Signs in torque - different analyses using different sign notations?

[mancity]

Homework Statement

Why are the sign notations different for the two derivations of MR(v_f-v_0)+I(w_f-w_0)=0?

Relevant Equations

MR(v_f-v_0)+I(w_f-w_0)=0.

In this derivation above, I have to account for the fact that v is translational and opposite to the sign of w, and similarly for v_0 and w_0, so the real equation should look something like this:

Now, what I don't understand is, in this second derivation using net change in linear and rotational angular momentum, we don't have to use the fact that v is opposite the sign of omega:

Is it somehow implied that, through the conservation of angular momentum, v would be opposite to w? I don't want to take this formula for granted, and a clarification of the "discrepancy" in these two approaches would greatly help me out. Thanks.

 

[kuruman]

Homework Statement: Why are the sign notations different for the two derivations of MR(v_f-v_0)+I(w_f-w_0)=0?
Relevant Equations: MR(v_f-v_0)+I(w_f-w_0)=0.

What is being derived here? What is the physical situation? Please explain the context.
It is certainly true that you can always express conservation of angular momentum as $$\Delta \vec{L}_{\text{linear}}+\Delta \vec{L}_{\text{translational}}=0.$$Note that this is a vector equation and that angular momentum must be expressed about a point that needs to be specified before you writing expressions in terms of the linear and angular velocities.

Is it somehow implied that, through the conservation of angular momentum, v would be opposite to w?

When angular momentum is conserved, the vector representing the translational velocity and the vector representing the rotational velocity cannot point in opposite directions. For rolling without slipping,$~\vec v=\vec{\omega}\times\vec r$. In this equation $\vec v$ is the linear velocity of a point on the rolling object at position $\vec r$ from the axis of rotation. This expression says that the linear velocity perpendicular to the angular velocity.

 

[mancity]

Thanks.

The context for my question is this problem, and its corresponding solution:

My logic was: here, the ball rotates in the counterclockwise direction, while the translational velocity of the ball is to the left (such that the point of contact of the ball with the ground rolls without slipping). But would that imply that instead of $\omega-\omega_0,$ we would have $-\omega-(-\omega_0)=\omega_0-\omega$, and that would be the "correct?" definition for the change in angular momentum for the first way of derivation.

My question is - why is it that for the second derivation, vec(delta L_trans)+vec(delta L_rot)=0, we can consider omega and omega 0 to be positive quantities, and not consider the fact that the rotational velocity is opposite that of the translational one?