How Does Pulling the String Affect the Speed of a Ball in Circular Motion?

[Koko$]

Homework Statement

Ball of mass=m on the frictionless table is connected to a string that passes in hole that is in table. The ball is set into a circular motion on a circle of radius = R. If the string pulled from the bottom, withe the force F, so that the radius of motion of ball would decrease, how would the final speed of ball change?

Homework Equations

I know that centripetal force sets the ball into motion, it is given by the formula:

$F_C = \frac{mv^2}{r}$
where
m - mass of the body(m in my example)
v - velocity in my example:
$v_0$
r - radius of path - R in my example
F - force applied

So if the volocity is $v_0$, mass is $m$ and the string that the ball is connected to, is pulled so that the radius of path is decreased, how would the final velocity change(the force F is applied to the string at its bottom)

The Attempt at a Solution

$F_C = \frac{mv^2}{r}$

Therefore:

$v= \sqrt {\frac {R_f F_C} {m}}$ where:

$R_f = R-r$ and $F_C =$ force applied at the bottom of string., so $F_C = F$

 

[Simon Bridge]

Homework Statement

Ball of mass=m on the frictionless table is connected to a string that passes in hole that is in table. The ball is set into a circular motion on a circle of radius = R. If the string pulled from the bottom, withe the force F, so that the radius of motion of ball would decrease, how would the final speed of ball change?

Homework Equations

I know that centripetal force sets the ball into motion,

No it doesn't - it keeps the ball in circular motion, but it does not "set the ball into motion". You can see this by pulling on the string of a stationary ball - would you expect circular motion to start?

Before you settle on your approach - it is useful to check your understanding of the physics.
Are there any other forces on the ball? i.e. is the ball rolling or slipping or a mixture of both?
Are there any other physical laws or rules that may be applied to this system?

$F_C = \frac{mv^2}{r}$
This equation tells you the constant speed for circular motion if the central force is Fc. For F>Fc, v must increase to keep the same radius. If v is unchanged, then r must decrease. You will see, if you try this, that r decreases ... but does v change as well? The math can balance out if v increases or decreases.
You have watched lots of things spiral down a hole in your life - what usually happens?

But there are other ways to approach things - i.e. is there a torque on the ball?

 

[Koko$]

The ball is set into motion on circular path of radius R. Ball is connected to a string, at the bottom of the string, the force acting downward is applied, so the radius(R) is decreased. I found similar problem in web, but there is no force applied, but the mass M.

So maybe I should just convert force into mass, aply to the $F_C$ formula and from this $F_C$ calculate the final velocity,.. but how much shorter will be the string(radius) if the force(some force - not given in numbers) is applied?

 

[rcgldr]

Consider the fact that for this problem, angular momentum is conserved.

 

[Simon Bridge]

The ball is set into motion on circular path of radius R. Ball is connected to a string, at the bottom of the string, the force acting downward is applied, so the radius(R) is decreased. I found similar problem in web, but there is no force applied, but the mass M.

Do not copy results from anywhere unless you understand them.

You appear to be trying to find an equation to apply.
That is a beginner technique and only works when there are not very many equations to memorize - i.e. when you don't know a lot of physics.
Use physics instead - the equations will follow.

When someone, trying to help you, asks a question - it is usually a good idea to try answering it:

i.e. is there a torque on the ball?

... if there is no external torque on the ball then there is no angular acceleration, so there is no change in the angular momentum. This is exactly the same as the regular F=ma case only with the rotational equivalents.

This sort of reasoning is called: "using the physics".

The torque is equal to the rate of change of angular momentum - if the torque is zero that means:

... angular momentum is conserved.