Calculating Time and Forces in Juggling Inside a Cone

[Kenny Lee]

Hi! Please follow the link; its a guy juggling balls in an inverted cone. What he does is he uses the cone as a surface, and he rolls the balls around him in circles. ITs really entertaining so this shouldn't be too much of a chore.

I thought I'd post it here because it demonstrates simple angular mechanics quite well. I was wondering however, how would we calculate the time it takes for the ball to reach the bottom of the cone, if let's say, the ball is projected with an initial horizontal velocity of u at a height L. Gravity accelerates the ball downwards. And to simplify, we neglect friction.

I've attached what I've managed to do, which isn't much... I wasn't sure if angular momentum is conserved... And wasn't so sure how to incorporate weight into the calculation.

Can someone help me solve the problem? Its killing me. I'm a 2nd year engineering student. Anyone would think I'd be able to handle a problem like this, but looky, I can't. I'm an idiot.

 

[Office_Shredder]

Calculating the time is probably tougher than you want do start with... finding the lower and upper bounds of motion is a good place to start.

Assuming you've worked with polar coordinate mechanics, if you look at it in r, theta and z, assuming the ball is rolled horizontally with speed v and the equation of the cone is z=r (or z=a*r, whatever), you can calculate the upper and lower bounds of motion assuming it's rolled at an initial height a by using the energy equation

 

[OlderDan]

That's a great video. The problem you are trying to solve will require a change in your thinking about including friction in the problem. If you follow Shredder's excellent suggestion you will come to realize that you will be waiting a very long time for a ball to reach the bottom without it.

 

[ChemGuy]

If there is no friction is it not the same as gravity between the two bodies. Angular momentum would be conserved. If there is no friction then it would rotate forever. The only way to remove energy from the angular momentum is to transfer it to friction. Think of how well it works with the hard rubber balls that roll and little (but some) friction. What would happen if you substituted blocks that have high friction!

Having said that I would think you could still develop an equation for the system that would involve the transer of energy from angular momentum to "friction".

 

[Kenny Lee]

That's all really good advice. I'll try it and see what I can come up with. Thanks.

//Oh I realized something. My working was really wrong. I shouldn't have taken the normal to be horizontal cause the wall of the cone is slanted hahah oops.

So as the ball spirals downwards, is it valid to use the equation for uniform circular motion? And where does the centripetal force come from? The normal reaction AND gravity - or just the normal? How do we incorporate gravity into the problem? Angular momentum is not conserved yea - cause of gravity?

And then I thought of using energy theory but what would the final state be; kinetic zero? Oh man there's too much going on, I'm in over my head. I give up. forget about it thanks though. I need to review my concepts.

 

[TheloniousMONK]

And then I thought of using energy theory but what would the final state be; kinetic zero? Oh man there's too much going on, I'm in over my head. I give up. forget about it thanks though. I need to review my concepts.

My first thought was to use energy, but there is so much to take into account. I think just looking at its motion in the curved path should get you close. I was able to come up with an equation for time. Let me know what you come up with and we can compare.

 

[OlderDan]

That's all really good advice. I'll try it and see what I can come up with. Thanks.

//Oh I realized something. My working was really wrong. I shouldn't have taken the normal to be horizontal cause the wall of the cone is slanted hahah oops.

So as the ball spirals downwards, is it valid to use the equation for uniform circular motion? And where does the centripetal force come from? The normal reaction AND gravity - or just the normal? How do we incorporate gravity into the problem? Angular momentum is not conserved yea - cause of gravity?

And then I thought of using energy theory but what would the final state be; kinetic zero? Oh man there's too much going on, I'm in over my head. I give up. forget about it thanks though. I need to review my concepts.

The centripetal force comes from the normal force. Gravity acts vertically, but that can be resolved into components parallel and perpendicular to the surface. OR the normal force, which is perpendicular to the surface, can be resolved into horizopntal and vertical components. If there is a net force parallel to the surface, or equivalenly a net force in the vertical direction, the ball goes higher or lower. If it goes lower, it speeds up because of energy conservation. If it goes higher, it slows down. If it speeds up as the radius gets smaller, what must happen to the normal force.?Remember that the components of the normal force are proportional (while the gravitational force is constant) so if the horizontal component of the normal force increases, so does the vertical component. The vertical component of the normal force can be greater or lesser than the gravitational force at different points on a balls path.