What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

Options

A. `sqrt(2gR)`
B. `sqrt(3gR)`
C. `sqrt(5gR)`
D. `sqrt(gR)`

Answer

C

Explanation

According to question, we have

Let the tension at point A be `T_(A)`. So, from Newton's second law
`T_(A)-mg=(mv_(c)^(2))/(R)`
Energy at point `A=(1)/(2)mv_(0)^(2)` ….(i)
Energy at point C is
`(1)/(2)mv_(c)^(2)+mgxx2R` ....(ii)
Applying Newton's second law at point C
`T_(c)+mg=(mv_(c)^(2))/(R)`
To complete the loop `T_(c)ge0`
So, `mg=(mv_(c)^(2))/(R)`
`rArr" "v_(c)=sqrt(gR)` ...(iii)
From Eqs. (i) and (ii) by conservation of energy
`(1)/(2)mv_(0)^(2)=(1)/(2)mv_(c)^(2)+2mgR`
`rArr" "(1)/(2)mv_(0)^(2)=(1)/(2)mgR+2mgR" "(becausev_(c)=sqrt(gR))`
`rArr" "v_(0)^(2)=gR+4gR`
`rArr" "v_(0)=sqrt(5gR)`