Two bodies of masses $‘ m ’$ and $‘ 2 m ’$ are moving with velocities of $‘ 2 v ’$ and $‘ v ’$ respectively. Calculate the ratio of their inertia.

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Solution

Step 1: Given Data
Mass $m_{1} = m$
Mass $m_{2} = 2 m$
Velocity $v_{1} = 2 v$
Velocity $v_{2} = v$
Let $r$ be the radius.
Step 2: Formula Used
Inertia $I = \frac{L}{ω}$
where $L$ is the angular momentum and $ω$ is the angular velocity.
Step 3: Calculate the ratio
Angular momentum of the first body,
$L_{1} = m_{1} v_{1} r = m . 2 v . r$
The angular velocity of the first body,
$ω_{1} = \frac{v_{1}}{r} = \frac{2 v}{r}$
Angular momentum of the second body,
$L_{2} = m_{2} v_{2} r = 2 m . v . r$
The angular velocity of the second body,
$ω_{2} = \frac{v_{2}}{r} = \frac{v}{r}$
Therefore, the ratio is,
$= \frac{I_{1}}{I_{2}} = \frac{\frac{L_{1}}{ω_{1}}}{\frac{L_{2}}{ω_{2}}} = \frac{\frac{m . 2 v . r}{\frac{2 v}{r}}}{\frac{2 m . v . r}{\frac{v}{r}}} = 1 : 2$
Hence, the required ratio is $1 : 2$ .

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