Question

Two thin circular discs of mass $m$ and $4m$, having radii of $a$ and $2a$, respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24}a$ through their centres. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point 'O' is $\overrightarrow{L}$ (see the figure). Which of the following statement(s) is(are) true?

• A. The magnitude of the z-component of $\overrightarrow{L}$ is $55{ma}^2\omega$
• B. The magnitude of angular momentum of centre of mass of the assembly about the point O is $81{ma}^2\omega$
• C. The centre of mass of the assembly rotates about the z-axis with an angular speed of $\omega /5$
• D. The magnitude of angular momentum of the assembly about its centre of mass is $17{ma}^2\omega /2$

Answer 1

Answer: (C), (D)

The correct options are
C The centre of mass of the assembly rotates about the z-axis with an angular speed of $\omega /5$
D The magnitude of angular momentum of the assembly about its centre of mass is $17{ma}^2\omega /2$

Let the angle of axis with horizontal be $\theta$.

$cos\theta =\frac{l}{\sqrt{l^2+a^2}}=\frac{\sqrt{24}}{5}$

Velocity of point P (center of lower disk) $a\omega =l\Omega$

Then $\Omega =\frac{a\omega }{l}$

Angular velocity of C.M. w.r.t z axis $=\Omega cos\theta$

${\omega }_{CM-z}=\frac{a\omega }{l}\frac{\sqrt{24}}{5}=\frac{\omega }{5}$

$L_{D-CM}=\frac{m{a}^2}{2}\omega +\frac{4m{\left(2a\right)}^2}{2}\omega =\frac{17m{a}^2\omega }{2}$