Answer to Question #13498 in Mechanics | Relativity for kinza rafiq
The law of conservation of angular momentum says that in the absence of
an external torque that may cause a system at rest to rotate, or cause an
already rotating system to change its motion, then the quantity known as
the total angular momentum of the system remains unchanged.
Total means just the sum of the individual angular momenta of a system.
Conservation means that the total angular momentum is constant.
Since angular momentum is a vector with both magnitude and direction,
constant total angular momentum means both the magnitude and direction
of the total angular momentum are constant.
(A) Example of the case where there is a change in the direction of the total
angular momentum when, in fact, the direction of the total angular
momentum must remain the same.
A system is initially at rest so its initial total angular momentum is zero.
After something happens within the system, a part of the system undergoes
rotational motion. Then, the total angular momentum of the system is no longer
zero. Since the change in the system came from within and not outside, there
was no external torque applied and the total angular momentum of the system
must remain unchanged. If it was initially at rest, then the total angular momentum
must remain zero because the absence of any rotation at the start signifies that
there is no total angular momentum initially present.
Now, total means the addition of at least two quantities. Since angular momentum
Is a vector with both magnitude and direction, addition of vectors means addition
of both magnitude and direction. Since the two additions take place independent
of each other, let us look at the addition of direction only.
Concerning the direction of the angular momentum, there are only two, namely
Counter clockwise rotation (+) and clockwise rotation( - ). If there was no direction
(zero direction) of rotation to speak of right at the start because the system is
initially at rest, so how can the addition of the directions of at least two angular
momenta give zero by cancelling each other out? By having one part of the system
rotate, say clockwise [ ( -) direction ], and having the rest of the system rotate in
the opposite direction, namely counter clockwise [ + direction ]. In that way, the
direction of the total angular momentum remains the same as before.
(B) Example of the case where there is a change in the magnitude of the total
angular momentum only because direction didn’t change when in fact the
total angular momentum must remain constant.
According to the formula for the angular momentum L :
(1) L = (mass in rotation) * (velocity in rotation)
the magnitude of the angular momentum depends on the mass in rotation as
well as in the velocity in rotation. Velocity in rotation is just the angular velocity.
It tells how fast an object turns through a certain angle as it rotates about a
particular axis of rotation. But mass in rotation is somewhat more difficult to
explain. Anyway, according to its definition, mass in rotation depends on three
& bull; the mass of the system,
& bull; the distribution of the mass about the axis of rotation,
& bull; the location of the axis of rotation
The larger the mass of the system for a given fixed distribution of the mass
about the axis of rotation, the greater the mass in rotation. The more the
distribution of mass about the axis of rotation for a given fixed amount of
mass, the greater also the mass in rotation. Between a thin rod (mass M,
length 2R) and a solid sphere (same mass M, radius R), the mass in rotation
of the solid sphere about an axis passing through its center is greater than
the mass in rotation of the thin rod about a vertical axis also passing through
its center. That’s because the mass of the sphere is more widely distributed
around the vertical axis than that of the rod.
Now, according to (1), the magnitude of the angular momentum remains
constant (as the law of conservation of angular momentum requires in the
absence of an applied external torque) if a decrease in one of the two
quantities there, say the mass in rotation, is followed by an increase in the
other quantity, namely the velocity, or speed, of rotation. That’s exactly
what happens when a spinning ice skater with a previously outstretched
arms folded the arms in other to reduce the mass in rotation. The action of
folding the arms doesn’t introduce an external torque because it is an
internal force in the first place. Since the total angular momentum must
remain fixed, a reduction in the mass in rotation is compensated by an
increase in the angular speed so the skater spins faster than before.
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