Conservation of Angular Momentum

The principle is this: If the net torque on a system is zero, the total angular momentum is constant. If you know about angular momentum and just want some typical applications, skip down several paragraphs.

Angular momentum of a rotating object is a function of its mass, its size and shape, and its rate of rotation. Recall that linear momentum is mv, and that for rotational motion the thing analogous to m is I, the inertia (or "moment of inertia") and the thing analogous to v is w , the rotational velocity in radians per second. Thus,

angular momentum L is given by L = Iw ,

where the direction of the L and w vectors is the same as the direction of advance of a screw or jar cover when it is rotated the way the object is rotating, lefty-loosey, righty-tighty, as they say. (This is a convention. It needs to be along the axis, because that is the unique direction line for a rotator, but the creatures from planet x might define it to be the opposite direction.)

Inertia is determined by mass and where the mass is located. You have greater inertia about a vertical axis when you stand with your feet apart and your arms outstretched sideways than when you are standing at attention. If you jump-spin the way figure skaters do, you will find that you can turn through a larger angle when your arms and legs are held in tight, reducing I, as soon as you leave the floor. Divers doing somersaults off a diving board can do more rotations if they go into the tuck position (chin on knees) after leaving the board than if they stay in the standing up position. More on this later.

To be exact, I = S mr², or in calculus, I = ò r²dm. (If the object can be regarded as a finite number of little lumps of mass, the sum works fine. For a continuous smear of mass, you need a sum of infinitely many infinitesimal masses: the integral. In both cases the r is the distance from mass to axis of rotation.

Here is another approach to the definition of angular momentum. Angular momentum of a particle is r x p, where r is distance vector from some reference point, and p is linear momentum mv. The cross product has magnitude r^ p = rp_^ and direction ^ to the plane defined by the r and p vectors. The direction is described above, in the paragraph following the introduction of L = Iw . For example, if you need to turn clockwise (as viewed from above your head) to watch a car go by, the car's angular momentum is down with respect to your location.

To continue: The above paragraph is for a particle. If you have more than one, the total angular momentum is the vector sum. A rotating object is infinitely many particles, each having mass dm and speed v = rw , where r is the distance to the axis of rotation, and the direction of v is ^ to r. Summing infinitely many of these, we have
L = ò r(rw )dm = [òr²dm]w = Iw .

At the beginning of this article it was stated that if the net torque on a system is zero, total angular momentum is constant. This comes from
S t = Ia in the same way that conservation of linear momentum comes from S F = ma, and most beginning college physics texts do the derivation. Here are some…

Applications:

1. The diver: (S)he starts with w _o leaning forward, body straight, arms above the head, inertia I_o. Then immediately goes into the tuck position, thighs against stomach, knees bent as much as possible, smaller inertia I. So the conservation of angular momentum principle says that I_o w _o = Iw . Given any three of the four quantities, you can calculate the 4^th.

2. The figure skater: same as the diver but with a vertical axis of rotation and different body orientation, but the same equation is valid if you can neglect friction during spins.

3. Satellite in non-circular (elliptical) orbit: If r is the distance from the center of the Earth to the satellite and q is the angle between v and r, conservation of momentum demands that rvsinq is constant. Conservation of energy demands that v²/2 - GM/r = constant. (In both of these equations, mass m of the satellite is canceled out. M is the mass of the Earth or whatever the satellite is orbiting, and we are assuming that the effect of all other planets, etc. is negligible. If you know r_a and v_a at the apogee (farthest), you can calculate r_p and v_p at the perigee (closest), for example. (q = 90^o at these extreme points.) Doing the math, we find that

v_p = GM/(r_av_a) + {[GM/(r_av_a)]² +v_a² - 2GM/r_a}^1/2

and r_p = r_av_a/v_p. Your instructor probably doesn't want you to use specialized formulas like this unless you derive them, because doing physics is not the process of plugging into equations. A big part of doing physics is figuring out what equation(s) apply.

Alternatively, given r_a and r_p you could find the two velocities. This is done on another page.

4. Sitting on a stool, which is capable of frictionless rotation, while holding a spinning wheel. Let's say the wheel is turning clockwise when viewed from above and you are at rest, holding it. If you grab the wheel to stop it, it will give you a clockwise rotation. If you then give the wheel a counterclockwise rotation, it will cause you to go faster clockwise. You could accomplish the same thing by simply turning the wheel over. It was going clockwise initially and the opposite when turned over. And turning it over will impart to you a clockwise rotation.

5. Jump onto a frictionless, motorless merry-go-round. Suppose it is initially at rest. If you are traveling toward the center, it will remain at rest. If your velocity has a tangential component, then you will give the merry-go-round some w . If you are small compared to the merry-g-r, you can be regarded as a point mass m. If the merry-g-r has inertia I before you jump on, conservation of angular momentum says that

r_^ mv = (I+mr²)w,

where r is the final r, not necessarily = r_^ . The dotted line to the center in the sketch is r_^.

6. Earth and Moon. By tugging on the tides, the Moon is decreasing the w of the Earth. Conservation of angular momentum of the Earth-Moon system says that the Moon must gain angular momentum. It also gains energy. (Earth rotates in a day, and it takes almost a month for the Moon to go around the Earth, so the tidal bulge on the Moon side is always leading the Moon.) The gain of energy is potential; it is moving farther away and slowing down, so it loses a little kinetic energy, but the net change is an increase of energy of the Moon. This process will continue until the Earth's spin w is equal to the Moon's orbit w , or until the Sun becomes a red giant and vaporizes Earth and Moon, billions of years from now.

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