Q7.25     Two discs of moments of inertia $I_{1}$ and $I_{2}$  about their respective axes (normal to the               disc and passing through the centre), and rotating with angular speeds $\omega _{1}$ and $\omega _{2}$               are brought into contact face to face with their axes of rotation coincident.              (a) What is the angular speed of the two-disc system?

Let the moment of inertia of disc I and disc II be I1 and I2 respectively.

Similarly, the angular speed of disc I and disc II be w1 and w2 respectively.

So the angular momentum can be written as :

$L_1\ =\ I_1 \omega_1$             and             $L_2\ =\ I_2 \omega_2$

Thus the total initial angular momentum is :      $=\ I_1 \omega_1\ +\ I_2 \omega_2$

Now when the two discs are combined the angular momentum is :

$L_f\ =\ (I_1\ +\ I_2)w$

Using conservation of angular momentum :

$I_1\omega_1\ +\ I_2\omega_2 =\ (I_1\ +\ I_2)w$

Thus angular velocity is :

$\omega \ =\ \frac{I_1\omega_1\ +\ I_2\omega_2}{I_1\ +\ I_2}$

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