θdθ=t0dt

3090(1cos2θ)=t0dt

[θsin2θ2]3090=[t]t0

t=30sin60290+sin1802

t=π634π2+0

t=1.48

t=1.48 s1.5 s

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