The motion. The bob moves in a horizontal circle of radius r = L sinθ, at a constant speed v. Its velocity keeps changing direction, so it is accelerating, and it needs a net force towards the centre of the circle: the centripetal force. There is no separate “centripetal force” to add to the diagram: it is the net of the forces that act on the bob.

The two forces are the weight mg (down) and the tension T (along the string, towards the pivot). The bob does not move up or down, so the vertical components balance. The horizontal component of the tension is the net force:

vertical: T cosθ = mg  ⇒  T = mg / cosθ
horizontal: T sinθ = mv²/r  ⇒  F_net = mg tanθ

Dividing the two equations gives tanθ = v²/(rg). With r = L sinθ this gives the speed for a given angle, v = √(g L sinθ tanθ), and the angle for a given speed. The faster the bob goes, the wider the cone and the bigger the tension. As the speed increases, θ gets closer to 90° but the string can never be horizontal, since the tension would need to be infinite to hold up the weight.

Mass does not matter for the angle, the speed or the period: change the mass and only the forces change (they are all proportional to m). The time for one revolution is

period = 2πr/v = 2π √(L cosθ / g)

which gets shorter as the cone gets wider.

The arrows are drawn to the same scale, relative to the biggest of the three forces. The view can be rotated by dragging it.