A particle of mass m is suspended from a ceiling through a string of length L. The particle moves in a horizontal circle of radius r. Find a. the speed of the particle and b. the tension in the string. Sch a system is called a conical pendulum.

Explanation

`theta`:angle made by the string with vertical
`sin theta=(r )/(L)`
The forces acting on the particle are `(i)` weight of particle `(mg)` and `(ii)` tension in the string.
Here, `T costheta` balance the weight and `T sin theta` provides neccessary centripetal force. `uarr:T cos theta=mg` ...`(i)`
`rarr:T sin theta=(mv^(2))/(r )` ....`(ii)`
`tan theta=v^(2)/(rg)impliesv=sqrt(rg tan theta)`
`T=(mg)/(cos theta)`
`v=sqrt(rg tan theta)=((rgxxr)/sqrt(L^(2)-r^(2)))^(1//2)=(rsqrt(g))/((L^(2)-r^(2))^1//4))`
`T=(mg)/(cos theta)=(mg L)/sqrt(L^(2)-r^(2))`