How Do You Calculate Maximum Displacement and Speed in a Standing Wave?

[xsc614]

Homework Statement

The wave function for a standing wave on a string is described by y(x, t) = 0.022 sin 4πx cos 54πt, where y and x are in meters and t is in seconds. Determine the maximum displacement and maximum speed of a point on the string at the following positions.

x=0.1m, 0.25m, 0.3m, 0.5m
Find Ymax and Vmax at these points on the string.

Homework Equations

A_n(x)=A_nsink_nx (Amplitude of a string vibrating in its nth node)
Wave equation for a standing wave in the nth harmonic motion: y_n(x,t) = A_nsin(k_nx)cos(w_nt+δ_n)
Kind of unsure on the equations for this problem.

The Attempt at a Solution

I assumed the maximum displacement would occur at t=0 when the cosine part of the equation equalled one. I plugged in the x values for the different points and solved for y, but did not get the correct answer. Read the entire chapter of the book and it is pretty vague and gives no example problems on calculating max displacement and max velocity like this

 

[diazona]

What you explained makes sense to me... can you show your work including the numbers?

 

[xsc614]

y(x, t) = 0.022 sin 4πx cos 54π

y(x, t) = 0.022 sin 4πx (1)

y(x, t) = 0.022 sin (4π(.1m)) (1)

y(x, t) = .021 m

Answer came back as incorrect. That π character is pi if you can't tell.

 

[xsc614]

Update: I used this method for all other points of (x) and got the right answers. Maybe webassign is just not accepting the right amount of sig figs or something...

I got 0 for x=0.25m and 0.5m for max displacement, so the velo's were 0 m/s... but since cos(54pi*t) =1, how do you solve for t to get the velocities for the other values of x?

 

[diazona]

From what I've heard, Webassign checks to see if you're within a certain percentage of the right answer, but it shouldn't mark you incorrect for having the wrong number of significant figures. You could try putting in one more decimal place and see if it helps.

For the velocity part of it: you have an equation for position. In general, when you have the position, how do you find velocity from it?