Is Vertical Circular Motion Ever Uniform?

[Sam McCrea]

Hi, I'm quite confused about vertical circular motion (particularly at minimum speed) and would appreciate any help.
I'm confused about velocity in a "loop the loop" situation. Say (theoretically) a car was going minimum speed around a loop (which I understand is sqrt of rg). Therefore the total energy must be the kinetic energy that allowed it to go that minimum speed at that point plus the gravitational potential energy. This therefore must be all the kinetic energy the car has at the bottom of the loop and if you rearrange this to find the velocity is a different value from the top speed. This must mean the velocity is getting smaller as it goes up the loop as energy is being changed however this must mean it is not uniform circular motion and centripetal force is not constant.

However on a ball attached to a string in vertical circular motion I read that as the ball nears the bottom of the loop the tension force increases which balances out the gravity force component making the centripetal force constant and from looking at the formula this must mean the speed is constant (velocity is the same value but constantly changing direction). However I read on another forum that it would still not be uniform circular motion and would be constantly changing speed (https://physics.stackexchange.com/q...form-circular-motion-can-it-really-be-uniform) but I do not understand the explanation. From this site here (http://www.schoolphysics.co.uk/age1...n/text/Motion_in_a_vertical_circle/index.html) tension is calculated from only one velocity value.As you can see I'm quite confused so can someone please tell me where I'm going wrong.
Thanks

 

[ehild]

Hi, I'm quite confused about vertical circular motion (particularly at minimum speed) and would appreciate any help.
I'm confused about velocity in a "loop the loop" situation. Say (theoretically) a car was going minimum speed around a loop (which I understand is sqrt of rg). Therefore the total energy must be the kinetic energy that allowed it to go that minimum speed at that point plus the gravitational potential energy. This therefore must be all the kinetic energy the car has at the bottom of the loop and if you rearrange this to find the velocity is a different value from the top speed. This must mean the velocity is getting smaller as it goes up the loop as energy is being changed however this must mean it is not uniform circular motion and centripetal force is not constant.

However on a ball attached to a string in vertical circular motion I read that as the ball nears the bottom of the loop the tension force increases which balances out the gravity force component making the centripetal force constant and from looking at the formula this must mean the speed is constant (velocity is the same value but constantly changing direction). However I read on another forum that it would still not be uniform circular motion and would be constantly changing speed (https://physics.stackexchange.com/q...form-circular-motion-can-it-really-be-uniform) but I do not understand the explanation. From this site here (http://www.schoolphysics.co.uk/age16-19/Mechanics/Circular motion/text/Motion_in_a_vertical_circle/index.html) tension is calculated from only one velocity value.As you can see I'm quite confused so can someone please tell me where I'm going wrong.
Thanks

What about an experiment? You connect something heavy (a key, for example) to a string and swing it around a vertical circle, so as it goes around with about uniform speed. Meanwhile you feel that the force your hand exerts on the string changes around the circle.

 

[haruspex]

making the centripetal force constant

But as you note, it will not be.
The text at the school physics link you posted is misleading. I will try to contact the author.

 

[haruspex]

What about an experiment? You connect something heavy (a key, for example) to a string and swing it around a vertical circle, so as it goes around with about uniform speed. Meanwhile you feel that the force your hand exerts on the string changes around the circle.

I don't think Sam has a problem with that. The issue is whether the speed would be uniform. The link he posted implies it would be.

 

[Sam McCrea]

I don't think Sam has a problem with that. The issue is whether the speed would be uniform. The link he posted implies it would be.

On that site there's a question at the bottom of that site (http://www.schoolphysics.co.uk/age16-19/Mechanics/Circular motion/text/Motion_in_a_vertical_circle/index.html) saying a plane is going at a constant speed and calculating the forces imposed upon him (presumably the centripetal force).

So it would be possible using thrust from the engine to achieve that constant speed and with a ball on a string scenario it would be impossible to achieve uniform circular motion as there's is no thrust from the string. Whereas if you had rod you could achieve it as it would apply torques. Is that correct?

 

[ehild]

So it would be possible using thrust from the engine to achieve that constant speed and with a ball on a string scenario it would be impossible to achieve uniform circular motion as there's is no thrust from the string. Whereas if you had rod you could achieve it as it would apply torques. Is that correct?

You can produce uniform circular motion. With trust, in case of a plane or with an appropriate external force. If you swing a ball on a string, and you keep the string with your fingers, you will notice that your hand moves, and the tip of your fingers go around a small circle. You can make your fingers move with uniform angular speed, and the ball would move with the same angular speed if the string is taut. It is worth trying.

 

[Sam McCrea]

You can produce uniform circular motion. With trust, in case of a plane or with an appropriate external force. If you swing a ball on a string, and you keep the string with your fingers, you will notice that your hand moves, and the tip of your fingers go around a small circle. You can make your fingers move with uniform angular speed, and the ball would move with the same angular speed if the string is taut. It is worth trying.

Sorry I don't quite get what you mean by your experiment but in summary you can never get a ball on a string (such as a pendulum) to undergo uniform vertical circular motion (and therefore constant a speed value) by itself?

 

[ehild]

Sorry I don't quite get what you mean by your experiment but in summary you can never get a ball on a string (such as a pendulum) to undergo uniform vertical circular motion (and therefore constant a speed value) by itself?

Not by itself. Something must overcome the tangential component of gravity. If you swing the ball with your hand, the motion of your hand influences the motion of the ball. It might be not a perfect circle it moves on. If I were you, I tried to move a key on a string by hand around a vertical circle, just to get the feeling.
On the other hand, the speed looks uniform if the change of potential energy is small with respect to the kinetic energy.
Assume a 0.5 m long string. The maximum change of PE is about 10 J. So the difference of the maximum and minimum v² is 20 J.
(v1-v2)(v1+v2)=20. With average speed of 10 m/s, the change of speed is 1 m/s, the speed changes between 9.5 and 10.5 m/s.

 

[haruspex]

But as you note, it will not be.
The text at the school physics link you posted is misleading. I will try to contact the author.

I contacted the author and the words "at a constant speed" have been inserted. No explanation as to how that might be achieved, nor why it would not generally be the case, so only a modest improvement.

 

[kuruman]

... so only a modest improvement.

Very modest improvement but still a misleading example. In this specific case, tension T₂ cannot possibly be as shown. Simply put, if the tension is horizontal and the weight vertical, their vector sum cannot be centripetal as it ought to be.

 

[CWatters]

Therefore the total energy must be the kinetic energy that allowed it to go that minimum speed at that point plus the gravitational potential energy. This therefore must be all the kinetic energy the car has at the bottom of the loop and if you rearrange this to find the velocity is a different value from the top speed. This must mean the velocity is getting smaller as it goes up the loop as energy is being changed however this must mean it is not uniform circular motion and centripetal force is not constant.

Correct.

However on a ball attached to a string in vertical circular motion I read that as the ball nears the bottom of the loop the tension force increases which balances out the gravity force component making the centripetal force constant and from looking at the formula this must mean the speed is constant (velocity is the same value but constantly changing direction).

The speed clearly wouldn't be constant. Consider a pendulum on earth. Pull it to one side and let it go. It accelerates as it falls then slows to a stop as it goes up the other side. The tension increases at as the pendulum falls because the speed is changing.

in summary you can never get a ball on a string (such as a pendulum) to undergo uniform vertical circular motion (and therefore constant a speed value) by itself?

Correct, although...

You could do it in an elevator undergoing freefall (or in space away from the Earth's gravity but then the definition of vertical is problematic) .

You could also force a rigid pendulum to move in a uniform vertical circle using a constant speed motor (but that's a different set up).