Answer to Question #281720 in Physics for Kar

Question

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.

Answer 1

Given:

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

The general equation of wave is given by

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

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

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

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

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

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

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

$$a_y(2,2)=-48000(cm/s^2)\cos [4π*2 – 200*2]\\=25214\: cm/s^2$$

(d) The propagation velocity of the wave

$$v=\omega/k=200/4\pi=15.9\: m/s$$