First slide

Rolling motion

Question

A disc of mass M and radius R is rolling with angular speed ω on a horizontal plane as shown in figure. The magnitude of angular momentum of the disc about the origin O is

Moderate

A

$\frac{1}{2} m R^{2} ω$
B

$\frac{3}{2} m R^{2} ω$
C

$\frac{5}{2} m R^{2} ω$
D

$\frac{1}{3} m R^{2} ω$

Solution

For plane motion of rigid body , if $\vec{r_{C}}$ be the position vector of its COM and $\vec{v_{C}}$ be the
$\large \\I_0=\frac{MR^2}{2}w+mvr\\\\=\frac{MR^2}{2}w+M.r^2.w=\frac32MR^2w$
Velocity of its COM , then
$\large \overrightarrow L_o=\overrightarrow r_c\;X\;M\overrightarrow r_c+I_c\overrightarrow w\\\\=(x_c\hat i+R \hat j)X M (wR\hat i)+(\frac12MR^2)w(-\hat k)\\\\=-MwR^2\hat k-\frac12MwR^2\hat k\\\\=-\frac32MR^2w\hat K\\\\ \therefore |\overrightarrow L_0| = \frac32MR^2w$

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