show the Relation between Rotational kinetic Energy and Moment of inertia..
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show the Relation between Rotational kinetic Energy and Moment of inertia..
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Don't give FALTU ANswer..Pls
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Kinetic energy of the first particle = ? m1v12
But v1=r1ω
∴ Kinetic energy of the first particle
=1/2 . m1(r1ω)2 =1/2 . m2(r2ω)2
Similarly,
Kinetic energy of second particle
=1/2 m2r22w2
Kinetic energy of third particle
= 1 / 2 . m3r32ω2 and so on.
The kinetic energy of the rotating rigid body is equal to the sum of the kinetic energies of all the particles.
∴ Rotational kinetic energy
= 1 / 2 . ( m1r12ω2 + m2r22ω2 + m3r32ω2 + ???. + mnrn2ω2)
= 1 / 2 . ω2 ( m1r12 + m2r22 + m3r32 + ???. + mnrn2)
In translatory motion, kinetic energy = 1 /2 mv2
Comparing with the above equation, the inertial role is played by the term ∑ mnrn2. This is known as moment of inertia of the rotating rigid body about the axis of rotation. Therefore the moment of inertia is I = mass ? (distance )2
Kinetic energy of rotation = 1/2 ω2I
When ω = 1 rad s-1, rotational kinetic energy
= ER = 1/2 (1)2I
(or) I = 2ER
It shows that moment of inertia of a body is equal to twice the kinetic energy of a rotating body whose angular velocity is one radian per second.
The unit for moment of inertia is kg m2 and the dimensional formula is ML2.