Rolling balls with different rotational inertia

[rolling]

I am stuck on problem presented about putting golf balls.

A stationary golf ball (mass=45g, dia=42mm, solid & homogeneous) is struck by a horizontal force (putter) and ignoring sliding immediately starts rolling on a level putting green. The ball eventually stops due to rolling resistance after rolling a distance X.

If a golf ball of the same mass and diameter, but is not solid - instead it is treated as a hollow sphere with all the mass at surface, is struck by the same force and rolls along the same level green, will the distance the ball rolls be more than X, less, or the same?

(As an added question if the ball is considered to have all its mass at the center of the ball, would it roll more than X, less, or the same?)

Thanks in advance for any help provided on this problem.

 

[PeroK]

Is this homework?

 

[rolling]

No it is not homework. Just a question my son asked (who is currently taking physics) that I (having not taken an engineering class in 20 years) felt I should be able to answer, but I can't and its driving me nuts.

 

[PeroK]

No it is not homework. Just a question my son asked (who is currently taking physics) that I (having not taken an engineering class in 20 years) felt I should be able to answer, but I can't and its driving me nuts.

Sounds like your son's homework!

I'm suspicious of the "ignore initial sliding", as the dynamics of an object going from linear motion to rolling motion depend on the moment of inertia.

I'm also suspicious of "force" as that should be "impulse".

I'd assume that the rolling resistance is the same in both cases.

You could straighten out the question by saying that both balls start rolling with the same linear speed (and hence same rate of rotation).

Then the answer is obvious - isn't it?

 

[PeroK]

Rolling resistance is not so simple:

https://en.wikipedia.org/wiki/Rolling_resistance

 

[rolling]

You said "If balls of the same mass are rolling at the same speed with the same rolling resistance." I would assume in the situation you presented the balls would travel the same distance.

But that does not intuitively seem correct in real life. It would seem to me, they would not roll at the same speed, because of the different moments of inertia. My initial guess at the problem was that they would roll the same distance but at different speeds.

But then I got to thinking that just like the initial force, the rolling resistance would impact each ball differently because of the different moments of inertia - and well it was beyond my math abilities currently (maybe when I was younger).

 

[PeroK]

But then I got to thinking that just like the initial force, the rolling resistance would impact each ball differently because of the different moments of inertia - and well it was beyond my math abilities currently (maybe when I was younger).

I'd encourage your son to do those calculations. If you give them the same impulse, then they have the same linear speed when they hit the rough surface and one of them loses less energy in the deceleration phase until they are rolling without slipping. The moment of inertia is the key factor.

That's why I don't like the question. It asks you to ignore the initial phase of attaining rolling without slipping, but that phase is different for the same reasons that the deceleration phase under rolling resistance is different.

A hollow sphere rolling without slipping certainly has more rotational KE than a solid sphere of the same mass rolling at the same speed. That's clear.

 

[rolling]

This is beyond the level of questions asked in his homework. And he has moved on, but it's nagging me. I guess I will have to go in the attic and dig out my old engineering books.

 

[PeroK]

This is beyond the level of questions asked in his homework. And he has moved on, but it's nagging me. I guess I will have to go in the attic and dig out my old engineering books.

It's all online these days!

 

[rolling]

And you have convinced me that I need to consider slipping.

 

[PeroK]

And you have convinced me that I need to consider slipping.

Edit: I think in general (with various assumptions about how rolling resistance works) that the object with the lesser MoI travels further over all.

 

[jbriggs444]

Edit: I think in general (with various assumptions about how rolling resistance works) that the object with the lesser MoI travels further over all.

With a different set of assumptions, the opposite it true. Rolling resistance normally behaves like a retarding torque. For a fixed initial rotation rate and a fixed rolling resistance, highest moment of inertia rolls longest.

The difficulty is with the assumptions -- do we assume an identical rotation rate with initial rolling without slipping? Or do we assume an identical launch speed with slipping until rolling without slipping is attained.

I read OP to indicate initial rolling without slipping. I read your post#7 to indicate initial sliding.

 

[PeroK]

With a different set of assumptions, the opposite it true. Rolling resistance normally behaves like a retarding torque. For a fixed initial rotation rate and a fixed rolling resistance, highest moment of inertia rolls longest.

The difficulty is with the assumptions -- do we assume an identical rotation rate with initial rolling without slipping? Or do we assume an identical launch speed with slipping until rolling without slipping is attained.

I read OP to indicate initial rolling without slipping. I read your post#7 to indicate initial sliding.

I was considering the full motion from receiving an initial impulse through the CoM, very quickly achieving rolling without slipping and then slowly decelerating under rolling resistance.

The object with the lower MoI should, therefore, start with a higher rolling speed and this appears to outweigh the greater deceleration. As far as I can model a common rolling resistance, that is.

 

[rolling]

So after looking at a great deal of confusing information online, I have come to these conclusion for a solid and hollow sphere.

1. Both the solid sphere and hollow sphere, when hit with the same impulse force will have the same starting linear velocity and rotational velocity.
2. During sliding, they will have the same negative linear acceleration but different rotational acceleration.
3. The solid sphere will reach true roll first and have a higher initial rolling velocity (both linear and rotational.)
4. During rolling, the hollow sphere will have the shorter time and distance in the rolling phase primarily due to a lower initial rolling velocity (both linear and rotational).
5. The solid sphere will roll a farther distance.

 

[PeroK]

1. Both the solid sphere and hollow sphere, when hit with the same impulse force will have the same starting linear velocity and rotational velocity.

Only if there is zero rotation. It's simplest to assume that the impulse is through the centre of mass.

2. During sliding, they will have the same negative linear acceleration but different rotational acceleration.
3. The solid sphere will reach true roll first and have a higher initial rolling velocity (both linear and rotational.)

I agree.

4. During rolling, the hollow sphere will have the shorter time and distance in the rolling phase primarily due to a lower initial rolling velocity (both linear and rotational).

This is the complicated part and depends how you model rolling resistance. If you model it as a brake with a common braking force, then the hollow sphere decelerates more slowly - as it has a greater MoI (moment of inertia). This ought to be the same for a more general model of rolling resistance.

5. The solid sphere will roll a farther distance.

Yes, because the slower deceleration does not fully compensate for the lower initial speed - but you have to calculate this. For a simple brake, the total distance is proportional to $\frac 1 {1 + k}$, where the MoI is $kMR^2$.