Solve rolling problem using concept of angular momentum

[guthix]

Homework Statement

A ball of mass M and radius R initially slides with speed vo but without rotating on a horizontal surface with friction.
Without knowing anything about the nature of the frictional force, find the speed of the ball when it begins to roll without slipping.

Homework Equations

moment of inertia of ball = (2/5)MR^2

The Attempt at a Solution

At first i try to solve the problem by setting up two equations
(1) v = vo - ft/M
(2) w = (5/2) ft/MR
and eliminating t to get v(final)=(5/7)vo

then i thought that if i know nothing about friction, i shouldn't have use "f" as well
and i tried to use another method to solve the problem by conservation of angular momentum

M(vo)R = MvR + Iw
substituting the final condition v=Rw
i still got v(final)=(5/7)vo

but my instructor told me that i can't apply conservation of angular momentum in this case
i was confused, even i use this method, i couldn't explain why this is correct
(i have used this method to solve many rolling problems and it gives at least "correct value of required answer" every time)

can anyone comment on the method i use?

 

[Doc Al]

then i thought that if i know nothing about friction, i shouldn't have use "f" as well

You may not know the value of the friction force or coefficient of friction, but you know that it exists. There's nothing wrong with calling it 'f'.

but my instructor told me that i can't apply conservation of angular momentum in this case

Did he give a reason why you can't apply it? There's nothing wrong with using conservation of angular momentum about some stationary point on the surface. (The torque due to friction will be zero about that point.)

 

[guthix]

Did he give a reason why you can't apply it? There's nothing wrong with using conservation of angular momentum about some stationary point on the surface. (The torque due to friction will be zero about that point.)

i told him if i take moment at the contact point then there is no external torque, and he ask me whether "that point" is moving or not, i don't know why he ask for this, i have never think about this so i didn't answer.

other classmates challenged me that "as the ball is rolling, the point does not exist on the surface; and the ball is sliding as well, the point does not exist on the ball, so, there exists no point that i take moment about."

then they began to talk about Coriolis force and I'm not familiar with that...

quite confused after listening to their arguments...

ps. i tried to compare conservation of linear momentum and angular momentum
when there is friction, for linear momentum
if we consider Earth in our system as well, linear momentum would then conserve
but for angular momentum
do i need to consider the Earth as well?
then my method seems to be incorrect by simply taking moment about the contact point...

 

[Doc Al]

i told him if i take moment at the contact point then there is no external torque, and he ask me whether "that point" is moving or not, i don't know why he ask for this, i have never think about this so i didn't answer.

The contact point is accelerating, so that's not a good point to use.

other classmates challenged me that "as the ball is rolling, the point does not exist on the surface; and the ball is sliding as well, the point does not exist on the ball, so, there exists no point that i take moment about."

You can take moments about any fixed point on the surface.

 

[guthix]

You can take moments about any fixed point on the surface.

you mean...i can take moment about, say, the initial contact point?

i just think it is difficult to understand, even the line of action of friction passes through it, i can't treat it as no external torque because the point is no longer related to the rotational motion of the ball...

and the ball is not rotating about any fixed point on the surface, i can still take moment about those points to solve the problem?

 

[Doc Al]

you mean...i can take moment about, say, the initial contact point?

Yes.

i just think it is difficult to understand, even the line of action of friction passes through it, i can't treat it as no external torque because the point is no longer related to the rotational motion of the ball...

I don't understand what you're saying. If you take the initial contact point as your axis, the torque due to friction will be zero.

and the ball is not rotating about any fixed point on the surface, i can still take moment about those points to solve the problem?

Sure.

 

[guthix]

um...then my method of using M(vo)R = MvR + Iw should be interpreted as "take moment about the initial contact point", right?

 

[Doc Al]

um...then my method of using M(vo)R = MvR + Iw should be interpreted as "take moment about the initial contact point", right?

Yes. If you take moments about the initial contact point (or any other fixed point on the surface), you'll get zero--which means that the angular momentum about that point is conserved.

You are adding two components to get the total angular momentum: Angular momentum of the center of mass about the axis plus the angular momentum about the center of mass.

 

[guthix]

Yes. If you take moments about the initial contact point (or any other fixed point on the surface), you'll get zero--which means that the angular momentum about that point is conserved.

You are adding two components to get the total angular momentum: Angular momentum of the center of mass about the axis plus the angular momentum about the center of mass.

wow, thanks very much
(i think) I've understand the problem

just one more minor question
for similar reason, is the total angular momentum of the Earth = angular momentum of Earth's CM about the Sun + angular momentum about the Earth's CM ?

 

[Doc Al]

just one more minor question
for similar reason, is the total angular momentum of the Earth = angular momentum of Earth's CM about the Sun + angular momentum about the Earth's CM ?

Yes. That's the total angular momentum of the Earth with respect to the Sun.

 

[guthix]

Yes. That's the total angular momentum of the Earth with respect to the Sun.

got it~thanks again :)