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For a particle of mass "m" moving in a central
force field, show that:
v²-i²+1²0²-h²[(du/d0)²+u²]
2

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Electron capture is a variant on beta-radiation. The lightest nucleus to decay by electron capture is 7Be -- beryllium-7. The daughter nucleus is 7Li -- lithium-7. The electron is transformed into a massless particle (a neutrino): e − + 7 B e + ⟶ 7 L i + ν The initial electron is bound in the atom, so the beryllium mass includes the electron. In fact, since the electron starts bound in the atom, a more-accurate statement of the nuclear reaction is probably: 7 B e ⟶ 7 L i + ν The masses are beryllium: 7.016929 u, and lithium: 7.016003 u, and refer to the neutral atom as a whole. (Use uc and uc2 as your momentum and energy units -- but carry them along in your calculation.) The initial beryllium atom is stationary. Calculate the speed of the final lithium nucleus in km/s. (You will make life much easier for yourself if you recognize that practically all the energy released goes into the lighter particle. c = 300,000 km/s)
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An ideal monatomic gas is used in a heat engine and is taken around the cycle shown on the PV diagram of the figure. In part C of the cycle, the gas experiences adiabatic expansion such that it obeys the relation pVY = constant, where y = 2, and f is the number of degrees of freedom of the molecules. Hint: you DO NOT need to know the number of molecules to solve this problem. 1. Find the pressure P2. 2. Compute the work done on the gas, the change in the internal energy, and the heat added to the gas in step A. 3. Compute the work done on the gas, the change in the internal energy, and the heat added to the gas in step B. 4. Compute the work done on the gas, the change in the internal energy, and the heat added to the gas in step C. 5. Compute the net work done BY the gas, the input heat added to the gas, and the net change in the internal energy of the gas. 6. Compute the efficiency of the heat engine. 7. Compute the efficiency of a Carnot engine operating between the same temperature…

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