## Angular Momentum Conserved & Spinning Wheel’s Constant Direction

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Angular momentum is a vector, so conservation of angular momentum tends to keep both the direction and angular speed of a spinning bicycle wheel constant.

Any bicycle rider knows that it is much easier to keep a bicycle balanced when it is moving than when the bike is not moving. Why? The key is **conservation of angular momentum** and the vector nature of angular momentum.

### Angular Momentum

Physicists measure the amount of spinning motion a rotating object has by its angular momentum. The angular momentum includes the spin rate, which is called angular velocity. The greater a rotating object’s angular velocity, the greater its angular momentum.

For linear motion, physicists define the momentum,

, of an object as its mass,pm, multiplied by its velocity,:v

=pm(where the bold variables indicate vector quantities)vPhysicists define the angular momentum,

, analogously as the moment of inertia,LI, multiplied by the angular velocity,:omega

=LIomega

The other quantity that determines an object’s angular momentum is its moment of inertia. In rotational motion, the moment of inertia is the analogy to mass. A rotating object’s moment of inertia is determined by the object’s mass and the distance from the mass to the rotational axis. A more massive rotating object will have a larger moment of inertia. If the mass is farther from the axis of rotation, the rotating object will also have a larger moment of inertia.

The angular momentum of a rotating object is its moment of inertia multiplied by its angular velocity.

The law of conservation of angular momentum states that if a rotating object, or system of objects, has no external torques, then its total angular momentum remains constant. In physics parlance, the angular momentum is conserved.

### Vectors and Angular Motion

Like momentum, angular momentum is a vector quantity. A vector is a quantity that includes both the amount (which physicists call the magnitude) and the direction of the quantity. For vector quantities, the direction matters. Changing either the magnitude or the direction of a vector quantity changes the quantity. Hence, changing the direction of the angular momentum vector changes the angular momentum, just as changing the moment of inertia or the angular speed does.

Like other rotational vectors, the vector direction of angular momentum is parallel to the axis of rotation. According to the right-hand rule, curving the fingers of the right hand to point in the direction of the rotation causes the perpendicular right thumb to point in the direction of the angular momentum vector along the rotational axis.

According to the conservation of angular momentum, the angular momentum of any object or system of objects having no external torques is conserved. For the total angular momentum of a rotating object to remain constant the direction, as well as the magnitude of the angular momentum, must remain constant.

### Easier to Balance a Moving Bike

When riding a bicycle, the spinning wheels have angular momentum. Unless there is some outside torque or force acting to change their angular momentum, the angular momentum of the wheels remains constant, as required by the conservation of angular momentum. Specifically, the direction of the bicycle wheels’ angular momentum tends to remain constant. Hence it takes a significant amount of external torque or force to change the direction that the wheels are spinning. The bicycle does not tip over easily and is easy to keep balanced.

When the bicycle is not moving and the wheels are not spinning, the wheels have zero angular momentum. It is therefore easy to change the direction that the wheels are spinning so balancing the bicycle is very difficult.

As a consequence of the law of conservation of angular momentum, it is easier to balance a bicycle when someone like Lance Armstrong is pedaling as hard as he can than when it is moving slowly. This fact however does not make it easier for a child just learning to ride to overcome the fear of falling off the bicycle.

### Applications of Angular Momentum Conservation

Figure skaters use this principle when they perform spins. Pulling their arms decreases their moment of inertia, so their angular velocity increases. The skater spins more rapidly. When the figure skater extends her arms again, her spin slows.

Angular momentum is a vector quantity, so direction matters. If the angular momentum remains constant, the direction of the rotation remains constant.

Lance Armstrong and other bike riders use this principle to remain balanced. The spinning wheels of moving bicycles remain pointed in the same direction if there are no external torques. Hence a moving bicycle is easier to balance than a stationary bicycle.

This principle also applies to gyroscopes, tops, and gyroscopic compasses.

Like the law of conservation of momentum, the law of conservation of angular momentum is a fundamental law of physics with no exceptions.