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There is never any harm in going to the force diagram, as other answers have done, but if your question is "how do I develop an intuition that angular momentum should be conserved", then you can develop that based on your existing intuition that linear momentum should be conserved.

Although you probably believe it already anyway, one way to understand "why" momentum is conserved comes straight from Newton's laws, which you may or may already consider intuitive. (2) says that rate of change of momentum is proportional to force (equal if we use sensible units), and (3) says that forces come in equal-magnitude but opposite-direction pairs. That is to say, two vectors whose sum is the zero vector. So, any time a force acts, there must be two rates of change of momentum with equal magnitude and opposite direction. These cancel out in total, so total momentum is conserved.

Granted, that is not an exciting or revelatory proof mathematically, but my purpose is to establish one intuition by showing its relation to other intuitions that you already have. So, what about angular momentum?

When a force acts, consider not its magnitude and direction, but the magnitude of its component perpendicular to the line drawn from the point of action of the force, to a fixed point (usually we look at the centre of rotation as our fixed point, but in fact any other point in the universe will do for Newtonian mechanics. Einstein will make it harder, but not for 300 years). Now, everything is in motion, so that special line is moving, and I will hand-wave the difference between instantaneous forces vs. integration over time. But actually we did the same hand-wave with conservation of momentum in the case of a force that changes over time, and it didn't do us any harm.

Now, nothing has changed in Newton's laws. We still have equal and opposite forces giving us equal and opposite rates of change of momentum. But if total momentum is conserved, that means the total of any component of momentum is conserved. That's just how vector quantities work. And contact forces act at a particular point, so the distance from the fixed point to the point of action of the forces is the same for both parts of our pair of forces. And what do you get if you multiply rate of change of the perpendicular component of momentum, by distance to the centre? Rate of change of angular momentum.

So, we have equal and opposite rates of change of angular momentum. Angular momentum is conserved. Hence your intuition should be that angular momentum is a "real thing", like momentum or energy, and it can't just go missing. If a spinning thing retracts towards the centre, it has to spin faster, "for the same reason that" if a moving stone gathers moss it has to move slower.

Working out the exact details of how the system achieves this conservation, then, is what the force diagram is for. But the intuition is that it has to, because otherwise there would be a violation somewhere of equal and opposite forces.

Beware also that I have ignored the case of gravity, or in fact any action at a distance. So, this intuition about how contact forces governing things on ropes must behave, unfortunately does not immediately account for situations where the two forces in our opposed pair are applied at different points.

There is never any harm in going to the force diagram, as other answers have done, but if your question is "how do I develop an intuition that angular momentum should be conserved", then you can develop that based on your existing intuition that linear momentum should be conserved.

Although you probably believe it already anyway, one way to understand "why" momentum is conserved comes straight from Newton's laws, which you may or may already consider intuitive. (2) says that rate of change of momentum is proportional to force (equal if we use sensible units), and (3) says that forces come in equal-magnitude but opposite-direction pairs. That is to say, two vectors whose sum is the zero vector. So, any time a force acts, there must be two rates of change of momentum with equal magnitude and opposite direction. These cancel out in total, so total momentum is conserved.

Granted, that is not an exciting or revelatory proof mathematically, but my purpose is to establish one intuition by showing its relation to other intuitions that you already have. So, what about angular momentum?

When a force acts, consider not its magnitude and direction, but the magnitude of its component perpendicular to the line drawn from the point of action of the force, to a fixed point (usually we look at the centre of rotation as our fixed point, but in fact any other point in the universe will do for Newtonian mechanics. Einstein will make it harder, but not for 300 years). Now, everything is in motion, so that special line is moving, and I will hand-wave the difference between instantaneous forces vs. integration over time. But actually we did the same hand-wave with conservation of momentum in the case of a force that changes over time, and it didn't do us any harm.

Now, nothing has changed in Newton's laws. We still have equal and opposite forces giving us equal and opposite rates of change of momentum. But if total momentum is conserved, that means the total of any component of momentum is conserved. That's just how vector quantities work. And contact forces act at a particular point, so the distance from the fixed point to the point of action of the forces is the same for both parts of our pair of forces. And what do you get if you multiply rate of change of the perpendicular component of momentum, by distance to the centre? Rate of change of angular momentum (well, OK, actually just the rate of change of the component of angular momentum in one direction, but I'm trying to avoid vector cross products).

So, we have equal and opposite rates of change of angular momentum. Angular momentum is conserved. Hence your intuition should be that angular momentum is a "real thing", like momentum or energy, and it can't just go missing. If a spinning thing retracts towards the centre, it has to spin faster, "for the same reason that" if a moving stone gathers moss it has to move slower.

