A particle of mass m is describing a circular motion of radius r with uniform speed. If L is the angular momentum of the particle about the axis of the circle, find the kinetic energy of the particle.
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Solution
Given: a particle of mass m is describing a circular motion of radius r with uniform speed. If L is the angular momentum of the particle about the axis of the circle
To find the kinetic energy of the particle.
Solution: Assume the particle has angular velocity ω.
We define the angular momentum L of the particle about the point as
$ L = r p sin\theta$
where r is the radius of the circular motion and p is its momentum.
For a particle moving in a circular path sinθ=1,
L=rp=rmv=mr2ω=Iω
where v is the velocity of the particle.
And velocity of the particle can be written as v=rω
and the moment of inertial of the particle moving in circle is mr2.
Hence, L=Iω........(i).
Then the kinetic energy of the particle moving in circle is
Ek=12mv2
Ek=12m(rω)2 (as v=rω)
⟹Ek=12mr2ω2
But the moment of inertial of the particle moving in circle is mr2.
⟹Ek=12Iω2=12(Iω)ω
⟹Ek=12Lω (from eqn(i))
Hence this is the kinetic energy of the particle moving in circle.