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A tornado touches down in Spring Hill, Kansas, May 20, 1957.
“Sure, and maybe the sun won't come up tomorrow.” Of course, the sun only appears to go up and down because the earth spins, so the cliche should really refer to the unlikelihood of the earth's stopping its rotation abruptly during the night. Why can't it stop? It wouldn't violate conservation of momentum, because the earth's rotation doesn't add anything to its momentum. While California spins in one direction, some equally massive part of India goes the opposite way, canceling its momentum. A halt to Earth's rotation would entail a drop in kinetic energy, but that energy could simply be converted into some other form, such as heat.
Other examples along these lines are not hard to find. A hydrogen atom spins at the same rate for billions of years. A high-diver who is rotating when he comes off the board does not need to make any physical effort to continue rotating, and indeed would be unable to stop rotating before he hit the water.
These observations have the hallmarks of a conservation law, but what numerical measure of rotational motion is conserved? Car engines and old-fashioned LP records have speeds of rotation measured in rotations per minute (r.p.m.), but the number of rotations per minute (or per second) is not a conserved quantity. For example, the twirling figure skater in figure a can pull her arms in to increase her r.p.m.'s.
The example of the figure skater suggests that this conserved quantity depends on distance from the axis of rotation. We'll notate this distance as r, since, for an object moving in a circle around an axis of rotation, its distance from the axis equals the radius of the circle.
Once we realize that r is a variable that matters, it becomes clear that the examples we've been considering were all examples that would be fairly complicated mathematically, because different parts of these objects' masses have different values of r. For example, the figure skater's front teeth are farther from the axis than her back teeth. That suggests that instead of objects with complicated shapes, we should consider the simplest possible example, which is a single particle, of mass m, traveling in a circle of radius r at speed v. Experiments show that the conserved quantity in this situation is
We call this quantity angular momentum. The symbol ± indicates that angular momentum has a positive or negative sign to represent the direction of rotation; for example, in a given problem, we could choose to represent clockwise angular momenta as positive numbers, and counterclockwise ones as negative. In this equation, the only velocity that matters is velocity that is perpendicular to the radius line; motion parallel to the radius line, i.e., directly in our out, is neither clockwise nor counterclockwise.
When the skater in figure a pulls her arms in, she is decreasing r for all the atoms in her arms. It would violate conservation of angular momentum if she then continued rotating at the same speed, i.e., taking the same amount of time for each revolution, because her arms would be closer to the axis of rotation and therefore have a smaller r (as well as a smaller v because they would be completing a smaller circle in the same time). This is impossible because it would violate conservation of angular momentum. If her total angular momentum is to remain constant, the decrease in angular momentum for her arms must be compensated for by an overall increase in her rate of rotation. That is, by pulling her arms in, she substantially reduces the time for each rotation.
b / Example 2: An early photograph of an old-fashioned long-jump.
c / Example 3.
◊ Conservation of plain old momentum, p, can be thought of as the greatly expanded and modified descendant of Galileo's original principle of inertia, that no force is required to keep an object in motion. The principle of inertia is counterintuitive, and there are many situations in which it appears superficially that a force is needed to maintain motion, as maintained by Aristotle. Think of a situation in which conservation of angular momentum, L, also seems to be violated, making it seem incorrectly that something external must act on a closed system to keep its angular momentum from “running down.”
◊ The figure is a strobe photo of a pendulum bob, taken from underneath the pendulum looking straight up. The black string can't be seen in the photograph. The bob was given a slight sideways push when it was released, so it did not swing in a plane. The bright spot marks the center, i.e., the position the bob would have if it hung straight down at us. Does the bob's angular momentum appear to remain constant if we consider the center to be the axis of rotation? What if we choose a different axis?
Discussion question B.
e / The boy makes a torque on the tetherball.
f / The plane's four engines produce zero total torque but not zero total force.
g / Example 5: the biceps muscle flexes the arm.
Discussion question C.
Force is the rate of transfer of momentum. The equivalent in the case of angular momentum is called torque (rhymes with “fork”):
Where force tells us how hard we are pushing or pulling on something, torque indicates how hard we are twisting on it.
Have you ever had the experience of trying to open a door by pushing on the wrong side, the side near the hinge? It's difficult to do, which apparently indicates that a given amount of force produces less torque when it's applied close to the axis of rotation. To try to pin down this relationship more precisely, let's imagine hitting a tetherball, e. The boy applies a force F to the ball for a short time t, accelerating the ball to a velocity v. Since force is the rate of transfer of momentum, we have
But ± mvr is simply the amount of angular momentum he's given the ball, so ± mvr/t also equals the amount of torque he applied. The result of this example is
where the plus or minus sign indicates whether torque would tend to create clockwise or counterclockwise motion. This equation applies more generally, with the caveat that F should only include the part of the force perpendicular to the radius line.
self-check: There are four equations on this page. Which ones are important, and likely to be useful later? (answer in the back of the PDF version of the book)
To summarize, we've learned three conserved quantity, each of which has a rate of transfer:
conserved quantity | rate of transfer
| ||
name | units | name | units |
energy | joules (J) | power | watts (W) |
momentum | kgunitunitdotmunitsunit | force | newtons (N) |
angular momentum | kgunitunitdotmunit2sunit | torque | newton-meters nunitunitdotmunit) |
Of course a force is necessary in order to create a torque --- you can't twist a screw without pushing on the wrench --- but force and torque are two different things. One distinction between them is direction. We use positive and negative signs to represent forces in the two possible directions along a line. The direction of a torque, however, is clockwise or counterclockwise, not a linear direction.
