A particle of mass 100 g, is made to describe a vertical circle of radius 1 m. Its instantaneous speed is `1ms^(-1)` when the string makes an sngle of `30^(@)` with the vertical Find the tension in the string at this position. Can the particle complete its circular path? `(g=10ms^(-2))`

Options

A. `0.66N` No it will not complete circular path
B. `0.966N` yes it will complete circular path
C. `0.1N` No it will not complete circular path
D. `0.966N` No it will not complete circular path

Answer

D

Explanation

Tension in the string when it make an angle `theta` with the vertical is
`T=(mv^(2))/®+mg cos theta`
Let speed at the lowest point be `v_(1)`
`v_(1)^(2)=v^(2)+2gr(1-costheta)`
`(v_(1))_(min)=sqrt(5gr)"chect if"v_(1)lt(v_(1))_(min).`
If yes, then it would not complete its vertical circular path.
The tension in the string. when it makes and angle `theta` with the vertical is
`T=(mv^(2))/(r)+mgcostheta`
`=(0.1xx1^(2))/(1)+0.1xx10xx0.866=0.966N`
let the speed at the lowest point be `v_(1)`
`v_(1)^(2)=v^(2).+2gr(1-costheta)`
`=1^(2)+2xx10xx1xx(1-0.866)`
`=1+20xx0.134`
`=3.68`
`v_(1)=sqrt3.68=1.91ms^(-1)`
`(v_(1))_(min)=sqrt(5gr)=sqrt(5xx10xx1)=7.07ms^(-1)`
`v_(1)lt(v_(1))_(min)implies` the particle would not be able to complete its circular path.