Solution: Here we are given a particle moving
in an elliptical trajectory under the influence of a central force
, which is
directed towards the center of rotation O.
(a)
- Angular
momentum is a vector quantity which is used as a
measure of the rotational momentum of a rotating body or system,
that is equal in classical physics to the product of the angular
velocity of the body or system and its moment of inertia with
respect to the rotation axis, and that is directed along the
rotation axis.
- Or, we can define the
angular momentum as the vector cross
product of the position vector of the particle and its linear
momentum which is also a vector.
- For a particle of mass m, moving with a certain velocity,
at a
certain position, with
respect to the axis of rotation, O, the angular
momentum, is
defined as:
where is the
linear momentum of the particle, which is defined as the momentum
of translation, being a vector quantity in classical physics, equal
to the product of the mass and the velocity of the center of mass.
It is given as -
- Putting this value in the equation for angular momentum, we
have
And its magnitude is given by
where is the angle
between and
.
(b) For a general case of motion:
- "The time rate of change of angular momentum of a
particle is equal to the vector sum of all torques acting on the
particle."
(c)
- The motion of any particle in such an elliptical trajectory is
only possible when it is subjected to a conservative force which is
acting towards the axis of rotation. Under such force, the energy
and the angular momentum of such a particle are always conserved.
One example of such a force is the force of gravity acting between
earth and the sun where sun orbits around the sun in a similar
elliptical path.
- So, applying the conservation of angular
momentum of the particle in positions 1 and 2, we get
where r_{1} and r_{2}
are the apogee and
perigee of the elliptical path and
So,
Since
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1 2 2 2
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ґр
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L1 = L2
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θ1 = θ2 =
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