When Is Angular Momentum Not Conserved?
Have you ever spun a bicycle wheel and felt how difficult it is to tilt it sideways? On top of that, or watched a gymnast tumble through the air and marveled at how they seem to control their rotation mid-flight? These are all examples of angular momentum in action—essentially, the rotational equivalent of linear momentum. But here's the thing: while angular momentum is often conserved in physics problems, it’s not always the case. So in fact, there are specific scenarios where it’s not conserved at all. So when does this happen, and why should you care?
What Is Angular Momentum?
Angular momentum is a measure of an object’s tendency to keep rotating. Think of it as rotational inertia in motion. In practice, just like linear momentum depends on mass and velocity, angular momentum depends on an object’s moment of inertia (how mass is distributed relative to the axis of rotation) and its angular velocity. The formula? It’s L = Iω, where L is angular momentum, I is the moment of inertia, and ω is angular velocity.
Easier said than done, but still worth knowing Small thing, real impact..
This concept isn’t just for spinning tops or planets. It applies to anything rotating—whether it’s a spinning ice skater, a tornado, or even the Earth orbiting the Sun. Plus, when there’s no external torque acting on a system, angular momentum stays constant. This is why a skater spins faster when they pull their arms in: their moment of inertia decreases, so their angular velocity must increase to conserve angular momentum.
But what happens when that conservation breaks down?
Why It Matters
Understanding when angular momentum is conserved—or not—isn’t just academic. Even so, it’s practical. Engineers use these principles to design stable satellites, athletes exploit them to optimize performance, and physicists rely on them to predict everything from galaxy formations to subatomic particle behavior Small thing, real impact. Which is the point..
If you misjudge when angular momentum is conserved, you could end up with a wobbly spacecraft, an inefficient engine, or a misunderstood physical system. So knowing the exceptions is just as important as knowing the rule.
How It Works: When Angular Momentum Isn’t Conserved
When There’s an External Torque
The golden rule is: angular momentum is conserved only when there’s no external torque. Torque is what causes rotational acceleration—essentially, a force that acts at a distance from the axis of rotation. If an external torque is present, it changes the system’s angular momentum Took long enough..
Imagine a door with a strong wind blowing against it. The wind applies a force at some distance from the hinges, creating a torque. This torque changes the door’s angular momentum, causing it to swing open or shut unexpectedly. In this case, the external torque (the wind) means angular momentum isn’t conserved.
Another example? A car’s engine. On top of that, when you press the gas pedal, the engine applies a torque to the wheels, changing their angular momentum. Without that torque, the wheels wouldn’t spin faster Worth keeping that in mind. Simple as that..
Friction and Other Dissipative Forces
Friction is a sneaky culprit. And it’s an external force that acts at the point of contact between surfaces, creating torque. Think of a spinning top slowing down on a table. The friction between the tip of the top and the table applies a torque, gradually reducing the top’s angular momentum until it stops.
Even air resistance can do this. A frisbee eventually comes to rest not because of some fundamental law, but because air molecules push against it, applying tiny torques that dissipate its spin And that's really what it comes down to..
Changing the Reference Point
Here’s a subtle but critical point: angular momentum depends on the reference point you choose. If you shift the reference point (say, from the center of mass to a different axis), the angular momentum of a system can change—even if no external torque is acting.
Take this: consider a person standing on a frictionless turntable holding a spinning bicycle wheel. If they tilt the wheel’s axis, the turntable itself will start rotating in the opposite direction. This is because the change in the wheel’s angular momentum (due to the tilt) creates an external torque on the system. The person and turntable are no longer a closed system in terms of angular momentum conservation Small thing, real impact. But it adds up..
Non-Rigid Systems
Angular momentum conservation assumes the system is rigid—that is, the mass distribution doesn’t change. But in the real world, systems are rarely perfectly rigid. When a system is flexible or deformable, internal forces can redistribute mass, altering the moment of inertia and thus the angular momentum And it works..
Some disagree here. Fair enough.
Take a cat landing on its feet. The cat twists its body mid-fall, shifting its internal structure. While the total angular momentum of the cat and Earth system is conserved, the cat’s own angular momentum isn’t because it’s not a rigid body. The Earth absorbs a tiny bit of angular momentum, but it’s negligible due to its enormous mass.
Not the most exciting part, but easily the most useful.
