A thin uniform rod of mass m, length L, area of cross section A and young's modulus Y rotates at angular velocity 'ω' in a horizontal plane about a vertical axis passing through one of its ends, then
A
tension in rod at distance 'r' from the axis of rotation is mw2(L2−r2)2L.
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B
elongation of rod is 23mw2L2AY.
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C
elongation of rod is mw2L23AY.
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D
elongation of small element of rod decreases linearly as we move away from axis.
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Solution
The correct option is C elongation of rod is mw2L23AY.
T=m′w2[r+(L−r)2]=mL(L−r)w2(L+r2)
T=mw2(L2−r2)2L
Y=TAdldr⇒dl=TAYdr
Δl=∫dl=∫L0mw22L(L2−r2)drAY
Δl=mw22LAY.23L3=mw2L23AY
Since tension does not decrease linearly with radius, elongation of small element also does not decrease linearly with radius.