**Problem : **

A skater spins in the counterclockwise direction, as seen from above. In what direction does the vector representing the angular momentum of the skater point?

To find the direction of the angular momentum we use the right hand rule in the same way we used it for angular velocity. Thus, if we look down on the skater, and curl our fingers around in the counterclockwise direction, our thumb points towards us. Thus the angular momentum of the skater is pointing upwards.

**Problem : **

A particle moves in a straight line past a point O, as shown below. At which point is the angular momentum maximum? If the distance between O and the line is 2 m, and the object has a mass of 2 kg and a velocity of 3 m/s, what is the maximum angular momentum of the particle with respect to O?

A body moving in a straight line by an axis O

One might think that the maximum angular momentum will be when the object is traveling in the tangential direction with respect to the radius. However, notice that the radius is smallest at the point when the object is traveling in the tangential direction. Since angular momentum varies with radius, it cannot be maximal at this point. We will show that at all points, the angular momentum of the particle is the same. Let's take another look at the figure, and calculate the angular momentum at some arbitrary point, P:

The angular momentum of the mass at an arbitrary point P

Notice that the thetas cancel, and this answer is valid for P anywhere on the line of travel of the particle. Thus we have shown that the angular momentum of the particle is the same in all places. This agrees with our theorem that a net torque is required to change the angular momentum of a particle.

**Problem : **

What is the angular momentum of a thin hoop of radius 2 m and mass 1 kg that is rotating at a velocity of 4 rad/s?

It can be easily shown, and has been established in other sections, that the
moment of inertia of a thin hoop is simply
*MR*
^{2}
. Thus the angular momentum is
easily calculable:

**Problem : **

Two particles travel in parallel directions, as shown below. What is the total angular momentum of the system with respect to O?

A two particle system. What is the angular momentum of the system with respect
to O?

Quite simply, the total angular momentum is zero. At every point while the two particles are traveling, one particle moves in the clockwise direction with respect to O and one moves in the counterclockwise direction. Also, at every point, both particles have the same distance to the axis, and angle between the radius and the velocity of the particle. Thus the two particles always have equal and opposite angular momenta, and the total momentum of the system is zero.

**Problem : **

Many times a spinning top will not only spin about its axis, but precess about a vertical axis, meaning its point of contact with the ground remains the same, but the top swings around the vertical axis at an angle. What is the direction of the change in angular momentum in this situation? Where does the torque come from which causes this change in angular momentum?

We begin by drawing a diagram of the spinning top:

A spinning top shown with its direction of spin, its direction of precession,
and the two forces acting upon the top

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