# A motion of a Conical Pendulum

A ball is attached to a string and swung so that it travels in a horizontal circle.

Suddenly the angular velocity is increased. Will the angle with the vertical increase, decrease or stay the same? Justify your answer.

###### Solution.

According to Newton’s Second Law for uniform circular motion, the net force acting on the ball equals mac.

Fnet = mac

The expression Fnet = mac is a vector equation so we can write it as two component equations: Fnet,x = mac and Fnet,y = 0. (The ball is not accelerating vertically.)

In the x-direction, there is only FT,x. Thus,

FT,x = mac

or

FT∙sin(α) = mac    (1).

The magnitude of the centripetal acceleration is given by

ac2r    (2).

We substitute equation (2) into equation (1) to obtain

FT∙sin(α) = mω2r    (3).

In the y-direction, Fnet,y = 0 becomes

FT,y – mg = 0

or

FT∙cos(α) = mg    (4).

We divide the equation (4) by equation (3) and obtain

cos(α)/sin(α) = g/(ω2r)    (5).

We can determine α by using the dimensions of the string and the circle radius.

sin(α) = r/l    (6).

We substitute equation (6) into equation (5) and we get

cos(α) = g/(ω2l)    (7).

We can see that the angle that the string makes with the vertical depends on the acceleration of gravity, the length of the string and the angular velocity.

Therefore, if the angular velocity is increased then the angle α between the string and the vertical will also increase.