A helium nucleus, also known as an a (alpha) particle,consists of two protons and two neutrons and has a diameter of 10-15 m= 1 fm. The protons, with a charge of +e, are subject to a repulsiveCoulomb force. Since the neutrons have zero charge, there must be anattractive force that counteracts the electric repulsion and keeps the protonsfrom flying apart. This so-called strong force plays a central role inparticle physics. (a) As a crude model, assume that an a particle consistsof two pointlike protons attracted by a Hooke’s-law spring with springconstant k, and ignore the neutrons. Assume further that in the absenceof other forces, the spring has an equilibrium separation of zero. Writean expression for the potential energy when the protons are separated bydistance d. (b) Minimize this potential to find the equilibrium separationd0 in terms of e and k. (c) If d0 = 1.00 fm, what is the value of k?(d) How much energy is stored in this system, in terms of electronvolts? (e) A proton has a mass of 1.67 * 10^-27 kg. If the spring were tobreak, the a particle would disintegrate and the protons would fly off inopposite directions. What would be their ultimate speed?
A helium nucleus, also known as an a (alpha) particle,
consists of two protons and two neutrons and has a diameter of 10-15 m
= 1 fm. The protons, with a charge of +e, are subject to a repulsive
Coulomb force. Since the neutrons have zero charge, there must be an
attractive force that counteracts the electric repulsion and keeps the protons
from flying apart. This so-called strong force plays a central role in
particle physics. (a) As a crude model, assume that an a particle consists
of two pointlike protons attracted by a Hooke’s-law spring with spring
constant k, and ignore the neutrons. Assume further that in the absence
of other forces, the spring has an equilibrium separation of zero. Write
an expression for the potential energy when the protons are separated by
distance d. (b) Minimize this potential to find the equilibrium separation
d0 in terms of e and k. (c) If d0 = 1.00 fm, what is the value of k?
(d) How much energy is stored in this system, in terms of electron
volts? (e) A proton has a mass of 1.67 * 10^-27 kg. If the spring were to
break, the a particle would disintegrate and the protons would fly off in
opposite directions. What would be their ultimate speed?
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