Answer
64.8k+ views
Hint: Find the time period of oscillation of the system. Using time period, find the relation between time period and frequency for both objects individually. Divide the squares of frequency and find out the solution.
Complete step by step solution:
The figure given consists of 2 masses attached to a given spring of spring constant k and length x. When the table is withdrawn from the second mass, the system begins to have increased oscillations of frequency \[{f_2}\] . Now we can first find the time period required for the oscillations to occur.
This is mathematically expressed as ,
\[T = 2\pi \times \sqrt {\dfrac{m}{k}} \]
Wherein T is the time period of oscillations, m is the mass of the object and K is the spring constant of the spring.
Now we know that frequency of an oscillation is inversely proportional to the time period of oscillations.
\[f = 1/T\]
\[ \Rightarrow f = 1/(2\pi \times \sqrt {\dfrac{m}{k}} )\]
\[ \Rightarrow f = \dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{m}} \]
For Object 1,
\[ \Rightarrow {f_1} = \dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{{{m_1}}}} \]
For Object 2,
\[ \Rightarrow {f_2} = \dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{{{m_2} + m1}}} \]
Note: Since the table is removed, the frequency experienced by the downgoing object will also experience the mass of the first object .
Now \[\dfrac{{{f_1}^2}}{{{f_2}^2}}\] will be
\[ \Rightarrow \dfrac{{{f_1}}}{{{f_2}}} = \dfrac{{\dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{{{m_1}}}} }}{{\dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{{{m_2} + m1}}} }}\]
Cancelling the like terms,
\[ \Rightarrow \dfrac{{{f_1}}}{{{f_2}}} = \dfrac{{\sqrt {\dfrac{k}{{{m_1}}}} }}{{\sqrt {\dfrac{k}{{{m_2} + m1}}} }}\]
Squaring on both sides we obtain,
\[ \Rightarrow \dfrac{{{f_1}^2}}{{{f_2}^2}} = \dfrac{{\dfrac{k}{{{m_1}}}}}{{\dfrac{k}{{{m_2} + m1}}}}\]
Removing spring constant k from the above equation,
\[ \Rightarrow \dfrac{{{f_1}^2}}{{{f_2}^2}} = \dfrac{{\dfrac{1}{{{m_1}}}}}{{\dfrac{1}{{{m_2} + m1}}}}\]
\[ \Rightarrow \dfrac{{{f_1}^2}}{{{f_2}^2}} = \dfrac{{{m_2} + m1}}{{{m_1}}}\]
Therefore, Option (A) is the correct answer to the following question.
Note:
The time period is defined as a time period required by a body expressing periodic motion to complete one period or one cycle. Frequency is defined as the repetition of the event per unit time. Frequency is measured in Hertz whereas the time period is measured in seconds. Frequency is inversely proportional to time period of the oscillation.
Complete step by step solution:
The figure given consists of 2 masses attached to a given spring of spring constant k and length x. When the table is withdrawn from the second mass, the system begins to have increased oscillations of frequency \[{f_2}\] . Now we can first find the time period required for the oscillations to occur.
This is mathematically expressed as ,
\[T = 2\pi \times \sqrt {\dfrac{m}{k}} \]
Wherein T is the time period of oscillations, m is the mass of the object and K is the spring constant of the spring.
Now we know that frequency of an oscillation is inversely proportional to the time period of oscillations.
\[f = 1/T\]
\[ \Rightarrow f = 1/(2\pi \times \sqrt {\dfrac{m}{k}} )\]
\[ \Rightarrow f = \dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{m}} \]
For Object 1,
\[ \Rightarrow {f_1} = \dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{{{m_1}}}} \]
For Object 2,
\[ \Rightarrow {f_2} = \dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{{{m_2} + m1}}} \]
Note: Since the table is removed, the frequency experienced by the downgoing object will also experience the mass of the first object .
Now \[\dfrac{{{f_1}^2}}{{{f_2}^2}}\] will be
\[ \Rightarrow \dfrac{{{f_1}}}{{{f_2}}} = \dfrac{{\dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{{{m_1}}}} }}{{\dfrac{1}{{2\pi }} \times \sqrt {\dfrac{k}{{{m_2} + m1}}} }}\]
Cancelling the like terms,
\[ \Rightarrow \dfrac{{{f_1}}}{{{f_2}}} = \dfrac{{\sqrt {\dfrac{k}{{{m_1}}}} }}{{\sqrt {\dfrac{k}{{{m_2} + m1}}} }}\]
Squaring on both sides we obtain,
\[ \Rightarrow \dfrac{{{f_1}^2}}{{{f_2}^2}} = \dfrac{{\dfrac{k}{{{m_1}}}}}{{\dfrac{k}{{{m_2} + m1}}}}\]
Removing spring constant k from the above equation,
\[ \Rightarrow \dfrac{{{f_1}^2}}{{{f_2}^2}} = \dfrac{{\dfrac{1}{{{m_1}}}}}{{\dfrac{1}{{{m_2} + m1}}}}\]
\[ \Rightarrow \dfrac{{{f_1}^2}}{{{f_2}^2}} = \dfrac{{{m_2} + m1}}{{{m_1}}}\]
Therefore, Option (A) is the correct answer to the following question.
Note:
The time period is defined as a time period required by a body expressing periodic motion to complete one period or one cycle. Frequency is defined as the repetition of the event per unit time. Frequency is measured in Hertz whereas the time period is measured in seconds. Frequency is inversely proportional to time period of the oscillation.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)