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

The situation is shown in figure. The angle `theta` made by the string with the vertical is given by sin`theta` = r/L ... (i)
The forces on the particle are
(a) the tension T along the string and
(b) the weight mg vertically downward
The particle is moving in a circle with a constant speed v. Thus, the radial acceleration towards the centre has magnitude `v^(2)`/r. Resolving the forces along the radial direction and applying. Newton’s second law,
Tsin `theta = m(v^(2)//r)" "`(ii)
As there is no acceleration in vertical directions, we have from Newton’s law,
Tcos`theta` = mg`" "`(iii)
`tan theta = (v^(2))/(rg) " or " v = sqrt(rg tan theta)`
and from (iii), T = `(mg)/(cos theta)`
Using, (i), `v = (r sqrt(g))/((L^(2) -r^(2))^(1//4)) " and " T - (mgl)/((L^(2) -r^(2))^(1//2))`