Can a Ball Roll in a Horizontal Circle Inside a Bowl?

[Nick886]

circular motion problem??

hey all, i have to talke app maths for a semester and it was all fine until we got to modeling, now haven't a clue? can anyone help with this please?
a ball rolls under gravity inside a smooth bowl with circular cross section and inner radius"a". show that the ball can roll in a horizontal circle at a height "h" above the bottom of the bowl provided
v^2 = gh(2a-h/a-h). thanks

 

[gneill]

What do you know about circular motion? Can you calculate centripetal acceleration?

Have you drawn a diagram of the setup, and a free-body diagram for the forces acting?

 

[Nick886]

ye i have done that, well tried! there is no friction because its a smooth bowl. so is it just gravity or am i missing something obvious?? Probably! i got as far as the cent acc and have $$\omega$$^2 = g(tan$$\vartheta$$ / a)

 

[gneill]

Hint: Consider for a moment, in an appropriate frame of reference where such things are not frowned upon, the sum of the centrifugal acceleration and the gravitational acceleration that the ball feels. The bowl's reaction is equal and opposite, so draw in the reaction acceleration. Here's the tricky bit. If this vector is not aligned with the surface normal of the bowl, that is, aligned with the radius vector of the bowl (curvature radius a) at the point of contact, then there will be a net acceleration away from the point of contact, either upward or downward, tending to shift the height of the ball.

If this acceleration vector must be aligned with the radius vector for stability of the rotation height, then a simple ratio follows.

 

[rwisz]

Sure, I can help with this! The first thing we need to understand is what is meant by "circular motion problem". A circular motion problem is any situation where an object moves in a circular path, and we need to use mathematical principles to analyze its motion. This type of problem is common in physics and engineering, as circular motion is a fundamental concept in many areas of science.

In the problem you have described, the ball is rolling in a circular path inside a bowl. This means that the ball is constantly changing direction and its velocity is always tangent to the circular path. The key to solving this problem is understanding the forces acting on the ball and using Newton's laws of motion to analyze its motion.

In this case, the only force acting on the ball is gravity. As the ball moves in a circular path, gravity is constantly pulling it towards the center of the bowl. This results in a centripetal force that keeps the ball moving in a circular path. We can use the equation for centripetal force, Fc = mv^2/r, to find the velocity of the ball at any point in its motion.

Next, we need to consider the height of the ball above the bottom of the bowl. This will affect the radius of the circular path and therefore the velocity of the ball. We can use trigonometry to find the relationship between the height of the ball and the radius of the circular path.

Finally, we can use the equation you provided, v^2 = gh(2a-h/a-h), to find the velocity of the ball at any given height above the bottom of the bowl. This equation takes into account the gravitational acceleration, the height of the ball, and the radius of the circular path.

I hope this helps to clarify what is meant by a "circular motion problem" and how to approach solving it. Remember to always analyze the forces acting on the object and use mathematical principles to find the solution. Good luck with your semester!