Problem 10.55 Protons and neutrons (together called nucleons) are held together
ID: 2077517 • Letter: P
Question
Problem 10.55 Protons and neutrons (together called nucleons) are held together in the nucleus of an atom by a force called the strong force. At very small separations, the strong force between two nucleons is larger than the repulsive electrical force between two protons hence its name. But the strong force quickly weakens as the distance between the protons increases. A well-established model for the potential energy of two nucleons interacting via the strong force is where a is the distance between the centers of the two nucleons, zo is a constant having the value ro J 2.0 x 10 15 m, and Uo 6.0 x 10 J. Quantum effects are essential for a proper understanding of nucleons, but let us innocently consider two neutrons as if they were small, hard, electrically neutral spheres of mass 1.67 x 10 27 kg and diameter 1.0 x 10 15 m. Suppose you hold two neutrons 4.0 x 10 m apart, measured between their centers, then release them.Explanation / Answer
mass of neutron m = 1.67e-27 kg
diameter D = 1.0e-15 m
Potential energy U = Uo (1-exp(-x/xo) ) , Uo = 6.0e-11 J and xo = 2.0e-15 m
intial the neutrons are held at distance of 4.0e-15 m apart
initial potential energy Ui = 6.0e-11*( 1- e-4/2 ) = 5.188 e-11 J
when the neutron crash together the minimum distance they can come close is the dia meter of each neutron = 1.0e-15 m
Final potential energy when the two neutrons are closest
Uf = 6.0e-11*(1- e-1/2) = 2.361 e-11 J
As the two neutrons crash upon they lose potential energy and gain Kinetic energy,
both neutron are of same size and mass and hecne the gina in KE of each neutron is exactly same, total KE of the two neutrons
mv2 = (5.188 -2.361) e-11 J
v = 1.693e+8 m/s
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