Answer
414.6k+ views
Hint: Firstly, the displacements of the spheres should be considered. The formulae for calculating the inertia and the torque should be used. Following, the angular momentum should be conserved. By using all the equations obtained, we need to derive the expressions for the amplitude of SHM of sphere B and the frequency of oscillation of the spheres.
Formula used:
\[\begin{align}
& I=\dfrac{5}{3}M{{R}^{2}} \\
& \tau =I\alpha \\
\end{align}\]
Complete step by step answer:
Consider the sphere A with the mean displacement of \[{{x}_{1}}\] has moved towards the left. Similarly, the sphere B with the mean displacement of \[{{x}_{2}}\] has moved towards the right. Consider the below figure.
Therefore, the value of the total extension will be equal to the value given by,
Total extension = \[{{x}_{1}}+{{x}_{2}}\].
Now we need to find the expression for the torque in terms of the momentum. Thus, we have,
The torque of A is given as follows.
\[{{\tau }_{A}}={{I}_{A}}{{\alpha }_{A}}\]
Now substitute the expressions of the torque and moment of inertia of a hollow sphere in the above equation. Therefore, we get,
\[K({{x}_{1}}+{{x}_{2}})r=\left( \dfrac{5}{3}m{{r}^{2}} \right){{\alpha }_{A}}\]
Cancel out the common terms
\[K({{x}_{1}}+{{x}_{2}})=\dfrac{5}{3}m(r{{\alpha }_{A}})\]
Where \[{{\alpha }_{A}}\] is the angular acceleration, thus, \[r{{\alpha }_{A}}\] will become linear acceleration.
So, we have,
\[K({{x}_{1}}+{{x}_{2}})=\dfrac{5}{3}m{{a}_{A}}\] …… (1)
Consider the moment of inertia of the solid sphere as well.
The angular momentum needs to be conserved. So, we get,
\[\dfrac{5}{3}m{{v}_{1}}r=\dfrac{7}{5}m{{v}_{2}}r\]
\[\Rightarrow 25{{v}_{1}}=21{{v}_{2}}\]
The \[{{v}_{1}}=\dfrac{d{{x}_{1}}}{dt},{{v}_{2}}=\dfrac{d{{x}_{2}}}{dt}\], so, we can write the above equation as,
\[25{{x}_{1}}=21{{x}_{2}}\]
Represent the above equation in terms of \[{{x}_{2}}\], so, we have,
\[{{x}_{2}}=\dfrac{25}{21}{{x}_{1}}\] ….. (2)
Substitute the equation (2) in the equation (1), so, we get,
\[K\left( {{x}_{1}}+\dfrac{25}{21}{{x}_{1}} \right)=\dfrac{5}{3}m{{a}_{A}}\]
Further continue the calculation.
\[\begin{align}
& \dfrac{46k{{x}_{1}}}{21}=\dfrac{5}{3}m{{a}_{A}} \\
& \Rightarrow {{a}_{A}}=\dfrac{46}{35}\dfrac{k}{m}{{x}_{1}} \\
\end{align}\]
In terms of displacement, we get the above equation as follows.
\[-\dfrac{{{d}^{2}}{{x}_{1}}}{d{{t}^{2}}}=\dfrac{46}{35}\dfrac{k}{m}{{x}_{1}}\]…… (a)
The negative sign indicates the opposite direction of the displacement compared to that of the acceleration.
The above equation represents the acceleration given by the standard form as follows.
\[\dfrac{{{d}^{2}}x}{d{{t}^{2}}}=-{{w}^{2}}{{x}_{1}}\]….. (b)
Upon comparing the equations (a) and (b) we can find the expression in terms of the angular frequency.
\[\begin{align}
& {{w}^{2}}=\dfrac{46}{35}\dfrac{k}{m} \\
& \Rightarrow w=\sqrt{\dfrac{46}{35}\dfrac{k}{m}} \\
\end{align}\]
The frequency is given by the formula.
\[\begin{align}
& f=\dfrac{1}{2\pi }w \\
& \Rightarrow f=\dfrac{1}{2\pi }\sqrt{\dfrac{46}{35}\dfrac{k}{m}} \\
\end{align}\]
Therefore, the frequency of oscillation of the spheres is \[\dfrac{1}{2\pi }\sqrt{\dfrac{46}{35}\dfrac{k}{m}}\].
