Question

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. Such a system is called a conical pendulum.

• A. Speed = rg tan Tension =
• B. Speed = Tension =
• C. Speed = Tension = mg
• D. None of these

Answer 1

The correct option is B

Speed =

Tension =

The situation is shown in figure. The angle θ made by the string with the vertical is given by sinθ = $\frac{r}{L}$ . . . . . . (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 $\frac{v2}{r}$. resolving the forces along the radial direction and applying Newton's second law,
$Tsin\theta =\frac{m{v}^2}{r}$
$TCos\theta =mg$
$\Rightarrow tan\theta =\frac{v^2}{rg}\Rightarrow v=\sqrt{rgtan\theta }$
From the figure,as $Tcos\theta =mg$
$T=\frac{mg}{cos\theta }=\frac{mgL}{\sqrt{L^2-r^2}}$