show the Relation between Rotational kinetic Energy and Moment of inertia..
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Rotational kinetic energy can be expressed as: Erotational=12Iω2 E rotational = 1 2 I ω 2 where ω is the angular velocity and I is the moment of inertia around the axis of rotation.
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Consider a rigid body rotating with angular velocity ω about an axis XOX′. Consider the particles of masses m1, m2, m3? situated at distances r1, r 2, r3? respectively from the axis of rotation. The angular velocity of all the particles is same but the particles rotate with different linear velocities. Let the linear velocities of the particles bev1,v2,v3 ? respectively.
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.
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.
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