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Question

Two balls A and B of mass m=1 kg each are joined together by a rigid massless rod of length l=2 m. A ball P of mass m=1 kg moving with speed (u=12 m/s) normal to AB, strikes end A of the rod and sticks to it. Then immediately after impact, the velocity of A+P in m/s with respect to COM will be



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Solution



COM of the system from point A(origin) is,
yCOM=m1y1+m2y2+m3y3m1+m2+m3
yCOM=m(0)+m(0)+m(l)m+m+m
yCOM=l3=23 m
Just after collision, the system starts moving with velocity vCOM=v (translational velocity) and with angular velocity ω about COM.
τext=0, applying angular momentum conservation for system about COM,
Li=Lf
mu (yCOM)=Ltrans+Lrot
[taking antilockwise sense of rotation as +ve]
mu(l3)=0+(Iω)
mu(l3)=Iω ...(i)
[Ltrans=0,because velocity vector passes through COM]

MOI of system about centre of mass is,
I=2m(l3)2+m(2l3)2
I=2ml23 ...(ii)
Substituting in Eq (i),
mu(l3)=2ml23ω
ω=u2l=122×2=3 rad/s

Velocity of (A+P) just after impact with respect to centre of mass of system C is the tangential velocity given as:
v=ωr ...(iii)
where r=l3, is the radius of circular path for (A+P) about COM of the system

v=(3×23)=2 m/s

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