The sum of the amplitudes of the spheres A and B will be equal to \[{{x}_{0}}\], because of the maximum position reached by the spheres A and B. So, we get,
\[{{x}_{0}}={{x}_{A}}+{{x}_{B}}\]
As the relation between the displacements is \[{{x}_{2}}=\dfrac{25}{21}{{x}_{1}}\], thus, the same relationship applies to the amplitudes of the spheres A and B.
\[\begin{align}
& {{x}_{0}}=\dfrac{21}{25}{{x}_{B}}+{{x}_{B}} \\
& \Rightarrow {{x}_{0}}=\dfrac{46}{25}{{x}_{B}} \\
\end{align}\]
Rearrange the terms of the above equation.
\[{{x}_{B}}=\dfrac{25}{46}{{x}_{0}}\]
Therefore, the amplitude of SHM of the sphere B is \[\dfrac{25}{46}{{x}_{0}}\].
Therefore, the amplitude of SHM of sphere B is \[\dfrac{25}{46}{{x}_{0}}\], thus, the option (A) is correct.
Therefore, the frequency of oscillation of the spheres is \[\dfrac{1}{2\pi }\sqrt{\dfrac{46}{35}\dfrac{k}{m}}\], thus, the option (B) is correct.
Note: In this case, the amplitude of SHM of the sphere B is asked, even they can ask for the amplitude of SHM of the sphere A. in that case, represent the value of displacement of the sphere B in terms of the displacement of sphere A. The frequency of oscillation of the spheres is asked, even, they may ask for the time period. In such a case, just reverse the expression of the frequency of oscillation of the spheres.
Formula used:
\[\begin{align}
& I=\dfrac{5}{3}M{{R}^{2}} \\
& \tau =I\alpha \\
\end{align}\]
Complete step by step answer:
Consider the sphere A with the mean displacement of \[{{x}_{1}}\] has moved towards the left. Similarly, the sphere B with the mean displacement of \[{{x}_{2}}\] has moved towards the right. Consider the below figure.
![seo images](https://www.vedantu.com/question-sets/54104c2e-987f-48d7-a1d8-8cf8edb44162768588717614936450.png)
Therefore, the value of the total extension will be equal to the value given by,
Total extension = \[{{x}_{1}}+{{x}_{2}}\].
Now we need to find the expression for the torque in terms of the momentum. Thus, we have,
The torque of A is given as follows.
\[{{\tau }_{A}}={{I}_{A}}{{\alpha }_{A}}\]
Now substitute the expressions of the torque and moment of inertia of a hollow sphere in the above equation. Therefore, we get,
\[K({{x}_{1}}+{{x}_{2}})r=\left( \dfrac{5}{3}m{{r}^{2}} \right){{\alpha }_{A}}\]
Cancel out the common terms
\[K({{x}_{1}}+{{x}_{2}})=\dfrac{5}{3}m(r{{\alpha }_{A}})\]
Where \[{{\alpha }_{A}}\] is the angular acceleration, thus, \[r{{\alpha }_{A}}\] will become linear acceleration.
So, we have,
\[K({{x}_{1}}+{{x}_{2}})=\dfrac{5}{3}m{{a}_{A}}\] …… (1)
Consider the moment of inertia of the solid sphere as well.
