physics homework help

Question

Suppose you were to reduce the radius of the circular path of a ball on a string by one-half without changing the speed. How would that affect the size of the inward force (tension) in the string? Show your work please. 

Answer 1

I assume you mean without changing the tangential speed v.

 

T₁ = mv²/r₁

 

At same tangential speed v but at r₁/2,

 

T₂ = mv²/(r₁/2) = 2mv²/r₁ = 2T₁

 

The tension is doubled if the radius is halved at constant tangential speed.  However, such a problem is not very realistic, because if you pull in the string to make the radius smaller, angular momentum is conserved, and the tangential speed will actually increase.  Conservation of angular momentum will make

 

mv₁r₁ = mv₂r₂ ⇒ v₁r₁ = v₂r₂

 

If r₂ = r₁/2, then

 

v₁r₁ = v₂(r₁/2) ⇒ v₂ = 2v₁

 

Then

 

T₂ = mv₂²/r₂ = m(2v₁)²/(r₁/2) = 8mv₁²/r₁ = 8T₁

 

So realistically, the inward force should increase by a factor of 8 if you reduce the radius by half.  But in this problem, as stated, with constant tangential speed v, the inward force is doubled.