The G string on a guitar is 69 cm long and has a fundamental frequency of 196 Hz. A guitarist can play different notes by pushing the string against various frets, which changes the string's length. The third fret from the neck gives B♭ (233.08 Hz); the fourth fret gives B (246.94 Hz).
How far apart are the third and fourth frets?
Express your answer to two significant figures and include the appropriate units.
Δd = 23 cm
When the eta string on the guitar has nodes at its ends, so the waves produced give rise to a standing wave, which can be described by the following expressions
Fundamental L = ½ λ
1st harmonic L = 2 ( λ / 2)
2nd harmonic L = 3 ( λ / 2)
Harmonic n L = n λ / 2
Where n is an integer
The speed of the rope is given by the ratio
v = λ f
This speed is constant since it depends on the tension and the linear density of the rope
Let's calculate the speed with the initial data
v = 0.69 196
v = 135.24 m / s
Let's look for the wavelength for the two frequencies
λ₁ = v / f₁
λ₁ = 135.24 / 233.08
λ₁ = 0.58022 m
λ₂ = v / f₂
λ₂ = 135.24 / 246.94
λ₂ = 0.54766 m
Let's replace in the resonance equation
Lₙ = n λ/2
For the third fret
m = 3
L₃ = 3 0.58022 / 2
L₃ = 0.87033 m
For the fourth fret
m = 4
L₄ = 4 0.54766 / 2
L₄ = 1.09532 m
The distance between the two frets is
Δd = L₄ –L₃
Δd = 1.09532 - 0.87033
Δd = 0.22499 m
Δd = 22.5 cm = 23 cm
b. The fret should be placed 25 cm far from the bridge.
c. So, the fraction of string at which the fret is placed is .
We are given,
The function representing the distance of a fret from the bridge by ,
where r = length of the root note string and n = number of notes higher than root note.
Now, Louis want to produce notes on a 50 com string. This gives r = 50.
1. It is required to produce notes which are 1 octave ( 12 notes ) higher than the root note. This gives that n = 12.
So, we get, r = 50 and n = 12 which gives us the function as,
b. Thus, the fret should be placed 25 cm far from the bridge.
Now, as the fret is placed 25 cm far on the string having length 50 cm.
c. So, the fraction of string at which the fret is placed is i.e. .
frets divide the neck into fixed segments at intervals related to a musical framework. On instruments such as guitars, each fret represents one semitone in the standard western system, in which one octave is divided into twelve semitones.
when jerome paddle in flowing river towards east the net speed of jerome will increase
so the net speed of jerome will be given as
now the speed of paul will be same as it is given as he is running on the ground
so here we can say since the speed of jerome is more so he will win the race
now as we know the distance so the time difference is given as
so jerome will reach first and the time difference will be 203.55 s or 3.4 min