Transverse Wave on a String: Speed & Acc., Wavelength, Period, Prop. Speed

[Husker70]

Homework Statement

A transverse wave on a string is described by the wave function
y = (.120m) sin(pie/8 + 4pie(t))

(a) Determine the transverse speed and acceleration of an element of the
string at t=0.200s for the point on the string located at 1.60m
(b) What are the wavelength, period, and speed of propagation of this wave?

Homework Equations

Part (a)
Velocity = -wA cos(kx-wt)
Acceleration = -w^2A sin(kx-wt)

Part B
Wavelength = k = 2pie/lambda
Period = 1/f
Speed = f(lambda)
w = 2pief
f = w/2pie = 2.0 rad/s

The Attempt at a Solution

Part (a)
V = -(4pie) (.120m)cos(pie/8 + 2.51) I don't know what to do from here
A = -(4pie)^2 (.120m) sin(pie/8 + 2.51) I don't know what to do from here

Part (b)
Wavelength = pie/8 = 2pie/lambda = 16m
Period = 1/2.0 s^-1 = .5s
Speed f(lambda) = (2.0s^-1)(16m) = 32m/s

If anyone can help I would appreciate it.
Thanks,
Kevin

 

[Redbelly98]

(.120m) sin(pie/8 + 4pie(t))

Hi,

Is that supposed to be
(.120m) sin((x)pie/8 + 4pie(t))
?
(An "x" is needed for this to be a wave.)

V = -(4pie) (.120m)cos(pie/8 + 2.51) I don't know what to do from here
A = -(4pie)^2 (.120m) sin(pie/8 + 2.51) I don't know what to do from here

Just plug the numbers into your calculator, using radian mode for the sine calculation. And again, please check if it should really be
(x)pie/8

Wavelength = pie/8 = 2pie/lambda = 16m

You got the right answer. However, what you wrote here is bad "math grammar". It says (among other things) that
Wavelength = pie/8
which is wrong. pie/8 is actually "k", which as you know is different than wavelength.

A better way to write this out would be something like this:

Wavelength:
k = pi/8 m^-1 = 2pi/lambda
lambda = 2pi / (pi/8 m^-1) = 16 mYour period and speed calculations look correct.

 

[thomate1]

Hi Kevin,

For part (a), you have correctly identified the equations for velocity and acceleration. To find the values at t=0.200s for the point at 1.60m, we need to plug in these values into the equations.

For velocity, we have:
V = -(4pie)(.120m)cos(pie/8 + 4pie(0.200s))
= -(4pie)(.120m)cos(1.57 + 2.51)
= -(4pie)(.120m)cos(4.08)
= -(4pie)(.120m)(-0.716)
= 0.344m/s

For acceleration, we have:
A = -(4pie)^2(.120m)sin(pie/8 + 4pie(0.200s))
= -(4pie)^2(.120m)sin(1.57 + 2.51)
= -(4pie)^2(.120m)sin(4.08)
= -(4pie)^2(.120m)(0.698)
= -0.357m/s^2

For part (b), you have correctly identified the equations for wavelength, period, and speed. However, your value for frequency (f) is incorrect. It should be 4pie rad/s, since the coefficient of t is 4pie in the wave function. Plugging in this value, we get:

Wavelength = 2pie/lambda = 4pie rad/s = 2pie/lambda
lambda = 8m

Period = 1/f = 1/4pie s = 0.159s

Speed = f(lambda) = (4pie rad/s)(8m) = 32m/s

Hope this helps! Keep up the good work in your studies.

Best,