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Grade 12th passMechanics

. The oxygen molecule, O2, has a total mass of 5.3010-26 kg and rotational inertia of 1.9410-46 kgm2 about an axis through the center perpendicular to the line joining the atoms. Suppose that such a molecule in a gas has mean speed of 500 m/s and that its rotational kinetic energy is two-thirds of its translational kinetic energy. Find its average angular velocity.

Profile image of joshua
6 Years agoGrade 12th pass
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1 Answer

Profile image of Arun
6 Years ago

Mass of an oxygen molecule, m = 5.30 × 10–26 kg

Moment of inertia, I = 1.94 × 10–46 kg m2
Velocity of the oxygen molecule, v = 500 m/s
The separation between the two atoms of the oxygen molecule = 2r
Mass of each oxygen atom = m/2
Hence, moment of inertia I, is calculated as:
(m/2)r2 + (m/2)r2 = mr2
r = ( I / m)1/2
(1.94 × 10-46 / 5.36 × 10-26 )1/2  =  0.60 × 10-10 m
It is given that:
KErot = (2/3)KEtrans
(1/2) I ω2 = (2/3) × (1/2) × mv2
mr2ω2 = (2/3)mv2
ω = (2/3)1/2 (v/r)
= (2/3)1/2 (500 / 0.6 × 10-10) = 6.80 × 1012 rad/sec