Question

A transverse wave is described by the function y(x,t)=2.3cos(4.7x+12t−π/2), where distance is measured in meters and time in seconds. How long does it take for the point at x = 0 to move from a displacement of 0 m to a displacement of 1.1 m?

Answer 1

Answer:

$\Delta t=4.988\ s$

Explanation:

Given:

• equation of transverse wave, $\rm y(x,t)=2.3\ cos(4.7x+12t-\frac{\pi}{2} )$

• initial position of point x=0, $y_i=0\ m$

• final position of point x=0, $y_f=1.1\ m$

Now putting initial condition in the wave equation:

$0=2.3\times cos(4.7\times 0+12t_i-\frac{\pi}{2} )$

$cos^{-1}(0)=12t_i-\frac{\pi}{2}$

encountering the first occurrence:

$\frac{\pi}{2} =12t_i-\frac{\pi}{2}$

$t_i=\frac{\pi}{12}=0.262\ s$ .........................(1)

Now putting final condition in the wave equation:

$1.1=2.3\times cos(4.7\times 0+12t_f-\frac{\pi}{2} )$

$cos^{-1}(\frac{1.1}{2.3} )=12t_f-\frac{\pi}{2}$

encountering the first occurrence:

$61.43 =12t_f-\frac{\pi}{2}$

$t_f=5.25\ s$ .........................(2)

Now time elapsed:

$\Delta t=t_f-t_i$

$\Delta t=5.25-0.262$

$\Delta t=4.988\ s$

Answer 2

Answer:

0.22 second

Explanation:

y = 2.3 Cos (4.7 x + 12 t - π/2)

at x = 0, y = 1.1 m , t = ?

Substitute the values in the given equation

1.1 = 2.3 Cos (4.7 x 0 + 12t - π/2)

Cos (12t - π/2) = 0.4783

(12t - π/2) = 1.07

12 t = 2.64

t = 0.22 second

Thus, the time taken is 0.22 second.