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 free-fall acceleration g.

v = √(Lsinθ tanθ)

Explanation:

From the diagram attached,

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 = Lsin θ

Thus, the centripetal acceleration is given as;

a = v²/r = v²/Lsin θ

Force = mass x acceleration

Thus,

Centripetal force = mv²/Lsin θ

Let's balance the vertical forces to obtain,

T cosθ - mg = 0

Thus, T cosθ = mg - - - - (eq1)

Similarly, let's balance the horizontal forces to obtain;

T sinθ = F = mv²/Lsin θ

So, T sinθ = mv²/Lsin θ - - - - (eq2)

Let's divide eq 2 by eq 1 to get;

Tsinθ/Tcosθ = (mv²/Lsin θ)/mg

tanθ = gv²/Lsinθ

Thus, v² =  Lsinθ tanθ

v = √(Lsinθ tanθ)