Answer to Question #281720 in Physics for Kar

Question #281720

3. A wave is represented by the equation y(x,t) = 1.2(cm) cos [4π(rad/m)x – 200(rad/s)t].


Find


(a) The transverse displacement of a particle of the medium at x = 2 m and at t = 2 s.


(b) The transverse velocity of the particle of the medium at the same position and time.


(c) The transverse acceleration at the same position and time.


(d) The propagation velocity of the wave.

1
Expert's answer
2021-12-21T12:16:10-0500

Given:

y(x,t)=1.2(cm)cos[4π(rad/m)x200(rad/s)t]y(x,t) = 1.2(cm) \cos [4π(rad/m)x – 200(rad/s)t]


The general equation of wave is given by

y(x,t)=y0cos[kxωt].y(x,t) = y_0 \cos [kx – \omega t].

(a) The transverse displacement of a particle of the medium at x = 2 m and at t = 2 s.


y(2,2)=1.2(cm)cos[4π22002]=0.63cmy(2,2) = 1.2(cm) \cos [4π*2 – 200*2]=-0.63\:\rm cm

(b) The transverse velocity of the particle of the medium at the same position and time


vy=y=240(cm/s)sin[4πx200t]v_y=y'=240(cm/s)\sin [4πx – 200t]vy(2,2)=240(cm/s)sin[4π22002]=204cm/sv_y(2,2)=240(cm/s)\sin [4π*2 – 200*2]\\=204\: cm/s

(c) The transverse acceleration at the same position and time

ay=vy=48000(cm/s2)cos[4πx200t]a_y=v_y'=-48000(cm/s^2)\cos [4πx – 200t]

ay(2,2)=48000(cm/s2)cos[4π22002]=25214cm/s2a_y(2,2)=-48000(cm/s^2)\cos [4π*2 – 200*2]\\=25214\: cm/s^2

(d) The propagation velocity of the wave

v=ω/k=200/4π=15.9m/sv=\omega/k=200/4\pi=15.9\: m/s


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