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A thin uniform rod is free to rotate abo...

A thin uniform rod is free to rotate about a fixed smooth horizontal axis as shown. A point mass hits horizontally with velocity `v_(0)` to the one end `B` of the rod. When it hits, it sticks to the rod, then :

A

Minimum value of `v_(0)` for the rod to rotate by an angle `(pi)/(2) is 2sqrt(gL)` .

B

Angular acceleration of the rod when the rod is horizontal is `(9g)/(8L)` .

C

Force applied by the axis on the rod in the horizontal state is `5 mg//6` .

D

for a small velue of `v_(0)` the rod performes small oscillations with a period of `(4)/(3) pi sqrt((l)/(g))`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
A thin ………….
(A) Conserving angular momentum about A
`mv_(0)L=((mL^(2))/(3)+mL^(2)) omega`
`implies omega = (3v_(0))/(4L)`
Now, applying energy conservation
`(1)/(2).(4mL^(2))(3)omega^(2)=mgL+(mgL)/(2)`
Put `omega`
`:. v_(0)= 2sqrt(gL)`

`mgL+(mgL)/(2)=(4mL^(2))/(3).alpha`
`alpha=(9g)/(8L)`

`mg+mg-N=2ma_(COM)`
`N=5mg//16`
(D) `T = 2pisqrt((I)/(mgL_("Distance of COM from axis")))`
`(2)/(3)pi sqrt((2l)/(g))`
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