How to Calculate When a Bowling Ball Will Begin to Roll Without Sliding

[w3390]

Homework Statement

A bowling ball is initially bowled so that it doesn't roll with a translational velocity Vo. Show that when the velocity of the ball drops to (5/7)Vo, the ball will begin to roll without sliding.

I= (2/5)MR^2
mass= M
radius= R

Homework Equations

V=wR

The Attempt at a Solution

I have had several unsuccessful attempts and my latest attempt has gotten me close to the right answer. The correct answer is (5/7)Vo, but I got sqrt(5/7)Vo. Here's what I did:

KE_T= KE_R
(1/2)MVo^2 = (1/2)MV^2 + (1/2)Iw^2
(1/2)Vo^2 - (1/5)R^2*w^2 = (1/2)V^2
Vo^2 - (2/5)R^2*w^2 = V^2
Vo^2 - (2/5)V^2 = V^2
Vo^2 = (7/5)V^2
V = SQRT(5/7)Vo

I am currently trying to figure out where I went wrong. Maybe I forget to account for friction. Any help would be much appreciated.

 

[Proofrific]

Check how you went from angular velocity (w) to translational velocity (v).

 

[w3390]

I'm sorry. Can you rephrase your question. I do not quite understand what you mean by check how I went from angular to translational.

 

[Proofrific]

Whoops, sorry. I meant $$v = r \omega$$, but you did that right. My bad.

I don't see any problems with what you did. It looks like you did it right. Perhaps the solution is wrong?

 

[w3390]

No. The solution is definitely correct. My instructor told all of us that if we got the answer I got that we made one particular mistake. Unfortunately, I cannot find it right now.

 

[Herr Malus]

It looks like an angular momentum approach would actually be easier here. I recall having tried the energy approach when I had to do this problem, and never being able to get it to work. When I switched to conservation of angular momentum though, it clicked very quickly.

 

[hikaru1221]

@w3390: Why do you think KE_T= KE_R? Energy is not conserved because before the ball reaches to the roll-without-sliding stage, it does slide. And what does it mean by "sliding"? Something about friction
@Herr Malus: Though the angular momentum of the ball about the lowest point is conserved, the reason behind this is not as simple as the "normal" angular momentum conservation law.

 

[Swap]

I tried to take into account of the friction but I got (sqrt5)/3.

 

[hikaru1221]

I tried to take into account of the friction but I got (sqrt5)/3.

Please show your work

 

[Swap]

ok distance traveled from the time the ball has been thrown and it starts to roll= (Vo^2-Vx^2)/2a. Vx being the final velocity, a being the accln due to friction.
Now I apply the conservation of energy theorem
0.5mVo^2=0.5mVx^2+0.5I(Vx/R)^2+ma(Vo^2-Vx^2)/2a
after solving this equation I get Vx=(sqrt5)/3.
The limitation of my answer is that I assumed the frictional force to be constant but I m not sure whether it is allowed or not.

 

[hikaru1221]

The limitation of my answer is that I assumed the frictional force to be constant but I m not sure whether it is allowed or not.

You can assume so to simplify the problem. You can say, that friction is constant is a special case, so the result got from this case should match the needed result.

ok distance traveled from the time the ball has been thrown and it starts to roll= (Vo^2-Vx^2)/2a. Vx being the final velocity, a being the accln due to friction.
Now I apply the conservation of energy theorem
0.5mVo^2=0.5mVx^2+0.5I(Vx/R)^2+ma(Vo^2-Vx^2)/2a
after solving this equation I get Vx=(sqrt5)/3.

I make bold the wrong part. The wrong thing is that the velocity of the point where friction exerts on is not equal to the velocity of the center (remember the formula dW = Fvdt, where v is the velocity of the point where the force F exerts on?). Therefore, the work done by friction is not equal to (friction force x distance traveled by the center).

P.S.: Why don't you try force analysis, and instead of using F=ma, you can use $$X=m\Delta v$$ where X is the impulse.

 

[w3390]

I don't see how using impulse will make the problem any easier. Also, when the ball is sliding why wouldn't the velocity of the point where friction acts be the same as the center of mass. If the velocity of the center of mass was different than the velocity of the contact point, wouldn't the ball begin to roll?

 

[Doc Al]

KE_T= KE_R
(1/2)MVo^2 = (1/2)MV^2 + (1/2)Iw^2

Mechanical energy is not conserved (as already pointed out).

Also, when the ball is sliding why wouldn't the velocity of the point where friction acts be the same as the center of mass. If the velocity of the center of mass was different than the velocity of the contact point, wouldn't the ball begin to roll?

The ball begins to roll immediately due to friction. (It both rolls and slides.) You are looking for the point where the motion becomes rolling without sliding.

 

[Herr Malus]

@Herr Malus: Though the angular momentum of the ball about the lowest point is conserved, the reason behind this is not as simple as the "normal" angular momentum conservation law.

The reason I could never get the energy method to work was because I was never able to come up with an expression for the Work done by friction. As for it not being "simple", I beg to differ. You need a point where all external torque can either be countered or is not a factor. You already pointed out that the lowest point of the ball satisfies this, justification would solve the problem.

 

[w3390]

@Herr Malus
So one of my options is to make my point of origin the contact point between the ball and the ground? I think this is what you mean by making torque not a factor because the radius component will be zero. Is that what you meant?

 

[Herr Malus]

Yes, if you are using the point of contact between the ball and the floor you should see that torque is not a factor (hence angular momentum is conserved).

First you need to figure out which forces experienced by the ball may be a source of torque.

 

[w3390]

Well I thought the only torque on the ball would be the friction force, but by making the contact point my origin I thought I eliminated that torque. The only other force acting on the ball would be gravity but that is antiparallel with the r component so it would just be zero. Am I wrong?

