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Question

A flat car o mass m0 starts moving to the right due to a constant horizontal force F. Sand spills on the flat car from a stationery hopper. The rate of loading is constant and equal to μ kg/s. find the time dependence of the acceleration of the flat car in the process of loading. The friction is negligibly small.
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A
Fm0(m0+μt)2
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B
Fm0(m0+μt)
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C
2Fm0(m0+μt)2
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D
Fm02(m0+μt)2
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Solution

The correct option is A Fm0(m0+μt)2
Answer is A.
Initial velocity of the flat car is zero. Let v be its velocity at time t and m its mass at that instant. Then
At t=0,v=0 and m=mo att=t,v=v and m=mo
Here, vr=v(backwards).
dmdt=μ
Therefore, Ft=vrdmdt=μv(backwards)
Net force on the flat car at time t is fnet=FFt
or mdvdt=Fμv...(i)
or (m0+μt)dvdt=Fμvorv0dvFμv=t0dtm0+μt
Therefore, 1μ[ln(Fμv)]v0=1μ[ln(m0+μt)]t0
ln(FFμv)=ln(m0+μtm0)
So, FFμv=m0+μtm0orv=Ftm0+μt
from Eq.(i), dvdt = acceleration of flat car at time t
or =Fμvma=⎜ ⎜ ⎜ ⎜FFμtm0+μtm0+μt⎟ ⎟ ⎟ ⎟ora=Fm0(m0+μt)2.
Hence, the time dependence of the acceleration of the flat car in the process of loading is Fm0(m0+μt)2.

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