Q: COMPREHENSION:

A sinusoidal wave is propagating in negative x-direction in a string stretched along x-axis. A particle of string

at x = 2 m is found at its mean position and it is moving in positive y direction at t = 1 sec. If the amplitude

of the wave, the wavelength and the angular frequency of the wave are 0.1 meter, π/4 meter and 4π rad/sec

respectively.

1 .The equation of the wave is

(A) y = 0.1 sin (4πt – 1)+ 8(x – 2))

(B) y = 0.1 sin (t-1)- (x – 2))

(C) y = 0.1 sin (4π(t -1)-8(x – 2))

(D) none of these

2. The speed of particle at x = 2 m and t = 1 sec is

(A) 0.2π m/s

(B) 0.6π m/s

(C) 0.4π m/s

(D) 0

3. The instantaneous power transfer through x = 2 m and t = 1.125 sec, is

(A) 10 J/s

(B) 4π/3 J/s

(C) 2π/3 J/s

(D) zero

Ans: 1.(A) ; 2. (C) ; 3.(D)

Solution: 1 . The equation of wave moving in negative x-direction, assuming origin of position at x = 2 and origin of time

(i.e. initial time) at t = 1 sec. y = 0.1 sin (4πt + 8x)

Shifting the origin of position to left by 2m, that is, to

x = 0. Also shifting the origin of time backwards by 1 sec,

that is to t = 0 sec. y = 0.1 sin [(4πt + 8(x – 2)]

2. As given the particle at x = 2 is at mean position at t = 1 sec. its velocity v = ωA = 4π × 0.1 = 0.4 π m/s.

3. Time period of oscillation T = 2π/ω sec. Hence at t = 1.125 sec, that is, at T/4 seconds after t = 1 second, the particle is at rest at extreme position. Hence instantaneous power at x = 2 at t = 1.125 sec is zero.