Answer
64.8k+ views
Hint In mechanics and physics, simple harmonic motion is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. It is vibratory motion in a system in which the restoring force is proportional to the displacement from equilibrium
Step by step solution
As it is Simple harmonic motion so the equation of motion will be
\[F = - kx\]
Or the equation can be written as
\[\dfrac{{vdv}}{{dx}} = - {\omega ^2}x\]
Now integrating the expression with boundary condition,
\[\int\limits_{{v_0}}^v {vdv} = - {\omega ^2}\int\limits_0^x {xdx} \]
After integrating the above equation we get
$\left[ {\dfrac{{{v^2}}}{2}} \right]_{{v_0}}^v = - {\omega ^2}\left[ {\dfrac{{{x^2}}}{2}} \right]_0^x$
Now we have to apply the limit in the above equation so we get
$\left[ {\dfrac{{{v^2}}}{2} - \dfrac{{v_0^2}}{2}} \right] = - {\omega ^2}\left[ {\dfrac{{{x^2}}}{2}} \right]$
Now we have to do further calculation then we get
$\dfrac{1}{2}\left[ {{v^2} - v_0^2} \right] = \dfrac{{ - {\omega ^2}{x^2}}}{2}$
Now after simplifying the above equation we get
$v = \sqrt {v{}_0^2 - {\omega ^2}{x^2}} $
Hence the correct answer is option is (B)
Note
A speed vector speaks to the pace of progress of the situation of an object.The greatness of a speed vector gives the speed of an article while the vector course gives its direction.Velocity vectors can be added or deducted by the standards of vector addition.In math and mechanics, a relocation is a vector whose length is the most limited good ways from the underlying to the last situation of a point P going through movement. It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point trajectory. A displacement may be identified with the translation that maps the initial position to the final position.
Step by step solution
As it is Simple harmonic motion so the equation of motion will be
\[F = - kx\]
Or the equation can be written as
\[\dfrac{{vdv}}{{dx}} = - {\omega ^2}x\]
Now integrating the expression with boundary condition,
\[\int\limits_{{v_0}}^v {vdv} = - {\omega ^2}\int\limits_0^x {xdx} \]
After integrating the above equation we get
$\left[ {\dfrac{{{v^2}}}{2}} \right]_{{v_0}}^v = - {\omega ^2}\left[ {\dfrac{{{x^2}}}{2}} \right]_0^x$
Now we have to apply the limit in the above equation so we get
$\left[ {\dfrac{{{v^2}}}{2} - \dfrac{{v_0^2}}{2}} \right] = - {\omega ^2}\left[ {\dfrac{{{x^2}}}{2}} \right]$
Now we have to do further calculation then we get
$\dfrac{1}{2}\left[ {{v^2} - v_0^2} \right] = \dfrac{{ - {\omega ^2}{x^2}}}{2}$
Now after simplifying the above equation we get
$v = \sqrt {v{}_0^2 - {\omega ^2}{x^2}} $
Hence the correct answer is option is (B)
Note
A speed vector speaks to the pace of progress of the situation of an object.The greatness of a speed vector gives the speed of an article while the vector course gives its direction.Velocity vectors can be added or deducted by the standards of vector addition.In math and mechanics, a relocation is a vector whose length is the most limited good ways from the underlying to the last situation of a point P going through movement. It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point trajectory. A displacement may be identified with the translation that maps the initial position to the final position.
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)