A uniform heavy rope hangs freely from a support. the support is made to oscillate horizontally with a small amplitude so as to send transverse waves down the rope.A standing wave is set up as these waves are reflected at the free end.Which of the following statements is / are correct?

1. the tension is the same throught the rope

2the wavelength is the same throughout the rope

3the frequency of oscillation is the same at all vibrating points of the rope

(please give your explanation of your ans)

If it is hanging freely the tension is highest at the top where the entire weight of the rope is on the hook. The tension at the bottom is zero.

The speed of the wave increases with the square root of tension, and is therefore highest at the top. Therefore the wavelength is short at the top for a given frequency (speed*period = length.
The frequency is the same up and down the rope. There is no way to store waves, the same number per unit time must occur at each point on the rope.

To determine which of the statements are correct, let's analyze each one:

1. The tension is the same throughout the rope.
This statement is not correct. As the transverse waves travel down the rope, they experience partial reflection at the free end, creating a stationary or standing wave. In a standing wave, there are regions of maximum (nodes) and minimum (antinodes) displacement. At the nodes, the rope does not oscillate, and therefore, the tension is zero. At the antinodes, the rope experiences maximum displacement and, consequently, maximum tension. Thus, the tension varies along the rope.

2. The wavelength is the same throughout the rope.
This statement is correct. In a standing wave, the wavelength is determined by the length of the rope. The standing wave is formed due to the superposition of the incident and reflected waves, resulting in constructive and destructive interference. The distance between consecutive nodes or antinodes represents half a wavelength, and this value remains constant along the entire length of the rope.

3. The frequency of oscillation is the same at all vibrating points of the rope.
This statement is correct. In a standing wave, the frequency is determined by the frequency of oscillation of the source, in this case, the support. As the waves travel down the rope and reflect, they transmit the same frequency from one end to the other. Hence, the frequency of oscillation remains constant at all vibrating points of the rope.

In summary, statement 2 and statement 3 are correct, while statement 1 is incorrect.