Working out the exact details of how the system achieves this conservation, then, is what the force diagram is for. But the intuition is that it has to, because otherwise there would be a violation somewhere of equal and opposite forces.

Beware also that I have ignored the case of gravity, or in fact any action at a distance. So, this intuition about how contact forces governing things on ropes must behave, unfortunately does not immediately account for situations where the two forces in our opposed pair are applied at different points.

There is never any harm in going to the force diagram, as other answers have done, but if your question is "how do I develop an intuition that angular momentum should be conserved", then you can develop that based on your existing intuition that linear momentum should be conserved.

Although you probably believe it already anyway, one way to understand "why" momentum is conserved comes straight from Newton's laws, which you may or may already consider intuitive. (2) says that rate of change of momentum is proportional to force (equal if we use sensible units), and (3) says that forces come in equal-magnitude but opposite-direction pairs. That is to say, two vectors whose sum is the zero vector. So, any time a force acts, there must be two rates of change of momentum with equal magnitude and opposite direction. These cancel out in total, so total momentum is conserved.

Granted, that is not an exciting or revelatory proof mathematically, but my purpose is to establish one intuition by showing its relation to other intuitions that you already have. So, what about angular momentum?

When a force acts, consider not its magnitude and direction, but the component of its magnitude perpendicular to a line drawn from the point of action of the force, to a fixed point (either the centre of rotation, or in fact any other point in the universe will do for Newtonian mechanics. Einstein will make it harder, but not for 300 years). Now, everything is in motion, so that special line is moving, and I will hand-wave the difference between instantaneous forces vs. integration over time. But actually we did the same hand-wave with conservation of momentum in the case of a force that changes over time, and it didn't do us any harm.

Now, nothing has changed in Newton's laws. We still have equal and opposite forces giving us equal and opposite rates of change of momentum. But if total momentum is conserved, that means the total of any component of momentum is conserved. That's just how vector quantities work. And forces act at a particular point, so the distance from the fixed point to the point of action of the forces is the same for both parts of our pair of forces. And what do you get if you multiply rate of change of the perpendicular component of momentum, by distance to the centre? Rate of change of angular momentum.

So, we have equal and opposite rates of change of angular momentum. Angular momentum is conserved. Hence your intuition should be that angular momentum is a "real thing", like momentum or energy, and it can't just go missing. If a spinning thing retracts towards the centre, it has to spin faster, "for the same reason that" if a moving stone gathers moss it has to move slower.

Working out the exact details of how the system achieves this conservation, then, is what the force diagram is for. But the intuition is that it has to, because otherwise there would be a violation somewhere of equal and opposite forces.

There is never any harm in going to the force diagram, as other answers have done, but if your question is "how do I develop an intuition that angular momentum should be conserved", then you can develop that based on your existing intuition that linear momentum should be conserved.

Although you probably believe it already anyway, one way to understand "why" momentum is conserved comes straight from Newton's laws, which you may or may already consider intuitive. (2) says that rate of change of momentum is proportional to force (equal if we use sensible units), and (3) says that forces come in equal-magnitude but opposite-direction pairs. That is to say, two vectors whose sum is the zero vector. So, any time a force acts, there must be two rates of change of momentum with equal magnitude and opposite direction. These cancel out in total, so total momentum is conserved.

Granted, that is not an exciting or revelatory proof mathematically, but my purpose is to establish one intuition by showing its relation to other intuitions that you already have. So, what about angular momentum?

When a force acts, consider not its magnitude and direction, but the magnitude of its component perpendicular to the line drawn from the point of action of the force, to a fixed point (usually we look at the centre of rotation as our fixed point, but in fact any other point in the universe will do for Newtonian mechanics. Einstein will make it harder, but not for 300 years). Now, everything is in motion, so that special line is moving, and I will hand-wave the difference between instantaneous forces vs. integration over time. But actually we did the same hand-wave with conservation of momentum in the case of a force that changes over time, and it didn't do us any harm.

Now, nothing has changed in Newton's laws. We still have equal and opposite forces giving us equal and opposite rates of change of momentum. But if total momentum is conserved, that means the total of any component of momentum is conserved. That's just how vector quantities work. And contact forces act at a particular point, so the distance from the fixed point to the point of action of the forces is the same for both parts of our pair of forces. And what do you get if you multiply rate of change of the perpendicular component of momentum, by distance to the centre? Rate of change of angular momentum.

So, we have equal and opposite rates of change of angular momentum. Angular momentum is conserved. Hence your intuition should be that angular momentum is a "real thing", like momentum or energy, and it can't just go missing. If a spinning thing retracts towards the centre, it has to spin faster, "for the same reason that" if a moving stone gathers moss it has to move slower.

Working out the exact details of how the system achieves this conservation, then, is what the force diagram is for. But the intuition is that it has to, because otherwise there would be a violation somewhere of equal and opposite forces.