External Forces Acting Off-Center
In linear motion, forces can be applied anywhere, but in rotational motion, the point of application matters. Even so, a force applied directly through the axis of rotation creates no torque and thus doesn’t affect angular momentum. But if that force is applied off-center, it creates torque, changing the system’s angular momentum Still holds up..
Imagine pushing a heavy crate across a floor. If you push straight toward the center, it slides without rotating. But if you push near the top edge, it both moves and spins. The off-center force creates torque, so angular momentum isn’t conserved for the crate alone.
Common Mistakes: What Most People Get Wrong
One common mistake is assuming that angular momentum is always conserved just because an object is spinning. But that’s only true in the absence of external torques. People often forget that even small forces—like air resistance
One common misconception is that the mere presence of a rotating body guarantees that its angular momentum will stay constant. Because of that, for instance, a spinning top that wobbles slowly is experiencing a torque from the gravitational force acting on its center of mass, which is offset from the rotation axis. Over time this torque drains the top’s angular momentum, causing it to lose speed and eventually fall. In real terms, in reality, any net external torque—no matter how slight—will alter the quantity, and the sign of that torque determines the direction of change. The same principle applies to satellites in orbit: atmospheric drag, though minute, exerts a tangential force that creates a tiny but measurable torque, gradually slowing the satellite’s spin and altering its orbital angular momentum No workaround needed..
Another frequent error involves treating the Earth as a fixed reference when analyzing rotational motion. When a object launches upward, the Earth’s rotation must be accounted for; otherwise, the apparent change in the object’s angular momentum seems paradoxical. Practically speaking, in an inertial frame, the planet’s own rotation contributes a substantial amount of angular momentum. In practice, engineers designing launch trajectories incorporate the planet’s rotational speed to see to it that the vehicle’s final angular momentum matches the desired orbit.
A related pitfall is the assumption that angular momentum is conserved in non‑inertial (accelerating) frames without introducing fictitious torques. When a system rotates while accelerating linearly, the apparent “centrifugal” force can be interpreted as an additional torque that must be included in the balance equation. Failing to add these pseudo‑torques leads to incorrect predictions, especially in rotating machinery that undergoes translational motion, such as turbines mounted on a moving ship.
Let’s examine a concrete illustration. And imagine a figure skater initially spinning with arms extended, possessing a certain moment of inertia (I_i) and angular velocity (\omega_i). Pulling the arms inward reduces the moment of inertia to (I_f) It's one of those things that adds up..
[ I_i ,\omega_i = I_f ,\omega_f . ]
If the skater’s initial angular momentum were mistakenly thought to change simply due to the altered arm position, the resulting prediction for (\omega_f) would be wrong. The correct outcome is a higher angular velocity proportional to the inverse ratio of the moments of inertia, a fact that underlies many performance tricks in gymnastics, figure skating, and even the operation of jet turbines.
Not the most exciting part, but easily the most useful.
In astrophysical contexts, the same precept applies. A collapsing star conserves its angular momentum, causing its rotation to speed up as its radius shrinks—a process that leads to the formation of rapidly spinning neutron stars or black holes. If one were to ignore the conservation law and assume the star’s spin would stay the same, the predicted rotational periods would be orders of magnitude off, undermining models of stellar evolution.
Practical engineers also encounter angular‑momentum misconceptions when designing rotating parts that are mounted on flexible supports. Now, if the designer assumes perfect isolation, the motor’s speed will drift, causing premature wear or loss of precision. A motor mounted on a vibrating base may appear to maintain a steady angular speed, yet the base’s motion introduces time‑varying torques through the mounting structure. Incorporating the support’s dynamics into the free‑body diagram restores proper angular‑momentum accounting Easy to understand, harder to ignore. Still holds up..
Finally, relativistic speeds introduce subtle corrections. And as an object approaches the speed of light, its relativistic mass increase modifies the moment of inertia, and the simple non‑relativistic expression for angular momentum no longer holds. In high‑energy particle accelerators, the angular momentum of bunches of electrons must be treated with relativistic dynamics to avoid systematic errors in beam steering.
In a nutshell, angular momentum is a powerful conservation law, but its correct application hinges on several nuanced considerations: the presence or absence of external torques, the choice of reference point, the rigidity of the system, and the inclusion of all forces—including minute ones like air resistance or mounting friction. By vigilantly checking these factors, scientists and engineers can avoid the most common mistakes and apply the principle reliably across everything from everyday playground toys to interstellar phenomena Took long enough..
