$Assertion :$ In simple harmonic motion, the velocity is maximum when the acceleration is minimum.
$Reason :$ Displacement and velocity of $S.H.M.$ differ in phase by $\frac{\pi }{2}$
$Reason :$ Displacement and velocity of $S.H.M.$ differ in phase by $\frac{\pi }{2}$
AIIMS 2014, Medium
Download our app for free and get started
At the middle point velocity of the particle under $SHM$ is maximum but acceleration is zero since displacement is zero. So Assertion is true.
We know that $x=a \sin \omega t$ $...(1)$
Where $x$ is displacement and a is amplitude.
Velocity $=\frac{d x}{d t}=a \omega \cos \omega t$
$=a \omega \cos (-\omega t)=a \omega \sin \left(\frac{\pi}{2}-(-\omega t)\right)$
$=a \omega \sin \left(\omega t+\frac{\pi}{2}\right)$ $...(2)$
From equation $( 1 )$ and $(ii)$ it is clear that
Velocity is ahead of displacement $(x)$ by $\frac{\pi}{2}$ angle.
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A force of $6.4\, N$ stretches a vertical spring by $0.1 \,m$. The mass that must be suspended from the spring so that it oscillates with a period of $\left( {\frac{\pi }{4}} \right)sec$. is ... $kg$View Solution
- 2Out of the following functions representing motion of a particle which represents $SHM$View Solution
$(A)\;y= sin\omega t-cos\omega t$
$(B)\;y=sin^3\omega t$
$(C)\;y=5cos\left( {\frac{{3\pi }}{4} - 3\omega t} \right)$
$(D)\;y=1+\omega t+{\omega ^2}{t^2}$
- 3A pendulum has time period $T$ in air. When it is made to oscillate in water, it acquired a time period $T' = \sqrt 2 T$. The density of the pendulum bob is equal to (density of water $= 1$)View Solution
- 4If the time period of a two meter long simple pendulum is $2\, s$, the acceleration due to gravity at the place where pendulum is executing $S.H.M.$ isView Solution
- 5The $S.H.M.$ of a particle is given by the equations $=2 \sin \omega t+4 \cos \omega t$. Its amplitude of oscillation is ........ unitsView Solution
- 6If $x = a\sin \left( {\omega t + \frac{\pi }{6}} \right)$ and $x' = a\cos \omega t$, then what is the phase difference between the two wavesView Solution
- 7$2$ particles $p$ and $q$ describe $SHM$ of same amplitude $a$ and same frequency $f$ along straight line, the maximum distance between the two particle $a\sqrt 2 $ . The initial phase difference between particle isView Solution
- 8A body oscillates with a simple harmonic motion having amplitude $0.05\, m .$ At a certain instant, its displacement is $0.01\, m$ and acceleration is $1.0 \,m / s ^{2} .$ The period of oscillation isView Solution
- 9The periodic time of a simple pendulum of length $1\, m $ and amplitude $2 \,cm $ is $5\, seconds$. If the amplitude is made $4\, cm$, its periodic time in seconds will beView Solution
- 10The $K.E.$ and $P.E.$ of a particle executing $SHM$ with amplitude $A$ will be equal when its displacement isView Solution