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Question

A non uniform rod OA of liner mass density λ=λ0×(λ0=const.) is suspended from ceiling with hinge joint O & light string as shown in figure. Find the angular acceleration of rod just after the string is cut
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A
2gL
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B
gL
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C
4g3L
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D
None of these
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Solution

The correct option is C 4g3L
Xcm=xdmdm=L0x×λoxdxL0λoxdx
=λoLox2dxλoLoxdx
Xcm=(x33)Lo(x22)Lo=23×L3L2=2L3
Moment of inertia about O
dI=dm×x2I=dmx2
Loλoxdx×x2λox3dx
λo4(x4)2oI=λo4×L4
& mass can be given as:
m=Loλoxdx=λo2(x2)L0=λ22×L2
So, torque about the center can be given as,
τ=mg×2L3=Iα
λ02×L2g×2L3=λ04×L4α
λogL33=λoαL44
λog3=λoαL4α=4g3L
Hence, the correct answer is option (C)

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