Sound and Wave - Standing Wave

Small-amplitude standing waves of wavelength λ occur on a string with tension T, mass per unit μ, and length L. One end of string is fixed and the other end is attached to a ring of mass M that slides on a frictionless rod, as shown in the figure above. When gravity is neglected, which of the following conditions correctly determines the wavelength? (You might want to consider the limiting cases M → 0 and M → ∞.)

A. μ/M = (2π/λ) cot (2πL/λ)
B. μ/M = (2π/λ) tan (2πL/λ)
C. μ/M = (2π/λ) sin (2πL/λ)
D. λ = 2L/n, n = 1, 2, 3, ...
E. λ = 2L/(n + ½), n = 1, 2, 3, ...

(GR9677 #85)

Solution:

Consider the limiting cases M → 0 and M → ∞.

D and E do not depend on M → FALSE

If M → 0 , μ/M → ∞

C. FALSE because sin (2πL/λ)  cannot go to infinity

tan α = sin α / cos α → can go to infinity if cos α = 0

cot α = cos α / sin α → can go to infinity if sin α = 0

If M → ∞ , μ/M → 0, ring will not move = nodes on left and right side
The only possible wavelength: λ = 2L

2πL/λ = 2πL/2L = π

sin π = 0 → cot π = ∞ and tan π = 0 

 

Answer: B 

Nuclear & Particle Physics - Bragg Diffraction

The longest wavelength X-ray that can undergo Bragg diffraction in a crystal for a given family of planes of spacing d is

A. d/4
B. d/2
C. d
D. 2d
E. 4d

(GR9277 #02)

Solution:

Bragg’s law: 2d sin θ = nλ

Maximum → sin θ = 1

and n = 1 (1st order)

λ = 2d

Answer: D

Nuclear & Particle Physics - X-rays

A beam of electrons is accelerated through a potential difference of 25 kV in an X-ray tube. The continuous X-ray spectrum emitted by the target of the tube will have a short wavelength limit of most nearly

A. 0.1 Å
B. 0.5 Å
C. 2 Å
D. 25 Å
E. 50 Å

(GR9277 #80)

Solution:

Energy: E = pc
Wavelength: λ = h/p = hc/E

E = 25 kV
h = 4.14 × 10⁻¹⁵ eV second
c = 3 × 10⁸ m/s
hc = 1.24 × 10⁻⁶ eVm

→ λ = ^{1.24 × 10−6}/_{25 × 10³}  = ½ × 10⁻¹⁰ m = 0.5 Å

Answer: B

Nuclear & Particle Physics - Photon Scattering

Photons of wavelength λ scatter elastically on free protons initially at rest. The wavelength of the photons scattered at 90^o is increased by

1. λ ⁄ 137
2. λ ⁄ 1836
3. ħ ⁄ (m_ec) where ħ Planck's constant, m_e the rest mass of an electron, and c the speed of light
4. ħ ⁄ (m_pc) where ħ Planck's constant, m_p the rest mass of a proton, and c the speed of light
5. zero

(GR8677 #45)

Solution:

Compton Scattering: the inelastic scattering of photons from charged particles.

θ = 90^o → cos θ = 0
For proton scattered at 90^o, Δλ = ħ ⁄ (m_pc)

Answer: D

Optics - Michelson Interferometer

A Michelson interferometer is configured as a wave-meter, as shown in the figure above, so that a ratio of fringe counts may be used to compare the wavelength of two lasers with high precision. When the mirror in the right arm of the interferometer is translated through a distance d, 100,000 interference fringes pass across the detector for green light and 85,865 fringes pass across the detector for red (λ = 632.82 nanometers) light. The wavelength of the green laser light is

A. 500.33 nm

B. 543.37 nm

C. 590.19 nm

D. 736.99 nm

E. 858.65 nm

(GR0177 #100)

Solution

λ_red = 632.82 nm

λ_red  λ_green → (D) and (E) are FALSE.

Constructive path difference = Nλ

N_gλ_g = N_rλ_r

λ_g = N_rλ_r / N_g

= (85,865 × 632.82) / 100,000

≈ (8.6 × 10⁴ × 6.3 × 10²) / 10⁵

= 541.8

Answer: B