Class 11PhysicsOscillationsPeriodic, Oscillatory motion and its characteristics and types of SHM and Equation of SHM
MediumShow that simple harmonic motion may be regarded as the projection of uniform circular motion along a diameter of the circle.
(N/A) Consider a particle moving in a circle of radius $R$ with a constant angular speed $\omega$ in a horizontal plane.
The position of the particle at any time $t$ can be represented by the angle $\theta = \omega t$ with respect to a reference diameter.
If we project the position of the particle onto a diameter of the circle,the displacement $y$ of the projection from the center of the circle at time $t$ is given by $y = R \sin(\omega t)$.
This equation $y = R \sin(\omega t)$ represents the displacement of a particle executing simple harmonic motion $(SHM)$.
As shown in the figure,as the particle moves along the circular path through points $A, B, C, D, E, F, G$,its projection on the vertical diameter moves back and forth between the extreme points $S$ and $Q$ through the center $O$. This oscillatory motion of the projection is simple harmonic motion.
The position of the particle at any time $t$ can be represented by the angle $\theta = \omega t$ with respect to a reference diameter.
If we project the position of the particle onto a diameter of the circle,the displacement $y$ of the projection from the center of the circle at time $t$ is given by $y = R \sin(\omega t)$.
This equation $y = R \sin(\omega t)$ represents the displacement of a particle executing simple harmonic motion $(SHM)$.
As shown in the figure,as the particle moves along the circular path through points $A, B, C, D, E, F, G$,its projection on the vertical diameter moves back and forth between the extreme points $S$ and $Q$ through the center $O$. This oscillatory motion of the projection is simple harmonic motion.
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Similar Questions
Which of the following functions of time represent $(a)$ simple harmonic,$(b)$ periodic but not simple harmonic,and $(c)$ non-periodic motion? Give period for each case of periodic motion ($\omega$ is any positive constant):
$(a)$ $\sin \omega t - \cos \omega t$
$(b)$ $\sin^3 \omega t$
$(c)$ $3 \cos (\pi/4 - 2 \omega t)$
$(d)$ $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t$
$(e)$ $\exp(-\omega^2 t^2)$
$(f)$ $1 + \omega t + \omega^2 t^2$
$(a)$ $\sin \omega t - \cos \omega t$
$(b)$ $\sin^3 \omega t$
$(c)$ $3 \cos (\pi/4 - 2 \omega t)$
$(d)$ $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t$
$(e)$ $\exp(-\omega^2 t^2)$
$(f)$ $1 + \omega t + \omega^2 t^2$
Medium
View Solution$A$ simple harmonic motion is represented by $\alpha \frac{d^2 x}{d t^2}+\beta x=0$. Its period is
DifficultMHT CET 2022
View Solution$A$ particle executes simple harmonic motion and is located at $x = a, b$ and $c$ at times $t_0, 2t_0$ and $3t_0$ respectively. The frequency of the oscillation is
DifficultJEE MAIN 2018
View Solution$A$ point particle of mass $0.1\, kg$ is executing $S.H.M.$ with an amplitude of $0.1\, m$. When the particle passes through the mean position,its kinetic energy is $8 \times 10^{-3} \, J$. Obtain the equation of motion of this particle if the initial phase of oscillation is $45^o$.
DifficultAIIMS 2013
View SolutionThe periodic time of $SHM$ is also measured by $T = 2 \pi \sqrt{\frac{\text{displacement}}{\text{acceleration}}}$. Can we say that the time period depends on the displacement?
Easy
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