Assignment 5

Problem 1:

Consider an atomic nucleus.
(a)  Why do nuclei tend to have approximately the same number of neutrons and protons?
(b)  As you increase element atomic number, why does the ratio of neutrons to protons increase?
(c)  Why do light nuclei release energy via fusion while heavy nuclei release energy via fission?

Problem 2:

For this one-dimensional problem let the x-axis point downward.  A vertical spring has a spring constant k = 48 N/m.  At t = 0 a force F(t) = (51 N)sin(4t) (t ≥ 0 with t in seconds) is applied to a 30 N weight which hangs in equilibrium at the end of the spring.  Neglecting damping, find the position of the weight at any time t.

Problem 3:

In 2017 ETH researchers succeeded in shortening the pulse duration of an X‑ray laser to only 43 attoseconds to produce the world's shortest laser pulse.  The soft X-ray laser pulse has a very large spectral bandwidth.  What is the uncertainty in the momentum of a photon in this pulse?

Problem 4:

Assume white light (λ = 400 nm - 650 nm) is incident at an angle of 60o with respect to the normal from air on a thin film of oil on water.  The thickness of the oil film is t = 100 nm, and the indices of refraction of water and oil are nwater = 1.33 and noil = 1.5, respectively.  Which wavelength(s) will dominate in the reflected light?

Problem 5:

The radioactive isotope 229Th is an α emitter with a half-life t½ Th = 7,300 y.  Its daughter, 225Ra, is a β emitter with a half-life t½ Ra = 14.8 d, which is much shorter than the parent's.
(a)  Show that the number of 225Ra nuclei obey the differential equation
dNRa/dt = NThTh - NRaRa,
where NTh is the number of 229Th nuclei and NRa is the number of 225Ra nuclei, and τTh and τRa are their mean lives, respectively.
(b)  Suppose that at time t = 0 there are NTh0 nuclei of the parent isotope 229Th and no nuclei of the daughter isotope 225Ra.  Find NRa for t > 0.
(c)  Show that after several years, NRa = NTh τRaTh.