Question

Two discs are rotating about their axes, normal to the plane of the discs and passing through the centre of the discs. Disc $D$ has $2\text{kg}$ mass and $0.2\text{m}$ radius and initial angular velocity of $50\text{rad/s}$. Disc $D_2$ has $4\text{kg}$ mass, $0.1\text{m}$ radius and initial angular velocity of $200\text{rad/s}$. The two discs are brought in contact face to face with their axes of rotation coincident. The final angular velocity (in $\text{rad/s}$) of the system is

• A. $50\text{rad/s}$
• B. $20\text{rad/s}$
• C. $100\text{rad/s}$
• D. $200\text{rad/s}$

Answer 1

The correct option is C $100\text{rad/s}$

MOI of disc $D$, about an axis passing through its centre and normal to its plane.
$I_1=\frac{M{R}^2}{2}=\frac{2\times (0.2{)}^2}{2}=0.04{\text{kg-m}}^2$
Initial angular velocity of disc $D_1$,
${\omega }_1=50\text{rad/s}$

MOI of disc $D_2$ about an axis passing through its centre and normal to the plane.
$I_2=\frac{M{R}^2}{2}=\frac{4\times (0.1{)}^2}{2}$
$=0.02{\text{kg-m}}^2$

Initial angular velocity of disc $D_2$,
${\omega }_2=200\text{rad/s}$

Total angular momentum of the two discs, initially is
$L_i=I_1{\omega }_1+I_2{\omega }_2$
When both discs are brought in contact, axes of rotation coincide.
Consider both discs as a system. Hence, final angular momentum of the system is,
$L_f=(I_1+I_2)\omega$
Here, $\omega$ is the final angular speed of the system.
According to the law of conservation of angular momentum, we get,
$L_i=L_f$
$I_1{\omega }_1+I_2{\omega }_2=(I_1+I_2)\omega$
$\Rightarrow \omega =\frac{\left(0.04\right)\times \left(50\right)+\left(0.02\right)\times \left(200\right)}{(0.04+0.02)}$
$\Rightarrow \omega =100\text{rad/s}$