Question

If two discs of moments of inertia $I_1$ and $I_2$, rotating about a collinear axis passing through their centres of mass and perpendicular to their planes with angular speeds ${\omega }_1$ and ${\omega }_2$ respectively in opposite directions are made to rotate combinedly along the same axis, then the magnitude of new angular velocity of the system is

• A. $\frac{I_1{\omega }_1+I_2{\omega }_2}{I_1+I_2}$
• B. $\frac{I_1{\omega }_1-I_2{\omega }_2}{I_1+I_2}$
• C. $\frac{I_1{\omega }_2+I_2{\omega }_1}{I_1+I_2}$
• D. $\frac{I_1{\omega }_1-I_2{\omega }_2}{I_1-I_2}$

Answer 1

The correct option is B $\frac{I_1{\omega }_1-I_2{\omega }_2}{I_1+I_2}$

Consider both discs as a system.
Since there is no external net toreque on the system, angular momentum about the rotational axis will be conserved.
Initial angular momentum $\left(L_i\right)=I_1{\omega }_1-I_2{\omega }_2$
(-ve because both are rotating in opposite directions)
Assume $\omega$ will be the common angular velocity when both are rotated combinedly.
Then, final angular momentum $L_f=I_1\omega +I_2\omega$
$L_f=(I_1+I_2)\omega$
(because both are rotating in same direction)
From $L_i=L_f$
$I_1{\omega }_1-I_2{\omega }_2=(I_1+I_2)\omega$
i.e $\omega =\frac{I_1{\omega }_1-I_2{\omega }_2}{I_1+I_2}$