Conical Pendulum

Tension, Free Body Diagrams and Speed

A 1kg mass hangs from a fixed point on a 1m long, light, inextensible, string. It has a velocity v that is entirely horizontal, and so it moves in a horizontal circle. This circle is centered on the vertical line below the point of support, and the string makes an angle of 45 degrees with the vertical at all times. This is known as a conical pendulum.

a) What is the tension in the string when the mass is moving?

To find the tension in the string, we look at the free body diagram. When the mass is in circular motion, the net force is exactly horizontal, so the vertical component of the tension must exactly cancel the weight of the mass.

T sin(45^o)=mg

T=14 N

b) Find the speed of the mass.

The centripetal force is the horizontal component of the tension:

F_c=mv²/R=Tcos(45^o)=10N

As the rope is 1m long, the radius of the circular motion is sin(45^o).

v²=7.1(m/a)²

v=2.66m/s