Angular Momentum of a sliding disc about a point on the floor

[naes213]

Hi everybody,

A seemingly straightforward example from lecture is causing me some confusion. The example was about calculating the angular momentum of a sliding disk (not rolling) about a point on the floor. The result given in lecture says the distance to the point on the floor is unrelated to the angular momentum:
$$\vec{L}=\vec{r} \times \vec{p}=\sum{\vec{r}_i \times \vec{p}_i}=\sum{\vec{r}_i \times m_i\vec{v}_i}=MRv$$

where M is the total mass of the disc, R is the radius of the disc, and v is the translational velocity of the sliding disc. Now my confusion comes in at the last equal sign. I think it should read:
$$\sum{\vec{r}_i \times m_i \vec{v}_i}=\sum{m_i r_i v_i \sin{\theta_i}}$$

where $\theta_i$ is the angle between each particles radius vector and the constant velocity vector. I don't see how this sum ends up as $MRv$ as was claimed in lecture.

I tried to write $\theta_i$ as a function of each ri and integrate over the disc, but didn't make progress. I know I can take the mi and vi out of the sum because they are the same for each i, but I still can't deal with the

$$\sum{r_i\sin{\theta_i}}$$

Any help would be greatly appreciated! Maybe I'm just missing something really obvious, I don't know.

Thanks,
Sean

 

[tiny-tim]

Hi Sean!

$$\vec{L}=\vec{r} \times \vec{p}=\sum{\vec{r}_i \times \vec{p}_i}=\sum{\vec{r}_i \times m_i\vec{v}_i}=MRv$$

where M is the total mass of the disc, R is the radius of the disc, and v is the translational velocity of the sliding disc.

No, R is the perpendicular distance from the point to the line of motion of the centre of mass.

∑ r_i x m_i v_i

= (∑ m_ir_i) x v since v_i = a constant, v

and then use ∑ m_i(r_i - r_o) = 0 by definition, where r_o is the centre of mass

 

[naes213]

Ok, I see. That makes sense. In my situation it just so happens that the perpendicular distance is equal to the radius of the disc. In a more general scenario this would be different. Thank you!