# Rolling motion of sphere (weird initial condition?)

• etothey1
So i guess my solution is incomplete then, and i must find out where the ball is actually rolling. Thanks!In summary, a spherical billiard ball with a radius a and mass M is placed on a flat surface with coefficient of friction u. It is given a sharp horizontal hit at the center, causing it to move with linear velocity Vo and angular velocity wo=0. The point of contact initially moves leftwards and the ball is not rolling at all. The friction force decreases linear velocity and increases angular velocity until the ball reaches a point where it is rolling without slipping. The relationship between the linear and rotational speeds must be just right for this to occur.
etothey1

## Homework Statement

A spherical billiard ball radius a and mass M is sitting on a flat surface. The coefficient of friction between the ball and table is u. The moment of inertia for the billiard ball is
2Ma^2/5. The ball is given a sharp horizontal hit at the centre and it starts moving with linear velocity Vo. and angular velocity wo=0. Vo is positive.
a, Which way is the point of contact moving? is it rolling or slipping?
b, Draw a diagram showing the forces, to they increase or decrease linear velocity? increase or decrease angular velocity?

## Homework Equations

Friction force = u*m*g
Parallel axis theorem, I= Icm + ml^2 (l is distance from centre)
Newtons second law, F=ma
Torque=rXF=dL/dt, F is force L is angular momentum.

## The Attempt at a Solution

a, point of contact moving leftwards (if ball is rolling to the right), ball is rolling.
b, Friction decreases linear velocity and increases angular velocity.
(However i believe there is an error in the question, regarding the wo=0 part. The angular frequency and linear velocity should decrease linearly together to 0. Therefore if angular velocity is 0 at t=0 then it immedietly spikes up and starts decreasing afterwards.

Anyway, here is my attempt at the solution, however i believe the error lies in the question which is that wo=0 and that it does not ask for the acceleration at a specific point of the ball, which would change the solution.

Any help appreciated!
http://imageshack.us/f/163/lastquestion.jpg
http://imageshack.us/photo/my-images/155/lastquestion2.jpg
http://imageshack.us/f/163/lastquestion.jpg
http://imageshack.us/f/155/lastquestion2.jpg/

Last edited:
etothey1 said:
a, point of contact moving leftwards (if ball is rolling to the right), ball is rolling.
If the ball moves to the right, how is the point of contact moving to the left? How can the ball be rolling, if ω0 = 0?
b, Friction decreases linear velocity and increases angular velocity.
OK.
(However i believe there is an error in the question, regarding the wo=0 part. The angular frequency and linear velocity should decrease linearly together to 0. Therefore if angular velocity is 0 at t=0 then it immedietly spikes up and starts decreasing afterwards.
Why do you think there's an error with having ω0 = 0?

Doc Al said:
If the ball moves to the right, how is the point of contact moving to the left? How can the ball be rolling, if ω0 = 0?

OK.

Why do you think there's an error with having ω0 = 0?

Because of the frictious force between the table and the sphere, the sphere will start rolling, because there will be a torque exerted on the sphere. The linear velocity of the sphere is related to its rolling motion.
Therefore, I must have made a mistake thinking that the ball was rolling, but with the condition that wo=0, the ball is not rolling at all?

Also about the point of contact is initially stationary but moves to the left?Thankful for further help, this question is a bit messy for me.

etothey1 said:
Because of the frictious force between the table and the sphere, the sphere will start rolling, because there will be a torque exerted on the sphere.
Yes, the friction force will start the sphere rolling. But at first it's not rolling, just translating.
The linear velocity of the sphere is related to its rolling motion.
Not always. In this case the sphere starts out with a linear velocity but no rolling at all.
Therefore, I must have made a mistake thinking that the ball was rolling, but with the condition that wo=0, the ball is not rolling at all?
Correct.

Also about the point of contact is initially stationary but moves to the left?
If the ball is translating to the right, the point of contact will move to the right.

Doc Al said:
Yes, the friction force will start the sphere rolling. But at first it's not rolling, just translating.

Not always. In this case the sphere starts out with a linear velocity but no rolling at all.

Correct.If the ball is translating to the right, the point of contact will move to the right.

Ok, So what would be wrong in my solution then? My solution says that the linear velocity decreases linearly and the rotational velocity increases linearly. The thing that bothers me is I find it very weird that when the ball has no linear velocity, it has rotational velocity.
Also, You said that the sphere will start rolling, but then you answered that it will not roll at all? Which one is it?

Thank you for the help.

Last edited:
So the only way that I can fix this at this point is if I immidietly after t=0 force the rotational velocity up so that it decreases linearly as well along with the translational velocity.

This will allow the rolling motion and the translational motion to cease at the same time.

If this is not the case, then the my current solution says that when the ball stops it translational velocity, it will stand still and spin.

Thankful for further help.

The way I understand it is this. Initially, the ball translates but does not rotate. The friction force slows the linear speed, while increasing the rotational speed. At some point, the relationship between the linear and rotational speeds will be just right and the ball will be rolling without slipping.

Doc Al said:
The way I understand it is this. Initially, the ball translates but does not rotate. The friction force slows the linear speed, while increasing the rotational speed. At some point, the relationship between the linear and rotational speeds will be just right and the ball will be rolling without slipping.

Yep, thank you very much for you're help.

## 1. What is rolling motion?

Rolling motion is a type of motion in which a spherical object moves along a surface while simultaneously rotating around its own axis.

## 2. How is the rolling motion of a sphere affected by initial conditions?

The rolling motion of a sphere can be affected by the initial conditions such as the angle at which it is released, the surface it is rolling on, and the direction of its initial velocity.

## 3. Can a sphere exhibit weird initial conditions during rolling motion?

Yes, a sphere can exhibit weird initial conditions during rolling motion, such as starting from an uneven surface or being released with a non-uniform initial velocity.

## 4. What factors affect the rolling motion of a sphere?

The factors that affect the rolling motion of a sphere include the shape and size of the sphere, the surface it is rolling on, the initial conditions, and external forces such as friction.

## 5. How can the rolling motion of a sphere be calculated?

The rolling motion of a sphere can be calculated using the principles of mechanics and kinematics, taking into account factors such as the sphere's mass, radius, and initial conditions. Alternatively, it can also be simulated using computer software.

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