# Answer to Question #13498 in Mechanics | Relativity for kinza rafiq

Question #13498

state the law of conservation of angular momentum applied

Expert's answer

The law of conservation of angular momentum states that when no external torque acts on an object or a closed system of objects, no change of angular momentum can occur. Figure skaters spinning, competitive divers twisting, pulsars spinning, riders balancing bicycles, gyroscopic compasses giving direction, and tops spinning are all applications of the law of conservation of angular momentum.

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

things, namely

& 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.

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

things, namely

& 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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