Sound and Wave - Wave Phenomena

Consider a particle moving without friction on a rippled surface, as shown above. Gravity acts down in the negative h direction. The elevation h(x) of the surface is given by h(x) = d cos (kx). If the particle starts at x = 0 with a speed v in the x direction, for what values of v will the particle stay on the surface at all times?

A.
B.
C.
D.
E. 

(GR9677 #83)

Solution:

Gravity pulls the particle down. 
Particle will stay on the surface if  a_max = g

For a wave, a_max = ω²A = ω²d
(A = d = amplitude)

Velocity of the particle: v =  λ / T 

From the picture: λ = 2π/k

v = 2π/kT  = ω/k
v² = ω²/k²
ω² = v²k²

a_max = v²k²d = g 
v² = g/k²d
v = √(g/k²d)

Answer: D 

Electromagnetism - Coulomb's Law

Two pith balls of equal mass M and equal charge q are suspended from the same point on long massless threads of length L as shown in the figure above. If k is the Coulomb's law constant, then for small values of θ, the distance d between the charged pith balls at equilibrium is

A. (2kq²L/Mg)^1/3
B. (kq²L/Mg)^1/3
C. (2kq²L/Mg)^1/2 
D. (kq²L/Mg)^1/2
E. L/4

(GR9277 #83)

Solution: 

In Equilibrium:  
T cos θ = Mg  
T sin θ = F_coulomb = ^kq²/_d²
with T = tension

For θ ≪, cos (θ → 0) → 1
T = Mg  
sin θ =  ^½d/_L

^Mgd/_2L = ^kq²/_d²
Mgd³ = 2kq²L
d = (2kq²L/Mg)^1/3

Answer: A

Nuclear & Particle Physics - EM Spectrum

A spectral line is produced by a gas that is sufficiently dense that the mean time between atomic collisions is much shorter than the mean lives of the atomic states responsible for the line. Compared with the same line produced by a low-density gas, the line produced by the higher-density gas will appear

A. The same
B. More highly polarized
C. Broader
D. Shifted toward the blue end of the spectrum
E. Split into a doublet

(GR8677 #83)

Solution:

Spectral lines = bright or dark lines correspond to the emission or absorption of light of a single wavelength.

High-density gas:

• Atoms experience frequent collisions and many other mechanisms that cause spectral lines to lose their sharpness.
• It produces a continuum spectrum = weaker, less distinct or broader spectral lines.

Low-density gas:

• Atoms do not experience many collisions (the energy is coming from the particles themselves).
• It produces sharp spectral lines: emission spectrum (by hot low-density gas) and absorption spectrum (by cold low-density gas).

Answer: C