Solutions to Sat Physics subject questions on waves with detailed explanations.

How long does it take a wave of frequency 0.2 Hz and wavelength 2 m to travel along a rope of length 4 m?
A) 2 s
B) 8 s
C) 0.8 s
D) 0.4 s
E) 10 s
Solution  Explanations
Let the wavelength be λ, frequency f and wave velocity v
λ = v T = v / f
v = f λ = 0.2 (2) = 0.4 m/s
time = length / velocity = 4 m / 0.4 m/s = 10 s

What is the frequency of a pendulum that swings at the rate of 45 cycles per minute.
A) 0.75 Hz
B) 1.3 Hz
C) 45 Hz
D) 2700 Hz
E) 60 Hz
Solution  Explanations
1 minute = 60 seconds
frequency = number of cycles / total time = 45 / 60 = 0.75 Hz

What is the wavelength of a wave of velocity 10 m/s and frequency 200 Hz?
A) 2000 m
B) 20 cm
C) 0.05 m
D) 0.05 cm
E) 5 mm
Solution  Explanations
wavelength = λ, velocity = v = 10 , frequency = f = 200
λ = v / f = 10 / 200 = 0.05 m

A standing wave is produced along a string of 100 cm whose ends are fixed. What is the wavelength of the wave if there are 3 nodes between the fixed ends of the string?
A) 300 cm
B) 40 cm
C) 20 cm
D) 50 cm
E) 33.3 cm
Solution  Explanations
n = total number of nodes = 3 + 2 (fixed ends do not move and are counted as nodes)
wavelength = λ length of string = L = 100 cm
L = (5  1) (λ / 2)
Hence λ = 2 L / 4 = 50 cm

A train arriving at a station, at 40 km/hr, whistles at a frequency of 600 Hz. What frequency will you hear if you were at that station?(speed of sound in air is 340 m/s)
A) 537
B) 680 Hz
C) 340
D) 600
E) 720
Solution  Explanations
Since the train is moving, the frequency that you will hear at the station will change due to the Doppler effect.
let fs = frequency of the whistle = 600 Hz, Vs = speed of sound in air = 340 m/s, and Vt = speed of the train = 40 m/s.
The frequency that will be heard = (Vs / (Vs  Vt)) fs = ( 340 / (340  40))600 = 680 Hz

The frequencies recorded at a fixed point of the sound of an ambulance sirene are f1 while the ambulance is approaching the fixed point and f2 while it is moving away from the same point. If Vs is the speed of sound in air, which of the formulas below may be used to find the speed Va of the ambulance?
A) Va = Vs ( f1 / (f1 + f2) )
B) Va = Vs (f1 + f2) / (f1  f2)
C) Va = Vs (f1 + f2) / (f1 + f2)
D) Va = Vs (f1  f2) / (f1 + f2)
E) Va = Vs (f1/(f1  f2))
Solution  Explanations
Let f0 be the frequency of the siren at the source and Vs the sound speed in the air. Due to Doppler effects, the frequencies f1 and f2 are given by
f1 = f0 * Vs / (Vs  Va) (ambulance approaching fixed point) (1)
f2 = f0 * Vs / (Vs + Va) (ambulance moving away from a fixed point) (2)
equation (1) gives: Vs f0 = f1 (Vs  Va)
equation (2) gives: Vs f0 = f2 (Vs + Va)
Combining the above, we can write
f1 (Vs  Va) = f2 (Vs + Va)
expand: f1 Vs  f1 Va = f2 Vs + f2 Va
solve for Va: Va = Vs (f1  f2) / (f1 + f2)

A string of length L is stretched out with a tension T. What happens to the velocity of the wave traveling down the string if the tension is quadrupled?
A) The velocity is multiplied by 2
B) The velocity is multiplied by 1 / 2
C) The velocity is multiplied by 4
D) The velocity is multiplied by 1 / 4
E) The velocity does not change
Solution  Explanations
Let v be the velocity for tension T and μ be the linear mass density.
The velocity v is given by : v = √(T/μ)
Let us quadruple T and find the new velocity: v2 = √(4 T/μ) = 2 √(T/μ) = 2 v
The velocity doubles

Which of the following statements is NOT correct?
A) Sound traveling through air is an example of a longitudinal wave.
B) Water waves may be considered as longitudinal and transverse waves
C) In a longitudinal wave, particles move in a direction parallel to the motion of the wave
D) In a transverse wave, particles move in a direction perpendicular to the motion of the wave
E) Electromagnetic waves cannot propagate in a vacuum (empty space)
Solution  Explanations
All statements are correct except E). Electromagnetic waves propagate in a vacuum.

If the square of the period of a simple pendulum is plotted against the length of the pendulum, the graph obtained is
A) a horizontal line
B) a vertical line
C) a parabola
D) a line through the origin with a positive slope
E) a line through the origin with a negative slope
Solution  Explanations
If L is the length of the pendulum, the formula for the period of the pendulum is given by.
T = 2π √(L/g) , g = 10 m/s^{2}
square both sides: T^{ 2} = 4 π^{2} L / g
T^{ 2} varies linearly with L and the graph of T^{ 2} as a function of L is a line through the origin with a positive slope.

A string, with fixed ends, of length 3 m vibrates as the third harmonic with 60 complete vibrations in 10 seconds. Find the speed of this wave.
A) 1 / 6 m/s
B) 12 m/s
C) 6 m / s
D) 18 m / s
E) 4 m / s
Solution  Explanations
There is a total of 4 nodes for the third harmonic. If L (= 3 m) is the length of the string, the relationship between L and the wavelength λ is given by
L = (4  1)( λ / 2)
which gives λ = 2 L / 3 = 6 / 3 = 2 m
T = 10 / 60 = 1 / 6 s
λ = v T which gives v = λ / T = 2 / (1 / 6) = 12 m/s