Question

A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still wear. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega$ . If the radius of the bottle is 2.5cm then $\omega$ is close to (density of water=1000kg/ ${{{m}}^{{3}}}$ )
(A) 3.75 rad/s
(B) 1.25 rad/s
(C) 2.50 rad/s
(D) $2.5\sqrt {10}$ rad/s

Answer 1

We need to simply apply the equation of SHM. On simplifying the equation of motion, we can get the solution to this problem

Formula used: Equation of SHM:
$\Rightarrow {{a = - }}\dfrac{{{{{d}}^{{2}}}{{x}}}}{{{{d}}{{{t}}^{{2}}}}}{{ = - }}{{{\omega }}^{{2}}}{{x}}$
${{\omega = }}\sqrt {\dfrac{{{k}}}{{{m}}}}$
Here, ${{a}}$ is the acceleration of the particle,
$\omega$ is the angular frequency
${{m}}$ is the mass of the object (in kilograms),
${{x}}$ is the deformation.

Complete step by step answer:
It is already known that the body is undergoing SHM. We are already aware that the volume of the bottle is 310 ml and radius of the bottle is 2.5cm.
Equation of SHM:
$\Rightarrow {{a = - }}\dfrac{{{{{d}}^{{2}}}{{x}}}}{{{{d}}{{{t}}^{{2}}}}}{{ = - }}{{{\omega }}^{{2}}}{{x}}$
$\Rightarrow {{\omega = }}\sqrt {\dfrac{{{k}}}{{{m}}}}$
And the equation of motion is:
$\Rightarrow {{x = Asin}}\left( {\omega {{t + }}\delta } \right)$
On simplifying the equation,
$\Rightarrow {{A}} \times \rho {{g = }}{{{F}}_{{{res}}}}$
$\Rightarrow \pi {{{r}}^{{2}}}\rho {{g}} \times {{x = }}{{{F}}_{{{res}}}}$
On solving further,
$\omega {{ = }}\sqrt {\dfrac{{\pi {{g}}}}{{{v}}}}$
$\Rightarrow \omega {{ = 2}}.{{5}} \times {{1}}{{{0}}^{{{ - 2}}}}\sqrt {\dfrac{{{{3}}.{{14}} \times {{10}}}}{{{{310}} \times {{1}}{{{0}}^{{{ - 6}}}}}}} {{ = 2}}.{{5}}\sqrt {{{10}}} {{rad/s}}$
So, we need to see from the above options, and select the correct value.
Thus, the correct answer is option D.

Additional Information
Simple harmonic motion, in physics, repetitive movement back and forth through an equilibrium, or central, position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side. The time interval of each complete vibration is the same. Whilst simple harmonic motion is a simplification, it is still a very good approximation. Simple harmonic motion is important in research to model oscillations for example in wind turbines and vibrations in car suspensions.

Note:
Some of the examples of SHM are Swings that we see in the park is an example of simple harmonic motion. The back and forth, repetitive movements of the swing against the restoring force is the simple harmonic motion. The pendulum oscillating back and forth from the mean position is an example of simple harmonic motion. The process of hearing is impossible without simple harmonic motion. The soundwaves that enter our ear causes the eardrum to vibrate back and forth.