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 ma_{c}.

F_{net} = ma_{c}

The expression F_{net} = ma_{c} is a vector equation so we can write it as two component equations: F_{net,x} = ma_{c} and F_{net,y} = 0. (The ball is not accelerating vertically.)

In the x-direction, there is only F_{T,x}. Thus,

F_{T,x} = ma_{c}

or

F_{T}∙sin(α) = ma_{c} (1).

The magnitude of the centripetal acceleration is given by

a_{c}=ω^{2}r (2).

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

F_{T}∙sin(α) = mω^{2}r (3).

In the y-direction, F_{net,y} = 0 becomes

F_{T,y} – mg = 0

or

F_{T}∙cos(α) = mg (4).

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

cos(α)/sin(α) = g/(ω^{2}r) (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/(ω^{2}l) (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.

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