How Can You Calculate the Ball's Speed Inside an Inverted Cone?

[abeltyukov]

Hi,

There is an inverted cone with height h. There is a ball rolling around inside creating a horizontal circle. The height from the tip of the inverted cone to the ball is y. a is given from the center of the cone to the outer edge.

The question asks to express the ball's speed in terms of a, h, y, and g.

I am confused as to why a is shown from the center of the cone to the edge. Is a in this case the same as acceleration?

I know a = w^2 * r as well as a = (v^2)/r
I have the forces in the y-direction: N = mg
I have the forces in the x-direction: (h-y)sin(theta) = (mv^2)/r

I believe I am screwing up on the x-direction forces. Any suggestions?


Thank you very much,
Alex

 

[Dorothy Weglend]

The normal force is perpendicular to the surface. That's not where you put it in your force equation. That will also help you fix the force equations for X.

Dorothy

 

[kyle8921]

Hello Alex,

Thank you for sharing your problem with me. From your description, it seems like you are trying to solve for the speed of the ball as it rolls around in a horizontal circle inside an inverted cone. To answer your question, a in this case is the distance from the center of the cone to the outer edge, also known as the radius. It is not related to acceleration, which is represented by the symbol a in physics equations.

To solve this problem, you can use the equations you mentioned: a = w^2 * r and a = (v^2)/r. However, I believe you may be overthinking it. Since the ball is rolling in a horizontal circle, the only force acting on it in the x-direction is the centripetal force (mv^2)/r. This force is equal to the horizontal component of the weight of the ball, which is (h-y)sin(theta). Therefore, you can set these two equations equal to each other and solve for v, the speed of the ball. This will give you the speed in terms of a, h, y, and g.

I hope this helps you with your problem. If you have any further questions, please don't hesitate to ask. Keep up the good work!

Best,