# A particle executes SHM on a line $8cm$ long. Its K.E and P.E will be equal when its distance from the mean position is

A. $4cm$

B. $2cm$

C. $\sqrt[2]{2}cm$

D. $\sqrt 2 $

Answer

Verified

277.5k+ views

**Hint:**SHM stands for Simple Harmonic Motion, which is characterised as a motion in which the restoring force is proportional to the body's displacement from its mean location. This restoring force often moves in the direction of the mean location. The acceleration of a particle in simple harmonic motion is given by \[a\left( t \right){\text{ }} = {\text{ }} - {\omega ^{2\;}}x\left( t \right)\] . Here, is the particle's angular velocity.

**Complete step-by-step solution:**

Simple harmonic motion is an oscillatory motion in which the particle's acceleration is directly proportional to its displacement from the mean position at any stage. It's a form of oscillatory motion that's a little different.

Simple Harmonic Motions (SHM) are oscillatory and periodic in nature, but not all oscillatory motions are SHM. The harmonic motion of all oscillatory motions, the most common of which is simple harmonic motion, is known as oscillatory motion (SHM).

The displacement, velocity, acceleration, and force in this form of oscillatory motion differ (with respect to time) in a way that can be represented by either sine (or) cosine functions, collectively known as sinusoids.

Now, coming to the question;

$K.E = P.E$

$ \Rightarrow \dfrac{1}{2}m{\omega ^2}({A^2} - {x^2}) = \dfrac{1}{2}m{\omega ^2}{x^2}$

$ \Rightarrow {A^2} - {x^2} = {x^2}$

$ \Rightarrow 2{x^2} = {A^2}$

$\therefore x = \dfrac{A}{{\sqrt 2 }}$

$2A = 8$

$A = \dfrac{8}{2} = 4cm$

$x = \dfrac{A}{{\sqrt 2 }}$

$x = \dfrac{4}{{\sqrt 2 }} = \sqrt[2]{2}$

**So, the correct option is: (C) $\sqrt[2]{2}cm$.**

**Note:**Simple Harmonic Motion is a very useful and important method for understanding the properties of sound waves, light waves, and alternating currents. Any non-simple harmonic oscillatory motion can be expressed as a superposition of several harmonic oscillatory motions of different frequencies.Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Define absolute refractive index of a medium

Which of the following would not be a valid reason class 11 biology CBSE

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What is meant by monosporic development of female class 11 biology CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How many meters are there in a kilometer And how many class 8 maths CBSE

What is pollution? How many types of pollution? Define it

Change the following sentences into negative and interrogative class 10 english CBSE

What were the major teachings of Baba Guru Nanak class 7 social science CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a labelled sketch of the human eye class 12 physics CBSE