Consider a standing wave formed on a string . It results due to the superposition of two waves travelling in opposite directions . The waves are travelling along the length of the string in the `x` - direction and displacements of elements on the string are along the `y` - direction . Individual equations of the two waves can be expressed as
`Y_(1) = 6 (cm) sin [ 5 (rad//cm) x - 4 ( rad//s)t]`
`Y_(2) = 6(cm) sin [ 5 (rad//cm)x + 4 (rad//s)t]` Here `x` and `y` are in `cm`.
Answer the following questions.
If one end of the string is at `x = 0` , positions of the nodes can be described as

Options

A. `x = n pi//5 cm , where n = 0 , 1, 2,…`
B. `x = n 2pi//5 cm , where n = 0 , 1, 2,…`
C. `x = n pi//5 cm , where n = 0 , 1, 3, 5,…`
D. `x = n pi//10 cm , where n = 0 , 1, 3,5,…`

Answer

A

Explanation

`y = y_(1) + y_(2) = ( 12 sin 5 x) cos 4 t`
Maximum value of `y` - position in `SHM`of an element of the string that is located at an antinode `= +- 12 cm ( sin 5 x = +- 1)`
For the position nodes amplitudeshould be zero.
So , `sin 5 x = 0 rArr 5 x = n pi`
`x = ( n pi)/(5)`
where `n = 0 , 1 ,2 , 3 ,.....`
Value of amplitude at `x = 1.8 cm`
`A = 12 sin ( 5 xx 1.8) = 4.9 cm`
At any instant say `t = 0` , instantaneous velocity of points on the string is zero for all points as at extreme position velocities of particles are zero.