91Ó°ÊÓ

1. The rotational inertia of first disk is,${\text{I}}_1=3.3\text{kg}.{\text{m}}^2$
2. The rotational inertia of second disk is$,{\text{I}}_2=6.6\text{kg}.{\text{m}}^2$
3. The angular speed of the first disk,${\text{Ó¬}}_1=450\mathrm{rev}/\min$
4. The angular speed of the second disk is,${\text{Ó¬}}_2=900\mathrm{rev}/\min$

Using the conservation law of the angular momentum we can find the angular speed of the system after coupling. Then by using the sign convention, we can find the angular speed of the system when the second disk is spinning clockwise.

Formula:

The law of conservation of angular momentum,${\mathbf{L}}_{\mathbf{i}}\mathbf{}\mathbf{=}{\mathbf{L}}_{\mathbf{f}}$

(a)

The law of conservation of angular momentum gives,${\text{L}}_{\text{i}}={\text{L}}_{\text{f}}$.

Angular momentum of the system before coupling = Angular momentum of the system after coupling

$\begin{array}{c}⇒{\text{I}}_1{\text{Ó¬}}_1+{\text{I}}_2{\text{Ó¬}}_2=({\text{I}}_1+{\text{I}}_2)\text{Ó¬}\\ ⇒\text{Ó¬}=\frac{{\text{I}}_1{\text{Ó¬}}_1+{\text{I}}_2{\text{Ó¬}}_2}{{\text{I}}_1+{\text{I}}_2}\\ ⇒\text{Ó¬}=\frac{\left(3.3\right)\left(450\right)+\left(6.6\right)\left(900\right)}{3.3+6.6}\\ ⇒\mathbf{Ó\neg }=750\mathbf{rev}/\mathbf{min}\end{array}$

(b)

If the second disk is spinning clockwise, its angular speed would be $−900\mathrm{rev}/\min$.

So, the angular speed of the system is

$\begin{array}{l}\text{Ó¬}=\frac{\left(3.3\right)\left(450\right)+\left(6.6\right)(−900)}{3.3+6.6}\\ \mathbf{Ó\neg }=−450\mathbf{rev}/\mathbf{min}\end{array}$

(c)

The minus sign indicates that$\stackrel{→}{\mathrm{Ó\neg }}$ is clockwise, that is, in the direction of the second disk’s initial angular velocity.