Find the slope of the string

Question

A certain string has a linear mass density of 0.25 kg/m and is stretched with a tension of 25 N. One end is given a sinusoidal motion with frequency 5 Hz and amplitude 0.01m. At time t=0, the end has zero displacement and is moving in the positive y-direction. (v=10 m/s, A=0.01m, ω=31.4 rad/s, T=0.2 s, λ=2.0 m, k=3.1 4/m).

 

Find the slope of the string at the point x=0.25m at time t=0.1s.

Answer 1

Hi Dalia,

 

I gave this question a shot, hopefully it'll be a good start for you.

 

The first thing I did was define an equation for the string. It's sinusoidal in space and time, so it should be something of the form: g(x,t) = Acos(wt-kx+phi). Defining the end of the string with initial motion as x=0, and having the motion move to the right can be defined as +x. Also, knowing that g(0,0) = 0 describes sin(), and initial motion in +y means no phase shift. Thus, the equation can be adjusted to: g(x,t) = Asin(wt-kx) for convenience.

 

These variables are given:

rho = 0.25 kg/m

F = 25 N

f = 5 Hz

A = 0.01 m

 

Using those, I filled in the remaining variables (these match your list):

w = 2pi*f = 31.4 rad/s

v = sqrt(F/rho) = 10 m/s

k = 2pi/lambda = 2pi*f/v = 3.14 /m

 

And to be more clear, I defined the variables like this: rho is the linear mass density, F is the tension force, f is the frequency, A is the amplitude, w is the angular frequency, v is the velocity, and k is the wave number.

 

After getting the equation to fully describe the string motion, I took the derivative with x to find slope (assuming "slope" means tangent in space, not time):

 

∂g(x,t)/∂x = -Ak*cos(wt-kx)

 

And plugging in all the variables gives:

 

∂g(x=0.25m,t=0.1s)/∂x = -(0.01m)(pi /m)*cos((10pi rad/s)(0.1s)-(pi /m)(0.25m))

= -0.01pi * cos(3/4*pi)

=  sqrt(2)/200 * pi

 

I hope that helps! It'd be great if you could double-check this before submitting, though, if it's homework. I can't guarantee it's right for what you're learning, and I'd rather not give you wrong info. Just a heads up :)