Electromagnetism - Oscilloscope

The figure above represents the trace on the screen of cathode ray oscilloscope. The screen is graduated in centimeters. The spot on the screen moves horizontally with a constant speed of 0.5 centimeter/millisecond and the vertical scale is 2 volts/centimeter. The signal is a superposition of two oscillations. Which of the following are most nearly the observed amplitude and frequency of these two oscillations? 

Oscillation 1 Oscillation 2 A. 5V, 250Hz 2.5V, 1000Hz B.  1.5V, 250Hz 3V, 1500Hz  C. 5V, 6Hz 2V, 2Hz  D. 2.5V, 83Hz 1.25V, 500Hz  E. 6.14V, 98Hz                1.35V, 257Hz

(GR9677 #28)

Solution:

The graph shows one big λ consists of 6 small λs.
λ_big ≈ 6 cm
λ_small ≈ 1 cm 

v = λf

Given: v = 0.5 cm/ms
f_big = v/λ_b = 0.5 cm/(6 cm ms) = 1/ (12 ms) = 10³/(12 s) = 83 Hz (Osc. 1)
f_small = v/λ_s = 0.5 cm/(1 cm ms) = 1/ (2 ms) = 10³/(2 s) = 500 Hz (Osc. 2)

Answer: D

Sound and Wave - Wave phenomena

A string consists of two parts attached at x = 0. The right part of the string (x  0) has mass μ_r per unit length and the left part of the string (x  0) has mass  mass μ_l per unit length. The string tension is T. If a wave of unit amplitude travels along the left part of the string, as shown in the figure above, what is the amplitude of the wave that is transmitted to the right part of the string?

A. 1
B. 

C. 

D. 

E. 0

(GR9677 #80)

Solution:

A. FALSE
Amplitude = 1, if  μ_l  = μ_r  but from the picture  μ_l ≠ μ_r
Since μ_l  μ_r suggests we expect  the transmitted amplitude will be less than 1

B. FALSE
The answer suggests that the transmitted amplitude will be bigger than 1, it should be less than 1
Let μ_l = 1, μ_r = 4
A = 2 / (1 + √¼) = 2/(3/2) = 4/3

C. TRUE
The answer suggests that the transmitted amplitude will be less than 1
Let μ_l = 1, μ_r = 4
A = 2(√¼) / (1 + √¼) = 1/(3/2) = 2/3

D. FALSE
The answer suggests that A = 0, if  μ_l  = μ_r

E. FALSE
Amplitude = 0 if  μ_r → ∞

Answer: C 

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