Question

A bob of mass m = 0.250 kg is suspended from a fixed point with a massless string of length L = 22.0 cm. You will investigate the motion in which the string traces a conical surface with half-angle theta = 24.0^\circ.What tangential speed v must the bob have so that it moves in a horizontal circle with the string making an angle 24.0^\circ with the vertical?

Answer 1

To solve the problem, it is necessary to use the concepts of gravitational force, centripetal force and trigonometric components that can be extrapolated from the statement.

By definition we know that the Force of Gravity is given by

$F_g=mg$

Where,

m= Mass

g = Gravitational Acceleration

The centripetal force is given by,

$F_c = \frac{mv^2}{R}$

Where,

m = Mass

v = Velocity

R = Radius

For the case described in the problem, the Force of gravity the net component would be given by sin?, While for the centripetal force the net component is in the horizontal direction, therefore it corresponds to the $cos\theta$

Then,

$F_g = mg sin\theta$

$F_c = \frac{mv^2}{r}cos\theta$

From the radius we have its length but not the net height, which would be given by

$r = L sin\theta$

So equating the equations we have to

$F_g = F_c$

$mg sin\theta=\frac{mv^2}{r}cos\theta$

$mg sin\theta=\frac{mv^2}{Lsin\theta}cos\theta$

Re-arrange to find v,

$v = \sqrt{\frac{gLsin^2\theta}{cos\theta}}$

Replacing with our values

$v = \sqrt{\frac{(9.8)(22*10^{-2})(sin^2 24)}{cos24}}$

$v = 0.624$

Therefore the tangential velocity of the mass is 0.624m/s