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Question

Sand kept inside the trolley drains out from its floor at a constant rate of 0.2 kg/s. A force 15 N is acting on the trolley as shown in figure. After time 5 sec, which of the following option(s) is/are true?


A
Mass of trolley is 1 kg
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B
Net force on trolley is 15 N
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C
Velocity of trolley is 75ln2 m/s
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D
Velocity of trolley is 75ln3 m/s
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Solution

The correct options are
A Mass of trolley is 1 kg
B Net force on trolley is 15 N
C Velocity of trolley is 75ln2 m/s
Initial mass of trolley, m0=2 kg
Rate of mass reduction, μ=dmdt=0.2 kg/s
Final mass of trolley after 5 s is
m=m0μt
m=2(0.2×5)
m=1 kg
Let, at t=5 sec, velocity of trolley be v.
Since sand is drained out from the trolley, its exit velocity will be the same as the velocity of trolley.


Relative velocity with which sand leaves the trolley
vst=vsvt=vv=0
When mass is reducing with time, then Ft acts opposite to the direction of vst
As velocity of sand w.r.t trolley is zero,
Ft=vstdmdt=0
Therefore, no thrust force will act,
i.e Fnet=F
ma=F
(m0μt)dvdt=F
v0dv=t0Fdtm0μt
v=F[ln(m0μt)μ]t0
v=Fμ[ln(m0μt)ln(m0)]
v=Fμlnm0μtm0=Fμlnm0m0μt
Substituting values of F=15 N, m0=2 kg, μ=0.2 kg/s, t=5 s
v=150.2ln22(0.2×5)
v=75ln2 m/s
Hence, net force on trolley is,
Fnet=F=15 N

(a), (b), (c) are correct choices.

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