Qstnid -4-which Of The Following Function Of Time Represent |A) Simple Harmonic,(b)periodic But Not Simple Harmonic, And Non-periodic Motion? Give Period For Each Case Of Periodic Motion (To Is Any Positive Constant).nullnullnullnullnullnullnullnullnullnullnullnull- &Nbsp; (1) (A) Sin Wt - Cos Wt (2) (B) Sin3 Wt (3) (C) 3 Cos -2 Cos (``\Pi``/4-2 Wt) (4) (D) Cos Wt + Cos 3 Wt + Cos 5 Wt (5) (E) ``E^{-w^2t^2}`` (6) (F) 1 + Wt + W2t2. (1) (A) Sin Wt - Cos Wt (2) (B) Sin3 Wt (3) (C) 3 Cos -2 Cos (``\Pi``/4-2 Wt) (4) (D) Cos Wt + Cos 3 Wt + Cos 5 Wt (5) (E) ``E^{-w^2t^2}`` (6) (F) 1 + Wt + W2t2. - 14: Oscillations

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NEET-XI-Physics

14: Oscillations

Which of the following function of time represent |a) simple harmonic,(b)periodic but not simple harmonic, and non-periodic motion? Give period for each case of periodic motion (to is any positive constant). nullnullnullnullnullnullnullnullnullnullnull page 2

Qstn# 4 Prvs-Qstn Next-Qstn

#4-0
Which of the following function of time represent |a) simple harmonic,(b)periodic but not simple harmonic, and non-periodic motion? Give period for each case of periodic motion (to is any positive constant).nullnullnullnullnullnullnullnullnullnullnullnull

(1) (a) sin wt - cos wt (2) (b) sin³ wt (3) (c) 3 cos -2 cos (``\pi``/4-2 wt) (4) (d) cos wt + cos 3 wt + cos 5 wt

(5) (e) ``e^{-w^2t^2}`` (6) (f) 1 + wt + w²t². (1) (a) sin wt - cos wt (2) (b) sin³ wt (3) (c) 3 cos -2 cos (``\pi``/4-2 wt) (4) (d) cos wt + cos 3 wt + cos 5 wt

(5) (e) ``e^{-w^2t^2}`` (6) (f) 1 + wt + w²t².
Ans : The function will represent a periodic motion, if it is identically repeated after a fixed interval of time and will represent S.H.M if it can be written uniquely in the form of a cos

(1) ``\sin \omega t - \cos \omega t``

=``\sin\omega t - \sin(\pi/2 -\omega t)`` ( as ``cos (\pi/2-\theta) = \sin \theta``)

(Again (`` \sin A - \sin B = 2\sin \frac{A-B}{2} \cos\frac{A+B}{2}``)
=``2 \cos \pi/4 \sin(\omega t - \pi/4)``

= ``\sqrt{2} \sin(\omega t - \pi/4)``
( shm equation ``x(t) = a\sin(\omega t + \alpha)``
comparing with shm,
A= ``\sqrt2`` and Time period T = ``2\pi/\omega`` and phase angle = ``\pi/4`` or ``( 2\pi-\pi/4) = 7\pi/4``.

(2) (3)
(4) (5) (e) e^-w2t2 . It is an exponential function which never repeats itself. Therefore it represents non-periodic motion.

(6) (f) 1 + wt + w²t² also represents non periodic motion. (1) ``\sin \omega t - \cos \omega t``

=``\sin\omega t - \sin(\pi/2 -\omega t)`` ( as ``cos (\pi/2-\theta) = \sin \theta``)

(Again (`` \sin A - \sin B = 2\sin \frac{A-B}{2} \cos\frac{A+B}{2}``)
=``2 \cos \pi/4 \sin(\omega t - \pi/4)``

= ``\sqrt{2} \sin(\omega t - \pi/4)``
( shm equation ``x(t) = a\sin(\omega t + \alpha)``
comparing with shm,
A= ``\sqrt2`` and Time period T = ``2\pi/\omega`` and phase angle = ``\pi/4`` or ``( 2\pi-\pi/4) = 7\pi/4``.

(2) (3)
(4) (5) (e) e^-w2t2 . It is an exponential function which never repeats itself. Therefore it represents non-periodic motion.

(6) (f) 1 + wt + w²t² also represents non periodic motion.

#4-1
(a) sin wt - cos wt
Ans : ``\sin \omega t - \cos \omega t``

=``\sin\omega t - \sin(\pi/2 -\omega t)`` ( as ``cos (\pi/2-\theta) = \sin \theta``)

(Again (`` \sin A - \sin B = 2\sin \frac{A-B}{2} \cos\frac{A+B}{2}``)
=``2 \cos \pi/4 \sin(\omega t - \pi/4)``

= ``\sqrt{2} \sin(\omega t - \pi/4)``
( shm equation ``x(t) = a\sin(\omega t + \alpha)``
comparing with shm,
A= ``\sqrt2`` and Time period T = ``2\pi/\omega`` and phase angle = ``\pi/4`` or ``( 2\pi-\pi/4) = 7\pi/4``.

#4-2
(b) sin³ wt
Ans :

#4-3
(c) 3 cos -2 cos (``\pi``/4-2 wt)
Ans :

#4-4
(d) cos wt + cos 3 wt + cos 5 wt

Ans :

#4-5
(e) ``e^{-w^2t^2}``
Ans : (e) e^-w2t2 . It is an exponential function which never repeats itself. Therefore it represents non-periodic motion.

#4-6
(f) 1 + wt + w²t².
Ans : (f) 1 + wt + w²t² also represents non periodic motion.