The angular momentum needs to be conserved. So, we get,
\[\dfrac{5}{3}m{{v}_{1}}r=\dfrac{7}{5}m{{v}_{2}}r\]
\[\Rightarrow 25{{v}_{1}}=21{{v}_{2}}\]
The \[{{v}_{1}}=\dfrac{d{{x}_{1}}}{dt},{{v}_{2}}=\dfrac{d{{x}_{2}}}{dt}\], so, we can write the above equation as,
\[25{{x}_{1}}=21{{x}_{2}}\]
Represent the above equation in terms of \[{{x}_{2}}\], so, we have,
\[{{x}_{2}}=\dfrac{25}{21}{{x}_{1}}\] ….. (2)
Substitute the equation (2) in the equation (1), so, we get,
\[K\left( {{x}_{1}}+\dfrac{25}{21}{{x}_{1}} \right)=\dfrac{5}{3}m{{a}_{A}}\]
Further continue the calculation.
\[\begin{align}
& \dfrac{46k{{x}_{1}}}{21}=\dfrac{5}{3}m{{a}_{A}} \\
& \Rightarrow {{a}_{A}}=\dfrac{46}{35}\dfrac{k}{m}{{x}_{1}} \\
\end{align}\]
In terms of displacement, we get the above equation as follows.
\[-\dfrac{{{d}^{2}}{{x}_{1}}}{d{{t}^{2}}}=\dfrac{46}{35}\dfrac{k}{m}{{x}_{1}}\]…… (a)
The negative sign indicates the opposite direction of the displacement compared to that of the acceleration.
The above equation represents the acceleration given by the standard form as follows.
\[\dfrac{{{d}^{2}}x}{d{{t}^{2}}}=-{{w}^{2}}{{x}_{1}}\]….. (b)
Upon comparing the equations (a) and (b) we can find the expression in terms of the angular frequency.
\[\begin{align}
& {{w}^{2}}=\dfrac{46}{35}\dfrac{k}{m} \\
& \Rightarrow w=\sqrt{\dfrac{46}{35}\dfrac{k}{m}} \\
\end{align}\]
The frequency is given by the formula.
\[\begin{align}
& f=\dfrac{1}{2\pi }w \\
& \Rightarrow f=\dfrac{1}{2\pi }\sqrt{\dfrac{46}{35}\dfrac{k}{m}} \\
\end{align}\]
Therefore, the frequency of oscillation of the spheres is \[\dfrac{1}{2\pi }\sqrt{\dfrac{46}{35}\dfrac{k}{m}}\].
The sum of the amplitudes of the spheres A and B will be equal to \[{{x}_{0}}\], because of the maximum position reached by the spheres A and B. So, we get,
\[{{x}_{0}}={{x}_{A}}+{{x}_{B}}\]
As the relation between the displacements is \[{{x}_{2}}=\dfrac{25}{21}{{x}_{1}}\], thus, the same relationship applies to the amplitudes of the spheres A and B.
\[\begin{align}
& {{x}_{0}}=\dfrac{21}{25}{{x}_{B}}+{{x}_{B}} \\
& \Rightarrow {{x}_{0}}=\dfrac{46}{25}{{x}_{B}} \\
\end{align}\]
Rearrange the terms of the above equation.
\[{{x}_{B}}=\dfrac{25}{46}{{x}_{0}}\]
Therefore, the amplitude of SHM of the sphere B is \[\dfrac{25}{46}{{x}_{0}}\].
Therefore, the amplitude of SHM of sphere B is \[\dfrac{25}{46}{{x}_{0}}\], thus, the option (A) is correct.
Therefore, the frequency of oscillation of the spheres is \[\dfrac{1}{2\pi }\sqrt{\dfrac{46}{35}\dfrac{k}{m}}\], thus, the option (B) is correct.
Note: In this case, the amplitude of SHM of the sphere B is asked, even they can ask for the amplitude of SHM of the sphere A. in that case, represent the value of displacement of the sphere B in terms of the displacement of sphere A. The frequency of oscillation of the spheres is asked, even, they may ask for the time period. In such a case, just reverse the expression of the frequency of oscillation of the spheres.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)