What tangential speed, v, must the bob have so that it moves in a horizontal circle with the string always making an angle θ from the vertical? express your answer in terms of some or all of the variables m, l, and θ, as well as the acceleration due to gravity g?

Refer to the diagram shown below.

v = the tangential speed.
r = the radius of the horizontal circle.
T = tension in the string.
θ =  the angle that the string makes with the vertical
m =  Bob's mass  (mg = the weight)
F =  centripetal force
l = the length of the string

From geometry,
r = l sin θ

The centripetal acceleration is

The centripetal force is


For vertical force balance,
T cosθ = mg                              (1)
For horizontal force balance,
         (2)
Divide (2) by (1).


   

The third hammer 8 pounds