A student plucks a fixed-end string, creating a standing wave with 6.00 nodes (including any nodes at the ends). The string is tuned such that waves on the string travel 360m/s.
How many wavelengths are on the string?
What is the wavelength of the waves on the string?
What is the frequency of the waves on the string?

1) 2.5 wavelengths

2) 0.208 m

3) 1731 Hz

Explanation:

1)

Standing waves are waves that do not propagate, but instead the particles of the medium just oscillate around a fixed position. Examples of standing waves are the waves produced on a string with fixed ends.

The points of a standing wave in which the amplitude of the oscillation is always zero are called nodes.

The two fixed ends of the string are two nodes. In this problem, we have a total of 6 nodes along the string: this means that there are 4 additional nodes apart from the two ends of the string.

Therefore, this also means that the string oscillate in 5 different segments.

One wavelength is equal to 2 segments of the oscillation: therefore, since here there are 5 segments, this means that the number of wavelengths that we have in this string is

2)

The wavelength of a wave is the distance between two consecutive crests (or throughs) of the wave.

The wavelength of a standing wave can be also measured as the distance between the nth-node and the (n+2)-th node: so, basically, the wavelength in a standing wave is twice the distance between two nodes:

where

is the wavelength

d is the distance between two nodes

Here the length of the string is

L = 0.520 m

And since it oscillates in 5 segments, the  distance between two nodes is

And therefore, the wavelength is

3)

The frequency of a wave is the number of complete oscillations of the wave per second.

The frequency of a wave is related to its speed and wavelength by the wave equation:

where

v is the speed

f is the frequency

is the wavelength

In this problem:

v = 360 m/s is the speed of the wave

is the wavelength

Therefore, the frequency is

force required to detach it is given by

charges are taken to be fundamental charge

now we have

so above is the minimum distance at which it will detach