Beware also that I have ignored the case of gravity, or in fact any action at a distance. So, this intuition about how contact forces governing things on ropes must behave, unfortunately does not immediately account for situations where the two forces in our opposed pair are applied at different points.

There is never any harm in going to the force diagram, as other answers have done, but if your question is "how do I develop an intuition that angular momentum should be conserved", then you can develop that based on your existing intuition that linear momentum should be conserved.

Although you probably believe it already anyway, one way to understand "why" momentum is conserved comes straight from Newton's laws, which you may or may already consider intuitive. (2) says that rate of change of momentum is proportional to force (equal if we use sensible units), and (3) says that forces come in equal-magnitude but opposite-direction pairs. That is to say, two vectors whose sum is the zero vector. So, any time a force acts, there must be two rates of change of momentum with equal magnitude and opposite direction. These cancel out in total, so total momentum is conserved.

Granted, that is not an exciting or revelatory proof mathematically, but my purpose is to establish one intuition by showing its relation to other intuitions that you already have. So, what about angular momentum?

When a force acts, consider not its magnitude and direction, but the component of its magnitude perpendicular to a line drawn from the point of action of the force, to a fixed point (either the centre of rotation, or in fact any other point in the universe will do for Newtonian mechanics. Einstein will make it harder, but not for 300 years). Now, everything is in motion, so that special line is moving, and I will hand-wave the difference between instantaneous forces vs. integration over time. But actually we did the same hand-wave with conservation of momentum in the case of a force that changes over time, and it didn't do us any harm.

Now, nothing has changed in Newton's laws. We still have equal and opposite forces giving us equal and opposite rates of change of momentum. But if total momentum is conserved, that means the total of any component of momentum is conserved. That's just how vector quantities work. And forces act at a particular point, so the distance from the fixed point to the point of action of the forces is the same for both parts of our pair of forces. And what do you get if you multiple rate of change of the perpendicular component of momentum, by distance to the centre? Rate of change of angular momentum.

So, we have equal and opposite rates of change of angular momentum. Angular momentum is conserved. Hence your intuition should be that angular momentum is a "real thing", just like momentum or energy, and it can't just go missing. If a spinning thing retracts towards the centre, it has to spin faster, "for the same reason that" if a moving stone gathers moss it has to move slower.

Working out the exact details of how the system achieves this conservation, then, is what the force diagram is for. But the intuition is that it has to, because otherwise there would be a violation somewhere of equal and opposite forces.

There is never any harm in going to the force diagram, as other answers have done, but if your question is "how do I develop an intuition that angular momentum should be conserved", then you can develop that based on your existing intuition that linear momentum should be conserved.

Although you probably believe it already anyway, one way to understand "why" momentum is conserved comes straight from Newton's laws, which you may or may already consider intuitive. (2) says that rate of change of momentum is proportional to force (equal if we use sensible units), and (3) says that forces come in equal-magnitude but opposite-direction pairs. That is to say, two vectors whose sum is the zero vector. So, any time a force acts, there must be two rates of change of momentum with equal magnitude and opposite direction. These cancel out in total, so total momentum is conserved.

Granted, that is not an exciting or revelatory proof mathematically, but my purpose is to establish one intuition by showing its relation to other intuitions that you already have. So, what about angular momentum?

When a force acts, consider not its magnitude and direction, but the component of its magnitude perpendicular to a line drawn from the point of action of the force, to a fixed point (either the centre of rotation, or in fact any other point in the universe will do for Newtonian mechanics. Einstein will make it harder, but not for 300 years). Now, everything is in motion, so that special line is moving, and I will hand-wave the difference between instantaneous forces vs. integration over time. But actually we did the same hand-wave with conservation of momentum in the case of a force that changes over time, and it didn't do us any harm.

Now, nothing has changed in Newton's laws. We still have equal and opposite forces giving us equal and opposite rates of change of momentum. But if total momentum is conserved, that means the total of any component of momentum is conserved. That's just how vector quantities work. And forces act at a particular point, so the distance from the fixed point to the point of action of the forces is the same for both parts of our pair of forces. And what do you get if you multiply rate of change of the perpendicular component of momentum, by distance to the centre? Rate of change of angular momentum.

So, we have equal and opposite rates of change of angular momentum. Angular momentum is conserved. Hence your intuition should be that angular momentum is a "real thing", like momentum or energy, and it can't just go missing. If a spinning thing retracts towards the centre, it has to spin faster, "for the same reason that" if a moving stone gathers moss it has to move slower.

Working out the exact details of how the system achieves this conservation, then, is what the force diagram is for. But the intuition is that it has to, because otherwise there would be a violation somewhere of equal and opposite forces.