Question

A uniform solid disc, is performing pure rolling on a fixed horizontal surface as shown in figure.

O, A and B are three points lying on y-axis where as C is top most point and D is bottom most point of the disc at the instant shown. Choose the correct option(s).

• A. Angular momentum of disc with respect to C and D are same
• B. Magnitude of angular momentum of disc with respect to A (among O, A, B, C and D) is minimum
• C. Magnitude of angular momentum of disc with respect to B (among O, A, B, C and D) is minimum
• D. Angular momentum with respect to O and C are same

Answer 1

Answer: (A), (B), (D)

The correct options are
A Angular momentum of disc with respect to C and D are same
B Magnitude of angular momentum of disc with respect to A (among O, A, B, C and D) is minimum
D Angular momentum with respect to O and C are same

$\overrightarrow{L_A}=m(\overrightarrow{R}\times \overrightarrow{V})+I_{com}\overrightarrow{\omega }\overrightarrow{L_A}=m{V}_{cm}\frac{R}{2}\hat{k}+\frac{m{R}^2}{2}\omega (-\hat{k})=0$

$\overrightarrow{L_B}=\frac{m{R}^2}{2}\omega (-\hat{k})$

Since C point is lying on body and point D is Instataneous centre of rotation, therefore for both point we can use,
$\overrightarrow{L}=I_{aboutthatpoint}\omega$
Since, $I_C=I_D=I_{com}+m(R{)}^2=\frac{3}{2}m{R}^2$
$\overrightarrow{L_C}=\frac{3}{2}m{R}^2\omega (-\hat{k})$ and
$\overrightarrow{L_D}=\frac{3}{2}m{R}^2\omega (-\hat{k})$

Now,
$\overrightarrow{L_O}=m{V}_{cm}R(-\hat{k})+\frac{m{R}^2}{2}\omega (-\hat{k})\overrightarrow{L_O}=\frac{3}{2}m{R}^2\omega (-\hat{k})$
Therefore, angular momentum with respect to O and C are same. Also, magnitude of angular momentum of disc with respect to B (among O, A, B, C and D) is minimum