Amplitude at various points on a string

[forestmine]

Homework Statement

A string with both ends held fixed is vibrating in its third harmonic. The waves have a speed of 192m/s and a frequency of 240 Hz. The amplitude of the standing wave at an antinode is .4cm.

Calculate the amplitude at points ont he string a distance of (i) 40cm, (ii) 20cm, and (iii)10 cm from the left end of the string.

Homework Equations

A*sin(kx)sin(ωt)
k=2$\pi$/λ
λ=v/f

The Attempt at a Solution

I'm assuming I need to use the equation for the position along a wave, given as A*sin(kx)sin(ωt). I'm thinking I need to set it up so that my only unknown is A. I started by solving for λ and then k. From there, I can find ω. I'm not sure, however, how to continue without knowing my time t. Also, when it says that the amplitude of the standing wave is .4cm, does that entail my maximum amplitude, and therefore all of the amplitudes I come up with should be points within this max displacement? And even if I have a given t, I still don't quite understand how to use the standing wave equation. In this form in particular, its solution entails the position, so is it simply A*sin(kx)sin(ωt)=the position at those given conditions.

Any direction would be greatly appreciated.

 

[Spinnor]

First you might come up with the proper k such that you get the wave in this link,

http://gbhsweb.glenbrook225.org/gbs/science/phys/mmedia/waves/harm3.html

I think you can assume .4cm = A, what else could it be?

I get .8m for lamda. Let sin(wt) = 1 and plug in your values of x.

Hope this helps.

Edit, if some values come back negative I'm not sure if they should be reported as positive or negative.

 

[forestmine]

I also get .8 for λ.

But I still don't understand how to compute each of the amplitudes. I'm assuming I should use Asin(kx)sin(ωt), but if I plug in the given amplitude of .4 cm, then what variable am I solving for?

Also, I don't understand why sin(ωt) should be equal to 1.

 

[Spinnor]

I also get .8 for λ.

But I still don't understand how to compute each of the amplitudes. I'm assuming I should use Asin(kx)sin(ωt), but if I plug in the given amplitude of .4 cm, then what variable am I solving for?

Also, I don't understand why sin(ωt) should be equal to 1.

They want you to find the amplitude which is the maximum value in one cycle. You plug .4 where A is, set sin(wt)=1 (you want the maximum displacement at the various values of x), and then plug in your values for lamda*x.

Hope this helps.

 

[asteriatic]

I would first clarify the given information and assumptions. Is the string assumed to be a perfect string with no mass, and are the ends of the string fixed at nodes (points of zero displacement)? Also, is the string assumed to be vibrating in the third harmonic, or is it actually vibrating in the third mode of vibration?

Assuming that the string is a perfect string with no mass and the ends are fixed at nodes, we can use the equation for the amplitude at an antinode, which is given by A = Ansin(kx), where An is the amplitude at the antinode, k is the wave number (k = 2π/λ), and x is the distance from the left end of the string.

To find the amplitude at a point on the string that is a distance of 40cm, 20cm, or 10cm from the left end, we can use the equation A = Ansin(kx), where x is the distance from the left end and An is the amplitude at the antinode (which is given as 0.4cm). We can calculate the wave number k using the equation k = 2π/λ, where λ is the wavelength. The wavelength can be calculated using the equation λ = v/f, where v is the wave speed (given as 192m/s) and f is the frequency (given as 240Hz).

Therefore, for a distance of 40cm from the left end, we have x = 40cm, An = 0.4cm, v = 192m/s, and f = 240Hz. Plugging these values into the equations, we get:
λ = v/f = 192m/s / 240Hz = 0.8m
k = 2π/λ = 2π / 0.8m = 7.85m^-1
A = Ansin(kx) = 0.4cm * sin(7.85m^-1 * 40cm) = 0.4cm * sin(314) = 0cm (if we assume that the amplitude is measured in cm, then the amplitude at this point is essentially zero)

Similarly, we can calculate the amplitude at distances of 20cm and 10cm from the left end using the same method.

In summary, to calculate the amplitude at various points on the string, we need to use the equation A =