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Question

# A long block A is at rest on a smooth horizontal surface and small block B whose mass is half of A, is placed on A at one end and projected along A with some velocity u. The coefficient of friction between the blocks is μ:

A
The blocks will reach the final common velocity u3
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B
The work done against friction is two -thirds of the initial kinetic energy of B.
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C
Before the blocks reach a common velocity, the acceleration of A relative to B is 23μg.
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D
Before the blocks reach a common velocity the acceleration of A relative to B is 32μg.
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Solution

## The correct option is D Before the blocks reach a common velocity the acceleration of A relative to B is 32μg.Let v be common velocity of the blocks Using conservation of linear momentum, since friction is internal for the 2 block system, m2u=(m+m2)v v=u3 Initial kinetic energy of the system ki=12(m2)u2 ki=14mu2 Using Work energy theorem, mu24−wf=12(3m2)u29 14mu2−112mu2=wf wf=3mu2−mu212=mu26=23×14mu2 wf=23ki Force of friction between blocks fk=μ(m2)g Acceleration of A to right. aA=μmg2(m)=μg2 Acceleration of B to left, aB=μmg2(m2)=μg Acceleration of A relative to B, aAB=aA−(−aB) aAB=μg2+μg aAB=3μg2

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