A transverse displacement wave is traveling on a string. It is described by y(x,

Question

A transverse displacement wave is traveling on a string. It is described by y(x, t) = Asin(kx – wt + 3p/4). Nearby, a thin, uniform rod of mass m is hung from one end and allowed to oscillate (through small angles only) about that end as a physical pendulum. Find expressions (using only known values) for each of the following:

a. Find the minimum straight-line distance between two points on the string whose acceleration magnitudes are each 2/3 of their maximum magnitudes (at the same moment).

b. If the pendulum has a frequency (Hz.) that is 1/100th of the wave’s frequency, find the length of the rod.

Assume the following known values: A, k, w, m, g

Explanation / Answer

y(x,t) = A sin (kx - wt + 3pi/4)

a) lets take t = 0

a = d^2 y(x,t)/ dt = - A w^2 sin(kx - wt + 3pi/4)

a_max = Aw^2

lets take t = 0

2 Aw^2 /3 = Aw^2 sin(kx + 3pi/4)

2/3 = sin(kx + 3pi/4)

when x = -3.086/k , -1.626/k , 0.0557/k, 1.515/k

deltax = 1.46/k or 1.68/k

so minimum distance = 1.46/k

b) angular frequency of pendulum: w' = sqrt[3g/2L]

and w' = w/100

sqrt[3g/2L] = w/100

3g / 2L = w^2 / 10^4

L = 15000g / w^2 ......Ans

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