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The end 'A' of a uniform rod AB of lengt...

The end `'A'` of a uniform rod `AB` of length `'l'` touches a horizontal smooth fixed surface. Initially the rod makes an angle of `30^(@)` with the vertical. Find the magnitude of displacement of the end `B` just before it touches the ground after the rod is released.

Text Solution

Verified by Experts

The correct Answer is:
`(sqrt(13)l)/(4)`
Displacement `=sqrt((3)/(4)l^(2)+(l^(2))/(16))`
`=(1)/(2)sqrt(3+(1)/(4))=(sqrt(13l))/(4)`

Alternate solution
Initially the rod is at rest
`:. U_(cm) ` or rod `=0`
All forces on rod, acti in vertical direction. Hence acceleration of centre of mass is vertically downwards.
`:.` centre of mass of rod moves vertically downwards.

`BP=l sin 60^(@)=(sqrt(3))/(2)l, C_(1)P=(l)/(2) cos 60^(@)`
`:. PB_(1)=B_(1)C_(1)-C_(1)P=(l)/(2)(1-cos 60^(@))=(l)/(4)`
`:. `Displacement of end `B` is `B B_(1)`
`=sqrt(BP^(2)+PB_(2)^(2))=sqrt(((sqrt(3))/(2)l)^(2)+((l)/(4))^(2))=(sqrt(13l))/(4)`
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Knowledge Check

  • A uniform rod AB of mass m and length l at rest on a smooth horizontal surface. An impulse P is applied to the end B . The time taken by the rod to turn through a right angle is :

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    D
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