The other difference between torque and force is a matter of leverage. A given force applied at a door's knob will change the door's angular momentum twice as rapidly as the same force applied halfway between the knob and the hinge. The same amount of force produces different amounts of torque in these two cases.
It is possible to have a zero total torque with a nonzero total force. An airplane with four jet engines, f, would be designed so that their forces are balanced on the left and right. Their forces are all in the same direction, but the clockwise torques of two of the engines are canceled by the counterclockwise torques of the other two, giving zero total torque.
Conversely, we can have zero total force and nonzero total torque. A merry-go-round's engine needs to supply a nonzero torque on it to bring it up to speed, but there is zero total force on it. If there was not zero total force on it, its center of mass would accelerate!
There are three forces acting on the forearm: the force from the biceps, the force at the elbow joint, and the force from the load being lifted. Because the elbow joint is motionless, it is natural to define our torques using the joint as the axis. The situation now becomes quite simple, because the upper arm bone's force exerted at the elbow has r=0, and therefore creates no torque. We can ignore it completely. In general, we would call this the fulcrum of the lever.
If we restrict ourselves to the case in which the forearm rotates with constant angular momentum, then we know that the total torque on the forearm is zero, so the torques from the muscle and the load must be opposite in sign and equal in absolute value:
where rmuscle, the distance from the elbow joint to the biceps' point of insertion on the forearm, is only a few cm, while rload might be 30 cm or so. The force exerted by the muscle must therefore be about ten times the force exerted by the load. We thus see that this lever is a force reducer. In general, a lever may be used either to increase or to reduce a force.
Why did our arms evolve so as to reduce force? In general, your body is built for compactness and maximum speed of motion rather than maximum force. This is the main anatomical difference between us and the Neanderthals (their brains covered the same range of sizes as those of modern humans), and it seems to have worked for us.
As with all machines, the lever is incapable of changing the amount of mechanical work we can do. A lever that increases force will always reduce motion, and vice versa, leaving the amount of work unchanged.
◊ You whirl a rock over your head on the end of a string, and gradually pull in the string, eventually cutting the radius in half. What happens to the rock's angular momentum? What changes occur in its speed, the time required for one revolution, and its acceleration? Why might the string break?
◊ A helicopter has, in addition to the huge fan blades on top, a smaller propeller mounted on the tail that rotates in a vertical plane. Why?
◊ The photo shows an amusement park ride whose two cars rotate in opposite directions. Why is this a good design?
Suppose a sunless planet is sitting all by itself in interstellar space, not rotating. Then, one day, it decides to start spinning. This doesn't necessarily violate conservation of energy, because it could have energy stored up, e.g., the heat in a molten core, which could be converted into kinetic energy. It does violate conservation of angular momentum, but even if we didn't already know about that law of physics, the story would seem odd. How would it decide which axis to spin around? If it was to spontaneously start spinning about some axis, then that axis would have to be a special, preferred direction in space. That is, space itself would have to have some asymmetry to it.
In reality, as I've already mentioned on page 15, experiments show to a very high degree of precision that the laws of physics are completely symmetric with respect to different directions. The story of the planet that abruptly starts spinning is an example of Noether's theorem, applied to angular momentum. We now have three such examples:
\begin{noethertable}
time symmetry & \noetherimplies & mass-energy
translation symmetry & \noetherimplies & momentum
rotational symmetry & \noetherimplies & angular momentum
\end{noethertable}
1. You are trying to loosen a stuck bolt on your RV using a big wrench that is 50 cm long. If you hang from the wrench, and your mass is 55 kg, what is the maximum torque you can exert on the bolt? (answer check available at lightandmatter.com)
2. A physical therapist wants her patient to rehabilitate his injured elbow by laying his arm flat on a table, and then lifting a 2.1 kg mass by bending his elbow. In this situation, the weight is 33 cm from his elbow. He calls her back, complaining that it hurts him to grasp the weight. He asks if he can strap a bigger weight onto his arm, only 17 cm from his elbow. How much mass should she tell him to use so that he will be exerting the same torque? (He is raising his forearm itself, as well as the weight.) (answer check available at lightandmatter.com)
3. An object is observed to have constant angular momentum. Can you conclude that no torques are acting on it? Explain. [Based on a problem by Serway and Faughn.]
4. (solution in the pdf version of the book) The figure shows scale drawing of a pair of pliers being used to crack a nut, with an appropriately reduced centimeter grid. Warning: do not attempt this at home; it is bad manners. If the force required to crack the nut is 300 N, estimate the force required of the person's hand.
5. Two horizontal tree branches on the same tree have equal diameters, but one branch is twice as long as the other. Give a quantitative comparison of the torques where the branches join the trunk. [Thanks to Bong Kang.]
6. (a) Alice says Cathy's body has zero momentum, but Bob says Cathy's momentum is nonzero.
Nobody is lying or making a mistake. How is this possible? Give a concrete example.
(b) Alice and Bob agree that Dong's body has nonzero momentum, but disagree about Dong's
angular momentum, which Alice says is zero, and Bob says is nonzero. Explain.
7. A person of weight W stands on the ball of one foot. Find the tension in the calf muscle and the force exerted by the shinbones on the bones of the foot, in terms of W,a, and b. (The tension is a measure of how tight the calf muscle has been pulled; it has units of newtons, and equals the amount of force applied by the muscle where it attaches to the heel.) For simplicity, assume that all the forces are at 90-degree angles to the foot. Suggestion: Write down an equation that says the total force on the foot is zero, and another equation saying that the total torque on the foot is zero; solve the two equations for the two unknowns.
