Comparative Study of Displacement Velocity and Acceleration
Category : JEE Main & Advanced
(1) All the three quantities displacement, velocity and acceleration show harmonic variation with time having same period.
(2) The velocity amplitude is \[\omega \] times the displacement amplitude
(3) The acceleration amplitude is \[{{\omega }^{2}}\] times the displacement amplitude
(4) In S.H.M. the velocity is ahead of displacement by a phase angle \[\pi /2\]
(5) In S.H.M. the acceleration is ahead of velocity by a phase angle \[\pi /2\]
(6) The acceleration is ahead of displacement by a phase angle of \[\pi \]
Various physical quantities in S.H.M. at different position :
| Graph | Formula | At mean position | At extreme position |
|
Displacement
|
\[y=a\sin \omega \,t\] | \[y=0\] | \[y=\pm a\] |
|
Velocity
|
\[v=a\omega \cos \omega \,t\] \[=a\omega \sin (\omega \,t+\frac{\pi }{2})\] or \[v=\omega \sqrt{{{a}^{2}}-{{y}^{2}}}\] | \[{{v}_{\max }}=a\omega \] | \[{{v}_{\min }}=0\] |
|
Acceleration
|
\[A=-a{{\omega }^{2}}\sin \omega \,t\] \[=a{{\omega }^{2}}\sin (\omega \,t+\pi )\] or \[\left| A\, \right|={{\omega }^{2}}y\] | \[{{A}_{\min }}=0\] | \[|{{A}_{\max }}|\] \[{{\omega }^{2}}a\] |
|
Force
|
\[F=-\,m{{\omega }^{2}}a\sin \omega \,t\] or \[F=m{{\omega }^{2}}y\] | \[{{F}_{\min }}=0\] | \[{{F}_{\max }}=\] \[\,m{{\omega }^{2}}a\] |
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