How Does Angular Velocity Affect the Ball's Position on a Basketball Hoop?

[Vrbic]

Homework Statement

A basketball player threw a ball with radius $a$ in such way that it rolls (without slipping) on a hoop of a basket with radius $R$. Let's define an angle $\theta$ as a angle between a plane of basket and line between the point of touch of the ball and the hoop and the center of the ball. What is the dependence between $\theta$ and angular velocity of rolling $\omega$?

Homework Equations

$F_g=mg$
$F_{od}=mv^2/r$

The Attempt at a Solution

I have some solution, but I don't know if it is right. I suppose there should be some balance between gravitational and centrifugal torque. It means I need to have same forces values of forces in incline coordinate system. I mean that x-axis will be aligned with line of indicating the angle $\theta$ and y-axis will be perpendicular to it. And origin in the center of the ball. Than I need to have projection of both forces to y-axis has to be same magnitude but opposite direction.
From that I have $\sin{\theta}F_{od}=\cos{\theta}F_g$ and result should be
$\omega=\sqrt{\cot{\theta}g/R }$
What do you mean about this solution?
I'm a bit suprised that I need not to know a radius of the ball.

 

[PeroK]

Homework Statement

A basketball player threw a ball with radius $a$ in such way that it rolls (without slipping) on a hoop of a basket with radius $R$. Let's define an angle $\theta$ as a angle between a plane of basket and line between the point of touch of the ball and the hoop and the center of the ball. What is the dependence between $\theta$ and angular velocity of rolling $\omega$?

Homework Equations

$F_g=mg$
$F_{od}=mv^2/r$

The Attempt at a Solution

I have some solution, but I don't know if it is right. I suppose there should be some balance between gravitational and centrifugal torque. It means I need to have same forces values of forces in incline coordinate system. I mean that x-axis will be aligned with line of indicating the angle $\theta$ and y-axis will be perpendicular to it. And origin in the center of the ball. Than I need to have projection of both forces to y-axis has to be same magnitude but opposite direction.
From that I have $\sin{\theta}F_{od}=\cos{\theta}F_g$ and result should be
$\omega=\sqrt{\cot{\theta}g/R }$
What do you mean about this solution?
I'm a bit suprised that I need not to know a radius of the ball.

What is the radius of the circluar motion?

 

[haruspex]

What is the radius of the circluar motion?

There are a couple of other issues.
ω Is the angular velocity of rolling, not its angular velocity around the hoop.
I strongly suspect we are supposed to take into account the gyroscopic effect. This means the net torque will not be zero.

 

[Vrbic]

What is the radius of the circluar motion?

Ah...I see. The radius is affected by the inclination of the ball. And I have to correct it by $a \sin{\theta}$. Do you agree?

 

[Vrbic]

There are a couple of other issues.
ω Is the angular velocity of rolling, not its angular velocity around the hoop.

Ok. That is also right and I can correct it by changing left hand side of solution by $a\omega/R$. Do you agree?

 

[Vrbic]

I strongly suspect we are supposed to take into account the gyroscopic effect. This means the net torque will not be zero.

In this case, I don't know what exactly do you mean or what and why should I do. Can you give me more hint?

 

[PeroK]

In this case, I don't know what exactly do you mean or what and why should I do. Can you give me more hint?

Have you been studying angular momentum?

 

[Vrbic]

Have you been studying angular momentum?

Yes, I know that such effects exist, but I'm not much familiar with them. I don't have much intuition in this. So far...

 

[PeroK]

Yes, I know that such effects exist, but I'm not much familiar with them. I don't have much intuition in this. So far...

This will be quite a complex problem, as you have the changing angular momentum of the rotating ball to take into account.

You might want to revise what you've learned about angular momentum before you tackle this.

 

[Vrbic]

This will be quite a complex problem, as you have the changing angular momentum of the rotating ball to take into account.

Why do you expect that angular momentum is changing? I believe that it is idealized case and the ball is rolling without friction and $\theta$ is also constant.

 

[Vrbic]

You might want to revise what you've learned about angular momentum before you tackle this.

I hoped that if I resolve such problem, I learn more about these things :) Should I use another concept? Try to construct Lagrangian and from it find equations of motion and from them look for my solution?
I don't want still look to theoretical books I would like to apply what I read (but it is possible that something I forgot or missed). Or what is your suggestion?

 

[PeroK]

Why do you expect that angular momentum is changing? I believe that it is idealized case and the ball is rolling without friction and $\theta$ is also constant.

The spin angular momentum of the ball is changing direction as it moves round the hoop. You need to learn about this before you can tackle this problem.

 

[Vrbic]

The spin angular momentum of the ball is changing direction as it moves round the hoop. You need to learn about this before you can tackle this problem.

Ok, and will you suggest me some good literature for this problem?

 

[PeroK]

Ok, and will you suggest me some good literature for this problem?

I've got the Kleppner and Kolenkow text on classical mechanics, which is very good, I think.

You may be able to find this one topic of spin AM of a rigid body covered online somewhere.

 

[haruspex]

idealized case and the ball is rolling without friction and θθ\theta is also constant.

Theta is constant, but there will be friction.

We already met the equation you need in the other thread. The component of "net" torque normal to the angular momentum is the product of the angular momentum and the precession rate. I put net in quotes because (I think this is right) you can treat the centrifugal pseudoforce as applying a torque, so the torque causing the precession is the resultant of gravity and centrifugal. (Do I have that right, PeroK?)

The rate of precession must be such that the ball endlessly repeats the same cycle. That allows you to relate it to ω and the radii.

 

[PeroK]

To be honest, I'm not familiar with this particular problem, but it will have the "millstone" effect, where there is an additional force needed to cause the change in spin angular momentum.

@Vrbic how a millstone works is that the changing angular momentum of the stone increases the force with which it grinds the corn. The circular motion gives you more than the weight of the stone that you would get if the stone simply rolled backwards and forwards.