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Standard X

Physics

Kinetic Energy

A uniform bar...

Question

A uniform bar of length $6 a$ and mass $8 m$ lies on a smooth horizontal table. Two point masses $m$ and $2 m$ moving in opposite direction in the same horizontal plane with speed $2 v$ and $v$ , respectively. strike the bar and stick to the bar after collision. Calculate.
Total energy.

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Solution

Since the given system has only rotatory motion $\Rightarrow E = \frac{1}{2} I ω^{2}$
Since $Y_{e x t} = 0$ angular momentum is conserved
$\Rightarrow m_{1} ν_{1} r_{1} + m_{2} ν_{2} r_{2} = (I_{1} + I_{2} + I_{3}) ω \Rightarrow 2 m ν_{a} + m (2 m ν) (2 a) = (2 m a^{2} + m (2 a)^{2} + (\frac{8 m \times 36 a^{2}}{12})) ω \Rightarrow 6 m ν_{a} = 30 m ω a^{2}$

$\Rightarrow$ Angular velocity $ω = \frac{v}{5 a}$
$ω = \frac{V}{5 a}$
$E = \frac{1}{2} (30 m a^{2} ({\frac{v}{5 a}}^{2} = \frac{3}{5} m ν^{2}))$

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A uniform bar of length 6a and mass 8m lies on a smooth horizontal table. Two point masses m and 2m moving in opposite direction in the same horizontal plane with speed 2v and v, respectively. strike the bar and stick to the bar after collision. Calculate.Total energy.

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