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You are holding one end of an elastic cord that is fastened to a wall 3.5 m away. You begin shaking the end of the cord at 5 Hz, creating a continuous sinusoidal wave of wavelength 1.0 m. How much time will pass until a standing wave fills the entire length of the string?

The question aims to find the time it takes for a wave generated in a string tied to a wall to have a standing wave.

The question depends on the concepts of waves generated in a string tied to a stationary object. A standing wave is generated when two waves with the same amplitude and wavelength have interference and move in opposite directions. A rope tied to a wall or a stationary rigid object will generate standing waves.

The waves generated in a string are called transverse waves. Transverse waves have the wave direction perpendicular to the oscillations of the string/rope. The velocity or speed of the wave oscillating in a string is given as:

\[ v = \lambda f \]

Also, frequency is given as:

\[ f = \dfrac{ 1 }{ T } \]

It also depends on the equation of motion as we need to calculate the time it takes a standing wave to fill the entire length of the cord. The equation for time is given as:

\[ t = \dfrac{ s }{ v } \]

Expert Answer

The given information about the problem is given as follows:

\[ Frequency\ of\ the\ Wave\ f = 5\ Hz \]

\[ Length\ of\ the\ String\ L = 3.5\ m \]

\[ Wavelength\ \lambda = 1\ m \]

The velocity of the wave in the string can be calculated by the formula, which is given as:

\[ v = f \lambda \]

Substituting the values, we get:

\[ v = 5 \times 1 \]

\[ v = 5\ m/s \]

The time that the wave will take to reach from one end to the other end is given by the equation of motion as:

\[ t’ = \dfrac{ L }{ v } \]

\[ t’ = \dfrac{ 3.5 }{ 5 } \]

\[ t’ = 0.7\ s \]

The total time taken by the standing wave to fill the entire length of the cord is given as:

\[ t = 2 \times t’ \]

\[ t = 2 \times 0.7 \]

\[ t = 1.4\ s \]

Numerical Result

The total time taken by the standing wave to fill the entire length of the cord is calculated to be:

\[ t = 1.4\ s \]

Example

A rope is tied to a steel block and is shaken from the other end. The length of the rope is 10m, and the wavelength of the wave generated is 1.5m. The frequency of the waves generated is 10 Hz. Find the time taken by the wave to reach from hand to the steel block.

The information given in the problem is as follows:

\[ Frequency\ of\ the\ Wave\ f = 10\ Hz \]

\[ Length\ of\ the\ String\ L = 10\ m \]

\[ Wavelength\ \lambda = 1.5\ m \]

The velocity of the wave in the string can be calculated by the formula, which is given as:

\[ v = f \lambda \]

Substituting the values, we get:

\[ v = 10 \times 1.5 \]

\[ v = 15\ m/s \]

The time that the wave will take to reach from one end to the other end is given by the equation of motion as:

\[ t = \dfrac{ L }{ v } \]

\[ t = \dfrac{ 10 }{ 15 } \]

\[ t = 0.67\ s \]

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