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.
Figure 7.104( c) shows the standing wave pattern at `t = 0` due to superposition of waves given by `y_(1)` and `y_(2)` in Figs.7.104(a) and (b) . In Fig. 7.104 (c ) , `N` is a node and `A` and antinode . At this instant say ` t = 0` , instantaneous velocity of points on the string named as A

Options

A. is different for different points
B. is zero for all points
C. changes with position of the point
D. is constant but not equal to zero for all points

Answer

B

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.