Noether’s Theorem for Angular Momentum

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:

symmetry conserved quantity

time symmetry ⇒ mass-energy

translation symmetry ⇒ momentum

rotational symmetry ⇒ angular momentum

Problem 4.

Problem 7.

Problems

Key_√

A computerized answer check is available online.

R A problem that requires calculus.

? A difficult problem.

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? ^√

2 A physical therapist wants her patient to rehabilitate his in-jured 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.) ^√ 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 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. . Solution, p. 182

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.

a/Albert Einstein.

Chapter 4

Relativity

Complaining about the educational system is a national sport among professors in the U.S., and I, like my colleagues, am often tempted to imagine a golden age of education in our country’s past, or to compare our system unfavorably with foreign ones. Reality intrudes, however, when my immigrant students recount the overemphasis on rote memorization in their native countries, and the philosophy that what the teacher says is always right, even when it’s wrong.

Albert Einstein’s education in late-nineteenth-century Germany was neither modern nor liberal. He did well in the early grades,¹ but in high school and college he began to get in trouble for what today’s edspeak calls “critical thinking.”

Indeed, there was much that deserved criticism in the state of physics at that time. There was a subtle contradiction between the theory of light as a wave and Galileo’s principle that all motion is relative. As a teenager, Einstein began thinking about this on an intuitive basis, trying to imagine what a light beam would look like if you could ride along beside it on a motorcycle at the speed of light. Today we remember him most of all for his radical and far-reaching solution to this contradiction, his theory of relativity, but in his student years his insights were greeted with derision from his professors. One called him a “lazy dog.” Einstein’s distaste for authority was typified by his decision as a teenager to renounce his German citizenship and become a stateless person, based purely on his opposition to the militarism and repressiveness of German society. He spent his most productive scientific years in Switzerland and Berlin, first as a patent clerk but later as a university professor.

He was an outspoken pacifist and a stubborn opponent of World War I, shielded from retribution by his eventual acquisition of Swiss citizenship.

As the epochal nature of his work became evident, some liberal Germans began to point to him as a model of the “new German,”

but after the Nazi coup d’etat, staged public meetings began, at which Nazi scientists criticized the work of this ethnically Jewish (but spiritually nonconformist) giant of science. When Hitler was appointed chancellor, Einstein was on a stint as a visiting professor at Caltech, and he never returned to the Nazi state. World War

1The myth that he failed his elementary-school classes comes from a misun-derstanding based on a reversal of the German numerical grading scale.

b/The first nuclear explo-sion on our planet, Alamogordo, New Mexico, July 16, 1945.

II convinced Einstein to soften his strict pacifist stance, and he signed a secret letter to President Roosevelt urging research into the building of a nuclear bomb, a device that could not have been imagined without his theory of relativity. He later wrote, however, that when Hiroshima and Nagasaki were bombed, it made him wish he could burn off his own fingers for having signed the letter.

Einstein has become a kind of scientific Santa Claus figure in popular culture, which is presumably why the public is always so tit-illated by his well-documented career as a skirt-chaser and unfaithful husband. Many are also surprised by his lifelong commitment to so-cialism. A favorite target of J. Edgar Hoover’s paranoia, Einstein had his phone tapped, his garbage searched, and his mail illegally opened. A censored version of his 1800-page FBI file was obtained in 1983 under the Freedom of Information Act, and a more com-plete version was disclosed recently.² It includes comments solicited from anti-Semitic and pro-Nazi informants, as well as statements, from sources who turned out to be mental patients, that Einstein had invented a death ray and a robot that could control the human mind. Even today, an FBI web page³ accuses him of working for or belonging to 34 “communist-front” organizations, apparently in-cluding the American Crusade Against Lynching. At the height of the McCarthy witch hunt, Einstein bravely denounced McCarthy, and publicly urged its targets to refuse to testify before the House Unamerican Activities Committee. Belying his other-worldly and absent-minded image, his political positions seem in retrospect not to have been at all clouded by naivete or the more fuzzy-minded variety of idealism. He worked against racism in the U.S. long be-fore the civil rights movement got under way. In an era when many leftists were only too eager to apologize for Stalinism, he opposed it consistently.

This chapter is specifically about Einstein’s theory of relativ-ity, but Einstein also began a second, parallel revolution in physics known as the quantum theory, which stated, among other things, that certain processes in nature are inescapably random. Ironically, Einstein was an outspoken doubter of the new quantum ideas that were built on his foundations, being convinced that “the Old One [God] does not play dice with the universe,” but quantum and rel-ativistic concepts are now thoroughly intertwined in physics.