Answer
Verified
333.8k+ views
Hint: Make use of the formula given for the speed of a wave and find out by what factor the speed of the pulse changes when it propagates from the bottom to the top of the rope. Then use the fact that speed of a wave is directly proportional to its wavelength.
Formula used:
$v=\sqrt{\dfrac{T}{\mu }}$
Complete step by step answer:
The velocity of propagation a transfer wave or pulse in a stretched rope is given as $v=\sqrt{\dfrac{T}{\mu }}$, where T is the tension in the rope and $\mu $ is the linear mass density of the rope.In this case, it is given that the rope is uniform. Therefore, its linear mass density is constant at point on the rope. However, the rope has some mass, which means that the tension will be different at every point on the rope.
Since the tension is different, the velocity of the pulse will be different at different points.
Let the velocity of the pulse at the bottom of the rope be v and the tension in the rope at this point is ${{T}_{1}}=mg$.
$\Rightarrow v=\sqrt{\dfrac{{{T}_{1}}}{\mu }}=\sqrt{\dfrac{mg}{\mu }}$
Let the velocity of the pulse at the top of the rope be v’ and the tension in the rope at this point is ${{T}_{2}}=mg+mg=2mg$ (because it has to balance the mass and well the rope).
$\Rightarrow v'=\sqrt{\dfrac{{{T}_{2}}}{\mu }}=\sqrt{\dfrac{2mg}{\mu }}$
$\Rightarrow v'=\sqrt{2}\left( \sqrt{\dfrac{mg}{\mu }} \right)=\sqrt{2}v$.
This means that the speed of the pulse increases by a factor of $\sqrt{2}$ when it goes from the bottom to the top of the rope.The wavelength of a pulse is directly proportional to the speed of the pulse. Therefore, if the speed increases by a factor of $\sqrt{2}$ then its wavelength will also increase by a factor of $\sqrt{2}$.
It is given that the wavelength of the pulse at the bottom of the rope is $\lambda $.
Therefore, the wavelength of the pulse at the top will be $\sqrt{2}\lambda $.But it said that this wavelength is equal to $a\lambda $. Then this means that $\sqrt{2}\lambda =a\lambda $.
$\Rightarrow a=\sqrt{2}$
$\therefore {{a}^{2}}=2$
Hence, the value of ${{a}^{2}}$ is 2.
Note: When the rope is massless, the tension is constant through the rope. When the rope has some mass with uniform distribution, the tension in the rope is not the same at every point. In this the tension at a point that is at distance x from the base of the rope is equal to $T=mg+\dfrac{x}{l}mg$.
Formula used:
$v=\sqrt{\dfrac{T}{\mu }}$
Complete step by step answer:
The velocity of propagation a transfer wave or pulse in a stretched rope is given as $v=\sqrt{\dfrac{T}{\mu }}$, where T is the tension in the rope and $\mu $ is the linear mass density of the rope.In this case, it is given that the rope is uniform. Therefore, its linear mass density is constant at point on the rope. However, the rope has some mass, which means that the tension will be different at every point on the rope.
Since the tension is different, the velocity of the pulse will be different at different points.
Let the velocity of the pulse at the bottom of the rope be v and the tension in the rope at this point is ${{T}_{1}}=mg$.
$\Rightarrow v=\sqrt{\dfrac{{{T}_{1}}}{\mu }}=\sqrt{\dfrac{mg}{\mu }}$
Let the velocity of the pulse at the top of the rope be v’ and the tension in the rope at this point is ${{T}_{2}}=mg+mg=2mg$ (because it has to balance the mass and well the rope).
$\Rightarrow v'=\sqrt{\dfrac{{{T}_{2}}}{\mu }}=\sqrt{\dfrac{2mg}{\mu }}$
$\Rightarrow v'=\sqrt{2}\left( \sqrt{\dfrac{mg}{\mu }} \right)=\sqrt{2}v$.
This means that the speed of the pulse increases by a factor of $\sqrt{2}$ when it goes from the bottom to the top of the rope.The wavelength of a pulse is directly proportional to the speed of the pulse. Therefore, if the speed increases by a factor of $\sqrt{2}$ then its wavelength will also increase by a factor of $\sqrt{2}$.
It is given that the wavelength of the pulse at the bottom of the rope is $\lambda $.
Therefore, the wavelength of the pulse at the top will be $\sqrt{2}\lambda $.But it said that this wavelength is equal to $a\lambda $. Then this means that $\sqrt{2}\lambda =a\lambda $.
$\Rightarrow a=\sqrt{2}$
$\therefore {{a}^{2}}=2$
Hence, the value of ${{a}^{2}}$ is 2.
Note: When the rope is massless, the tension is constant through the rope. When the rope has some mass with uniform distribution, the tension in the rope is not the same at every point. In this the tension at a point that is at distance x from the base of the rope is equal to $T=mg+\dfrac{x}{l}mg$.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill the blanks with proper collective nouns 1 A of class 10 english CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers