Centripetal Force and radius of the circle

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adjacent
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When the radius of the circle decrease the object moves faster(With the same force)
I believe this is a misconception .In common sense the object revolves around more,This leads us to think that it is moving fast but actually the object moves the same distance.Am I right?
 

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When the radius of the circle decrease the object moves faster(With the same force)
This is wrong, if "the same force" refers to centripetal (or centrifugal) force. Check its formula to see it.

If you pull a circling object towards the center, it will increase its velocity (and increase the required force for a circular orbit a lot).

If you have something else in mind, please explain it.
 
  • #3
rcgldr
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During the time that the object is pulled inwards, a portion of the force is in the direction of the path of the object, which increases the speed of the object. Angular momentum is conserved. The work done on the object = the tension (as a function of radius) x change in radius, which equals the increase in KE (1/2 m v^2) of the object. Angular momentum is conserved. For example, if the radius is reduced to 1/2 the original radius, the speed is doubled and the tension increased by a factor of 8. This image is an example, the short inward lines are perpendicular to the path, while the long lines are in the direction of the inwards force:

hole.jpg
 
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When the radius of the circle decrease the object moves faster(With the same force)
I believe this is a misconception .In common sense the object revolves around more,This leads us to think that it is moving fast but actually the object moves the same distance.Am I right?

It's surprisingly to difficult to reduce the radius while keeping the centripetal force constant; rcgldr's post explains why. Note especially his comment about conservation of angular momentum; if angular momentum is conserved the centripetal force will be greater after the radius is reduced.

Counterintuitvely, if you consider the centripetal force equation with F and m constant
[tex]F=\frac{mv^2}{r} \Rightarrow v \propto \sqrt{r}[/tex]
the speed decreases with the radius - but the only way you can make that happen is to apply some external action to reduce the angular momentum and the kinetic energy of the system.
 
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This is wrong, if "the same force" refers to centripetal (or centrifugal) force. Check its formula to see it.

If you pull a circling object towards the center, it will increase its velocity (and increase the required force for a circular orbit a lot).

If you have something else in mind, please explain it.
In my text book it was stated:The centripetal force increase if,
-mass of the object is increased
-Speed of the object is increased
-Radius of the circle is reduced

By looking at these one may consider,as radius decrease the centripetal force(C.F) increase.If the C.E increase the speed of the object is increased.Therefore as radius decrease,the speed of the object increase.
I know that the object will be accelerating but it will have the same speed.It increase its velocity by changing direction continuously. My question is how do the speed change?Provide an explanation and is my explanation correct?
Note:I dont know about momentum and functions and Centripetal force equation all that yet Im 9th grade
 
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By looking at these one may consider,as radius decrease the centripetal force(C.F) increase.
At the same velocity, indeed.
If the C.E increase the speed of the object is increased.Therefore as radius decrease,the speed of the object increase.
I know that the object will be accelerating but it will have the same speed.
Not necessarily (valid for all 3 statements).
It increase its velocity by changing direction continuously.
A change in direction does not have to change the speed.
My question is how do the speed change?
That depends on the setup. See rcgldr's post for a common situation.
 
  • #7
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I dont know about momentum and functions and Centripetal force equation all that yet; Im in 9th grade
No problem, it's not that hard as long as we stick with the simple case: We have a weight tied to the end of a string, we're spinning it about our head, we won't worry about gravity pulling the weight down, and we can change the radius by pulling in on the string to reduce the radius or letting the string out to increase the radius, and only consider the situation when the weight is moving in a perfect circle (it may do some funny stuff while the radius is actually changing), the string has no mass and doesn't stretch.

In this ideal case, the tension in the string (this is the centripetal force) will be
[tex]F=\frac{mv^2}{r}[/tex]
where m is the mass, v is its speed, and r is the radius of the circle. This can be calculated from Newton's second law (or you can take my word for it)

The quantity called angular momentum is given (in this simplified case only, but that's good enough here) by [itex]L=mvr[/itex] and it will remain constant as we pull the string in or let it out, causing both v and r to change. This is "The Law of Conservation of Angular Momentum".

You know the initial radius and speed so you can calculate the initial value of L and F, and you can use the angular momentum formula to see how the speed changes with the radius (remember, L cannot change in this setup unless you deliberately spin the thing faster), and then the force formula to see how the force changes.
 
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  • #8
rcgldr
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how does the speed change?
I answered that in my previous post. During the transtion where the radius is decreased, the path of the object is no longer a circle. In the image I posted above, I use a spiral like path. During this transition when the path is not circular, a component of the force is in the same direction as the path of the object, changing it's speed. The speed is increased if the radius is decreased, and the speed is decreased if the radius is increased.

The path doesn't have to be a spiral. You could adjust the radius inwards and outwards periodically so that the path is an ellipse. The objects speed would be fastest when the radius was smallest and slowest when the raidus was largest. Again it's the component of force in the direction of the path of the object that is causing it's speed to change. Using the fact that angular momentum is conserved makes it easier to calculate the speeds (versus using calculus to determing the force versus object speed versus radius to use the force to determine acceleration in the direction of the object and change in speed).
 

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