Answer to Question #133549 in Physics for Josh

Question

A conical pendulum consists of a bob (small mass at end of massless rope) that moves in a horizontal circle at constant speed (the cord sweeps out a cone as the bob rotates). Here the angle θ = 18.92°, the length s = 2.59 m, and the bob has an unknown mass.
(a) Draw a free body diagram of the bob.
(b) Using your diagram, sum the forces on the bob(you should have
two equations).
(c) Solve your system to find the centripetal acceleration of the bob.
(d) Find the speed of the bob.
(e) Find the period of the motion.

Answer 1

(a) The diagram is below:

(b) Sum the forces:

$$F\text{ cos}\theta=mg,\\ F\text{ sin}\theta=m\frac{v^2}{R}.$$

(c) Solve for the centripetal acceleration by dividing the second equation by the first one:

$$a_c=\frac{v^2}{r}=g\text{ tan}\theta.$$

(d) Find the speed:

$$v=\sqrt{gr\text{ tan}\theta}=\sqrt{gl\text{ sin}\theta\text{ tan}\theta},\\ v=1.68\text{ m/s}.$$

(e) Find the period:

$$T=\frac{2\pi r}{v}=2\pi\sqrt{\frac{l\text{ cos}\theta}{g}}=3.14\text{ s}\approx\pi\text{ s}.$$