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 

Thermal Physics - Conduction Electron

The Fermi temperature of Cu is about 80,000 K. Which of the following is most nearly equal to the average speed of a conduction electron in Cu? 

A. 2 × 10⁻² m/s
B. 2 m/s
C. 2 × 10² m/s
D. 2 × 10⁴ m/s
E. 2 × 10⁶ m/s

(GR9277 #23)

Solution:

T_F = 8 × 10⁴ K
m_e = 9.11 × 10⁻³¹ kg
k = 1.38 × 10⁻²³ J/K

Fermi energy = Kinetic energy



Answer: E

Classical Mechanics - Wave equation

The equation where A, T, and λ are positive constants, represents a wave whose

A. Amplitude is 2A
B. Velocity is in the negative x–direction
C. Period is T/λ
D. Speed is x/t
E. Speed is λ/T

(GR8677 #04)

Solution:

(A) FALSE.
Amplitude  = maximum displacement = y_max = A.

(B) FALSE.
For a wave traveling to the right (positive x-direction): y = ƒ(x − vt)
(x − vt) = constant.
As t increases, x must increases to keep (x − vt) = constant

For a wave traveling to the left (negative x-direction): y = ƒ(x + vt)
As t increases, x decreases to keep (x + vt) = constant.

The problem gives:  y = ƒ(vt − x)
As t increases, x must increases to keep (x − vt) = constant.
The waves is traveling to the right.

(C) FALSE.
The unit of T/λ (second/meter) does not match with the unit is period (second).

(D) FALSE.
The speed of  the wave is λ/T.

(E) TRUE.
See D.

Answer: E