Find the maximum transverse tension in a rope supporting a travelling wave

[knowlewj01]

Homework Statement

The tansverse displacemrnt of a rope, is given by a function of x and t ( in m and sec) by:

y(x,t) = 0.04 sin(0.21x - 8t)

if the rope is 7m long and has a mass 0.4kg
Find:
a. the tension in the rope
b. the maximum transverse component of the tension

Homework Equations

$$y=Asin(kx-\omegat)$$

The Attempt at a Solution

part a.

we know that:

$$v = \frac{\omega}{k} = \sqrt{\frac{F}{\mu}}$$
and $$\mu = \frac{0.4}{7}=0.057$$

so $$Tension = F = 82.93N$$

i have checked this, and this is correct so far.

part b.

find the maximum transverse tension.

i think you have to find the se3cond time drivative of the given function and maximise it to find the acceleration then use F=ma, this is how far i got:

second time derivative = a

$$a = - 0.04 \times 8^2 sin( 0.21x - 8t)$$

so acceleration is maximum when periodtic part sin is +/- 1

so $$a(max) = 0.04 \times 8^2 = 2.56 m/s^2$$

i think using f=ma with this acceleration should give the transvese component of tension but I'm unsure as to what m is, any ideas?

 

[aim1732]

You should rather focus on physical implications of what max. acc. would mean. It is that the particle on the rope is at it's max. height which is the amplitude. Tension at this point would be 2T*sin theta where theta would be the the angle subtended by the curve of the wave. That can be approximated to tan theta which would be dy/dx(partial actually).
For understanding how, you should refer to how they calculate the velocity of the wave on the string.