Question

A 1.30-m string of weight 0.0121N is tied to the ceiling at its upper end, and the lower end supports a weight W. Neglect the very small variation in tension along the length of the string that is produced by the weight of the string. When you pluck the string slightly, the waves traveling up the string obey the equation
y(x,t)=(8.50mm)cos(172rad?m?1x?2730rad?s?1t)
Assume that the tension of the string is constant and equal to W.

 

How much time does it take a pulse to travel the full length of the string?

 

What is the weight W?

 

How many wavelengths are on the string at any instant of time?

 

What is the equation for waves traveling down the string?

a) y(x,t)=(8.50mm)cos(172rad?m?1x?2730rad?s?1t)

b) y(x,t)=(8.50mm)cos(172rad?m?1x+2730rad?s?1t)

c) y(x,t)=(10.5mm)cos(172rad?m?1x+2730rad?s?1t)

d) y(x,t)=(10.5mm)cos(172rad?m?1x?2730rad?s?1t)

 

 

Answer 1

Concepts and reason

The concepts used to solve this problem are progressive waves in the string.

First use the relation between angular speed and wave number to calculate velocity.

Then use the velocity and length of the string to calculate the time taken by the wave.

Then use the relation between velocity of transverse wave, mass, and length of the string to calculate the weight of the support.

Then use the relation velocity of transverse wave to calculate the wavelength and use the wavelength and length of the string to calculate the number of the wavelength are in the string.

Finally use the general equation of the wave travelling in the string to find the equation of the wave travelling down in the string.

Fundamentals

Expression for the wave travelling up in the string is,

Here, is the displacement along position of distance and time, A is the amplitude, x is the distance, k is the wave number, is the angular speed, and t is the time.

Expression for the velocity is,

Here, v is the velocity.

Expression for the time taken by the wave travelling is,

Here, l is the length of the string.

Substitute for v.

Tension in the string is equal to the weight of the support.

Here, T is the tension in the string and W is the weight of the support.

Expression for the weight of the string is,

Here, w is the weight of the string and g is the acceleration due to gravity.

Rearrange the equation to get the mass of the string,

Expression for the mass density is,

Here, m is the mass of the string and is the mass density.

Substitute for m.

Expression for the velocity of transverse wave is,

Substitute W for T and for .

Rearrange the above equation to get weight of the support,

Expression for the wave length is,

Here, is the wavelength.

Expression for the number of the wavelength in the string is,

Expression for the wave travelling down in the string is,

(1)

General expression for the wave travelling up in the string is,

Given expression for the wave travelling up in the string is,

Expression for the time taken by the wave travelling is,

Substitute for , for , and for .

(2)

Expression for the velocity is,

Substitute for , and for .

Expression for the weight of the support is,

Substitute for , for , for g, and for

(3)

Expression for the wave length is,

Substitute for .

Expression of number of the wavelength are in the string is,

Substitute for and for .

(4)

The incorrect options are,

(a).

(c).

(d).

Because, is the equation for the wave travelling up in the string and the other two has different amplitude

So the correct option is

Because it supports the general equation .

Ans: Part 1

Time taken by the pulse to travel full length of the string is .

Part 2

The weight of the support is .

Part 3

The number of the wavelength are in the string is 36.

Part 4

The equation for the wave travelling down in the string is,

.