A long string carries a wave; a 5.40-m segment of the string contains three complete wavelengths and has a mass of 180 g. The string vibrates sinusoidally with a frequency of 45.0 Hz and a peak-to-valley displacement of 19.0 cm.
how do you go about writing a formula in the form of A sin(kx+wt)??
Although you don't say so, the formula you are trying to fit indicates a traveling wave, not a standiung wave.
I don't see why you need the mass of the string. It does have a role in determining the wave speed, but so does the tension, which you do not provide.
Your wavelength is 5.4/3 = 1.8 m
Your angular frequency is
w = 2 pi f = 282.74 rad/s
Your amplitude is A = 9.5 cm
k = 2 pi/(wavelength) = 3.49 m^-1
the mass is important for part b. they must have just not included it. but it asks for the power in part b. you need the mass to find the inertial factor(mass/length)
To write the formula for the wave in the form of A sin(kx + wt), where A is the amplitude, k is the wave number, x is the position, w is the angular frequency, and t is the time, we need to determine the values of these variables based on the given information.
1. Amplitude (A): The peak-to-valley displacement of the wave is given as 19.0 cm. The amplitude (A) is equal to half of this value since the wave oscillates symmetrically around the equilibrium position. Therefore, A = 19.0 cm / 2 = 9.5 cm.
2. Wave number (k): The wave number (k) is related to the wavelength (λ) by the formula k = 2π / λ. We are given that the segment of the string contains three complete wavelengths (λ), and the length of the segment is 5.40 m. Thus, the wavelength can be calculated as λ = 5.40 m / 3.
To find the wave number (k), we substitute this value of wavelength into the formula:
k = 2π / (5.40 m / 3) = 3π / 5.4 m.
3. Angular frequency (w): The angular frequency (w) is related to the frequency (f) by the formula w = 2πf. The frequency is given as 45.0 Hz. Therefore,
w = 2π × 45.0 Hz = 90π Hz.
Now that we have the values for amplitude (A), wave number (k), and angular frequency (w), we can write the formula for the wave as:
y(x, t) = A sin(kx + wt).
Substituting the values we calculated:
y(x, t) = 9.5 cm sin((3π / 5.4 m) x + 90π Hz t).
Note: Remember to convert the units of wavelength, amplitude, and position to be consistent with each other.