91Ó°ÊÓ

By using the concept of moment of inertia of the discrete particle system you can find percentage decrease in rotational inertia for the given conditions. The formula of above is given as follows

Moment of inertia for discrete particle system

$\mathit{l}\mathbf{=}\mathbf{∑}\mathit{m}{\mathit{r}}^{\mathbf{2}}$

For a system with discrete particles, total moment of inertia is given by

$l=∑m{r}^2$

As we have three particles, moment of inertia of total system

$l_T=m_l{r}_1^2+m_2{r}_2^2+m_3{r}_3^2$

Since $m_1=m_2=m_3=m$

$l_T=m\left(r_1^2+r_2^2+r_3^2\right)$ …(1)

Distance of the innermost particle from the axis r₁ = d

Distance of the innermost particle from the axis r₂ = 2d

Distance of the innermost particle from the axis r₃ = 3d

Hence equation 1 become,

$\begin{array}{rcl}{l}_T& =& m\left(d^2+{\left(2d\right)}^2+{\left(3d\right)}^2\right)\\ & =& m{d}^2\left(1+4+9\right)\\ & =& m\left(d^2+4d^2+9d^2\right)\end{array}$

$l_T=14m{d}^2$ …(2)

To find the percentage of decrease in rotational inertia after removing innermost particle,

$\begin{array}{rcl}{l}_i& =& m\left(r_2^2+r_3^2\right)\\ l_T& =& m\left(4d^2+9d^2\right)\\ l_i& =& 13m{d}^2\end{array}$

Percentage of decrease in rotational inertia

$=\frac{l_T-l_i}{l_T}×100$

Percentage of decrease in rotational inertia

$=\frac{14m{d}^2-13m{d}^2}{14m{d}^2}×100$

Percentage of decrease in rotational inertia
$$\begin{gathered} =\frac{1}{14}×100 \\ =7.14\% \end{gathered}$$

Hence the percentage of decrease in rotational inertia is, 7.14% or 7.1%.

To find the percentage of decrease in rotational inertia after removing outermost particle,

$\begin{array}{rcl}{l}_o& =& m\left(r_1^2+r_2^2\right)\\ & =& m\left(d^2+4d^2\right)\\ & =& 5m{d}^2\end{array}$

Percentage of decrease in rotational inertia

$=\frac{l_T-l_o}{l_T}×100$

Percentage of decrease in rotational inertia

$=\frac{14m{d}^2-5m{d}^2}{14m{d}^2}×100$

Percentage of decrease in rotational inertia

$$\begin{gathered} =\frac{9}{14}×100 \\ =64.3\% \end{gathered}$$

Hence the percentage of decrease in rotational inertia is, 64.3% or 64%.