 

[hikaru1221]

The reason I could never get the energy method to work was because I was never able to come up with an expression for the work done by friction. As for it not being "simple", I beg to differ. You need a point where all external torque can either be countered or is not a factor. You already pointed out that the lowest point of the ball satisfies this, justification would solve the problem.

The work done by friction can be computed: $$dW = (v-\omega R)Fdt = (v-\omega R)mdv$$. From here, we need force analysis to derive the relation between v and omega. I'll leave the force analysis part for someone else.

As for it not being "simple", I beg to differ. You need a point where all external torque can either be countered or is not a factor. You already pointed out that the lowest point of the ball satisfies this, justification would solve the problem.

Torque = 0 is just one reason. The general formula is NOT: torque = d(angular momentum)/dt. Though this formula is popular, it's only true under some certain conditions.

 

[Herr Malus]

Well I thought the only torque on the ball would be the friction force, but by making the contact point my origin I thought I eliminated that torque. The only other force acting on the ball would be gravity but that is antiparallel with the r component so it would just be zero. Am I wrong?

Correct, Gravity is acting parallel and produces no torque. Friction acts at the point of contact and produces no torque. This gives you the justification to set up the equation initial angular momentum equals final angular momentum.

@hikaru1221
-Educate me. What are the conditions?

 

[hikaru1221]

I don't remember the exact formula, but the condition is $$\vec{v}_{center-of-mass}\times\vec{v}_{A}=0$$, where A is the point we consider to calculate angular momentum. You can prove this with prudence.
The problem is even more complicated, as there can be 3 kinds of points, which correspond to 3 kinds of motion, at the point of contact. But I'll not discuss this here, as this may confuse the OP. Anyway, caution must be taken when applying the angular momentum conservation law.

 

[Doc Al]

Correct, Gravity is acting parallel and produces no torque. Friction acts at the point of contact and produces no torque. This gives you the justification to set up the equation initial angular momentum equals final angular momentum.

Note that the point of contact is accelerating, which makes it poor choice as a reference point if you wish to apply angular momentum conservation. But you can certainly choose a fixed point on the ground as your reference for computing torques and angular momentum. (Which is probably what you did anyway.)

 

[w3390]

Maybe I am missing something very fundamental, but if I am to use conservation of angular momentum, initially there is none and later there is L=Iw^2. This has to be wrong as setting that equal to zero gets me nowhere. Sorry that I am still missing the point here.

 

[Doc Al]

Maybe I am missing something very fundamental, but if I am to use conservation of angular momentum, initially there is none and later there is L=Iw^2. This has to be wrong as setting that equal to zero gets me nowhere. Sorry that I am still missing the point here.

You are incorrect that initially there is no angular momentum. (And the angular momentum about the center of mass is Iω, not Iω².)

As hikaru1221 points out, there are some subtleties involved with using conservation of angular momentum. Another way to solve the problem is just to apply Newton's 2nd law (for translation and rotation) and a bit of kinematics. (It's longer than using conservation, but it may be easier to understand.)

 

[hikaru1221]

@w3390: First, I confirm that angular momentum is conserved.
Note that you're calculating angular momentum w.r.t. the point of contact, therefore, the initial L is not zero and the final L is not Iw (not Iw^2!). You may want to have a look at this:
http://en.wikipedia.org/wiki/Angular_momentum#Angular_momentum_simplified_using_the_center_of_mass

 

[w3390]

Okay. So using the contact point as my origin, I will have a torque when the ball is initially rolled of I*alpha. The only force that is causing a torque would be the ball's mass and acceleration. I have found the acceleration of the center of mass to be -(mu)*g. I then said:

Tnet = I*alpha
R*(Ma) = I*alpha
-R*(mu)*g = (2/5)*R^2*alpha

alpha = -(5/2)*(1/R)*(mu)*g

Is this the angular acceleration once the ball begin rolling? I'm not sure if I am on the right track. I am trying to use Doc Al's advice of Newton's Second Law.

 

[Doc Al]

You're partially on the right track. Think of it this way. The angular speed of the ball increases while the translational speed decreases. Hint: Solve for the time it takes for those speeds to meet the condition for rolling without slipping.

You are using the ball's center of mass as your reference for calculating torque, not the point of contact.

 

[w3390]

Okay. Well to roll without slipping means v=wr. I have already figured out that the ball accelerates at a=-(mu)*g. Then, kinematically, Vf=Vo-(mu)*g*t. So the time it takes for the translational speed to equal the rotational speed will be:

t=(wR-Vo)/-(mu)*g or t= (Vo-wR)/(mu)*g

 

[Doc Al]

Well to roll without slipping means v=wr. I have already figured out that the ball accelerates at a=-(mu)*g.

OK.

Then, kinematically, Vf=Vo-(mu)*g*t.

Good. Now write a similar equation for ω as a function of time. Combine all equations to solve for the time. Once you have the time, use it to find Vf.

You're almost there.

 

[w3390]

Oh my god. It just hit me. All my equations just came together. Thank you all so much for help.

 

[Swap]

Here is the correct solution I just figured out.
fR=I(alpha) ...(f=frictional force)
maR=(2/5)mR^2*(a/R) ( where a is the translational accln.)
m(Vx-Vo)=(2/5)mR^2*((Vx-Vo)/R)
from here u will get Vx=(5/7)Vo... =p

 

[Doc Al]

Now solve it using conservation of angular momentum.

 

[carlosOviedo]

OK.

Good. Now write a similar equation for ω as a function of time. Combine all equations to solve for the time. Once you have the time, use it to find Vf.

You're almost there.

Do you mean using d(theta)/dt?

 

[Doc Al]

Do you mean using d(theta)/dt?

I don't understand your question. ω ≡ dθ/dt, if that's what you mean.