Question

A rope of length l is clamped at both ends. which one of the following is not a possible wavelength for standing waves on this rope?

a) l/2
b) 2l/3
c) l
d) 2l
e) 4l g

Answer 1

The correct answer is option (b)
`\( (2l)/(3) \)`, as it does not correspond to a whole number of half-wavelengths within the length of the rope.

The possible wavelengths for standing waves on a rope clamped at both ends must satisfy the condition that the wavelength (λ) corresponds to a whole number of half-wavelengths within the length of the rope (l). Mathematically, this is expressed as:

`\[ \lambda = (2l)/(n) \]`

where
`\( n \)` is a positive integer (1, 2, 3, ...).

Let's evaluate the given options:

a)
`\( (l)/(2) \):` This is valid for
`\( n = 1 \)`, so it is a possible wavelength.

b)
`\( (2l)/(3) \):` This is valid for
`\( n = (3)/(2) \),` which is not a positive integer. Therefore, option (b) is not a possible wavelength for standing waves on the rope.

c)
`\( l \)` : This is valid for
`\( n = 2 \),` so it is a possible wavelength.

d)
`\( 2l \):` This is valid for
`\( n = 1 \)` , so it is a possible wavelength.

e)
`\( 4l \)` : This is valid for
`\( n = (1)/(2) \)`, which is not a positive integer. Therefore, option (e) is not a possible wavelength for standing waves on the rope.

Therefore, the correct answer is option (b) \( \frac{2l}{3} \), as it does not correspond to a whole number of half-wavelengths within the length of the rope.