Question

A body executing linear S.H.M has a velocity of 3cm/s when its displacement is 4cm and a velocity of 4cm/s when its displacement is 3cm. (i) Find amplitude and period of oscillation. (ii) if the mass of the body is 100g, calculate total energy of oscillation.

Answer 1

In this question, we first use the relation between the velocity v and displacement x and also amplitude a. from this we get the first part of our question by putting the given values. Next, we find the energy by using the energy formula. Also, we study the basics of simple harmonic motion.
Formula used:
$v = \omega \sqrt {{a^2} - {x^2}}$
$E = \dfrac{1}{2}ma{\omega ^2}$

Complete answer:
(i).Here, we have the relation between the velocity v and displacement x, which is given as:
\eqalign{ & v = \omega \sqrt {{a^2} - {x^2}} \cr & 3 = \omega \sqrt {{a^2} - 16} \cr}
\eqalign{& \Rightarrow 9 = {\omega ^2}({a^2} - 16) \cr & \Rightarrow 16 = {\omega ^2}({a^2} - 9) \cr}
Simplifying the equation:
\eqalign{ & \Rightarrow \dfrac{9}{{16}} = \dfrac{{{a^2} - 16}}{{{a^2} - 9}} \cr & \Rightarrow 9{a^2} - 81 = 16{a^2} - 256 \cr}
\eqalign{ & \Rightarrow {a^2} = \dfrac{{175}}{7} \cr & \therefore a = \sqrt {25} \cr}
Therefore, we get the required amplitude of the body.
Now, to find the period of oscillation given by T:
\eqalign{ & 3 = \omega \sqrt {25 - 16} \cr & \Rightarrow 3 = \omega \sqrt 9 \cr & \Rightarrow \omega = 1 \cr}
substituting the given values we get:
\eqalign{ & \Rightarrow \dfrac{{2\pi }}{T} = 1 \cr & \Rightarrow T = 2\pi \cr & \Rightarrow T = 2 \times 3.14 \cr & \therefore T = 6.28\sec \cr}
Therefore, we get the required period of oscillation as 6.28 sec.

(ii).Now, to solve for the total energy of oscillation E, the equation is given by:
\eqalign{ & E = \dfrac{1}{2}ma{\omega ^2} \cr & \Rightarrow E = \dfrac{1}{2} \times 0.1 \times {1^2} \times 5 \times {10^{ - 2}} \cr & \therefore E = 2.5 \times {10^{ - 3}}Jule \cr}
Therefore, we get the required total energy E of oscillation from above value.

Additional information:
We know that the frequency is defined as the number of waves that pass a fixed point in unit time. It can also be defined as the number of cycles or vibrations undergone during one unit of time. The S.I unit of frequency is Hertz or Hz and the unit of wavelength is meter or m. Furthermore we also know the S.I unit of time which is given by second or s. Two waves are said to be coherent if they are moving with the same frequency and have constant phase difference.
Simple harmonic motion is defined as a special type of periodic motion where the restoring force (force applied in the opposite direction) on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. Here, the motion is sinusoidal in time and demonstrates a single resonant frequency.
Also, a simple harmonic progressive wave is a wave that continuously advances in a given direction without the change of form and also, the particles of the medium perform simple harmonic motion about their mean position with the same amplitude and period, when the waves pass over them.

Note:
All particles of medium perform S.H.M. when the wave passes through the medium. All particles vibrate with the same amplitude. It should be remembered that in the simple harmonic progressive wave the particles in the medium show simple harmonic motion. There are other systems as well which
shows simple harmonic motion like- simple pendulum.