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

(c )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 2nd law at point C
`T_(c )+mg=(mv_(c )^(2))/(R)`
To complete the loop `T_(c )le0`
so, `mg=(mv_(c )^(2))/(R)`
`impliesv_(c )=sqrt(gR)` .......(iii)
From Eqs. (i) and (ii) by conservation of energy
`1/(2)mv_(0)^(2)=1/(2)mv_(c )^(2)+2mgR`
`implies1/(2)mv_(0)^(2)=1/(2)mgR+2mgRxx2`
`(thereforev_(c )=sqrt(gR))`
`impliesv_(0)^(2)=gR+4gR`
`impliesv_(0)sqrt(5gR)`