SIMPLE HARMONIC MOTION -

(definition is on the previous page “periodic and oscillatory motion”)

PHASE

It determines the status of the particle in SHM. It is always expressed in angle. It is a time-dependent quantity.

Let the particle execute in SHM with time period T and at t=0, the particle is at mean position i.e x=0.

From fig

$\fn_cm \large Time$	$\fn_cm \large Displacement$	$\fn_cm \large Phase$
$\fn_cm \large at\;\;t=0$	$\fn_cm \large x=0$	$\fn_cm \large 0^0$
$\fn_cm \large at\;\;t=\frac{T}{4}$	$\fn_cm \large x=A$	$\fn_cm \large \frac{\pi}{2}$
$\fn_cm \large at\;\;t=\frac{2T}{4}=\frac{T}{2}$	$\fn_cm \large x=0$	$\fn_cm \large \pi$
$\fn_cm \large at\;\;t=\frac{3T}{4}$	$\fn_cm \large x=-A$	$\fn_cm \large \frac{3\pi}{2}$
$\fn_cm \large at\;\;t=\frac{4T}{4}=T$	$\fn_cm \large x=0$	$\fn_cm \large 2\pi$

From the above concept, we can write the relation between displacement from the mean position with time.

$\fn_cm \large x=A \sin (\frac{2\pi}{T}.t)\;\;\;\;\;\;\;\;\;\;\frac{2\pi}{T}.t=\omega t\rightarrow Phase$

$\fn_cm \large \left [ x=A\sin \omega t \right ]$

where $\fn_cm \large \omega$ is the angular frequency of S.H.M. From fig, the particle has the same phase if their phase difference is $\fn_cm \large 2\pi$

PHASE CONSTANT (INITIAL PHASE)

What happens when the particle starts executing from other than the mean position. In such a case, can we use the same equation?

Definitely No! the equation will be modified a little bit. Due to this a concept of phase constant is introduced.

i.e the general equation of SHM is

$\fn_cm \large \left [ x=A\sin(\omega t+\phi ) \right ]$

where $\fn_cm \large (\omega t+\phi)\rightarrow phase$ and $\fn_cm \large \phi\rightarrow phase\;constant$

This phase constant depends on the choice of the instant t=0.

for example:

a. If we take t=0, when it passes from the mean position and going +ve direction

$\fn_cm \large i.e\; phase=0$

$\fn_cm \large \Rightarrow \omega t+\phi=0$

$\fn_cm \large at\;t=0,\; \phi=0$

$\fn_cm \large \therefore equation\;become\;\;$ $\fn_cm \large \left [ x=A\sin \omega t \right ]$

b. If we take t=0, when it is at +ve extreme position

$\fn_cm \large i.e\;phase=\frac{\pi}{2}$

$\fn_cm \large \Rightarrow \omega t+\phi=\frac{\pi}{2}$

$\fn_cm \large at\;t=0,\; \phi=\frac{\pi}{2}$

$\fn_cm \large \therefore equation\;become\;\;$ $\fn_cm \large \left [ x=A\sin (\omega t+\phi) \right ]\;\;\;or\;\;\;\left [ x=A\cos \omega t \right ]$

c. If we take t=0, when it passes from the mean position and going -ve direction

$\fn_cm \large i.e\;phase=\pi$

$\fn_cm \large \Rightarrow \omega t+\phi=\pi$

$\fn_cm \large at\;t=0,\; \phi=\pi$

$\fn_cm \large \therefore equation\;become\;\;$ $\fn_cm \large \left [ x=A\sin (\omega t+\pi) \right ]\;\;\;or\;\;\;\left [ x=-A\sin \omega t \right ]$

d. If we take t=0, when it is at -ve extreme position

$\fn_cm \large i.e\;phase=\frac{3\pi}{2}$

$\fn_cm \large \Rightarrow \omega t+\phi=\frac{3\pi}{2}$

$\fn_cm \large at\;t=0,\; \phi=\frac{3\pi}{2}$

$\fn_cm \large \therefore equation\;become\;\;$ $\fn_cm \large \left [ x=A\sin (\omega t+\frac{3 \pi}{2}) \right ]\;\;\;or\;\;\;\left [ x=-A\cos \omega t \right ]$

NOTE:

a. We know that the equation of SHM is

$\fn_cm \large x=A\sin(\omega t+\phi)$

we can write here

$\fn_cm \large x=A[\sin \omega t \cos \phi+\cos \omega t \sin \phi]$

$\fn_cm \large x=A\cos \phi \sin \omega t+A \sin \phi \cos \omega t$

$\fn_cm \large [x=C \sin \omega t+D \cos \omega t]$

i.e For the motion of a particle to be simple harmonic, Its displacement x must be expressible in either of the following forms:

$\fn_cm \large x=A\sin \omega t+B\cos \omega t,x=A\cos(\omega t+ \phi),x=A\sin(\omega t+ \phi)$

This function is also periodic.

A combination of two simple harmonic motions with arbitrary amplitudes and phases is not necessarily periodic. It is periodic only if the frequency of one motion is an integral multiple of the other’s frequency. However, a periodic motion can always be expressed as a sum of infinite number of harmonic motions with appropriate amplitudes.

for example, the function $\fn_cm \large x=\sin \omega t+\cos 2\omega t+\sin \4 \omega t$ is a periodic function with a time period $\fn_cm \large \frac{2 \pi}{\omega}$ but this complete function is not in SHM (individually all three represent SHM of the different time periods)

b. Two SHM may have same $\fn_cm \large \omega$ and $\fn_cm \large \phi$ but different amplitudes A and B.

c. Two SHM may have the same amplitude (A) and ω but different phase constant

d. Two SHM may have the same Amplitude (A) and Φ but different ω

1. Which of the following functions of time represent (a) periodic and (b) non-periodic motion? Give the period for each case of periodic motion [ω is any positive constant].
(i) sin ωt + cos ωt
(ii) sin ωt + cos 2 ωt + sin 4 ωt
(iii) $\fn_cm e^{-\omega t}$
(iv) log (ωt)

2. 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 (ω is any positive constant):
(a) sin ωt – cos ωt
(b) $\fn_cm \sin^3\omega t$
(c) 3 cos (π/4 – 2ωt)
(d) cos ωt + cos 3ωt + cos 5ωt
(e) exp (– $\fn_cm \omega^2 t^2$ )
(f) $\fn_cm 1+\omega t+ \omega ^2 t^2$

3. Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic? Give the period for each case.
(1) sin ωt – cos ωt
(2) $\fn_cm \sin^2 \omega t$

4. A body oscillates with simple harmonic motion according to the following equation:

$\fn_cm \large x=0.40 \cos(0.70 t- 0.30)m$

Find (i) the amplitude (ii) the frequency (iii) the time-period and (iv) the phase constant

5. The displacement of a particle is given by :

$\fn_cm \large x=4 \cos(3 \pi t+ \pi)$

where x is in meter and t is in second.

Determine (a) the frequency and period of motion (b) the amplitude of the motion (c) the phase constant and (d) the position of the particle at t=0.

6. The equation of SHM is given as:

$\fn_cm \large x=6\sin 10\pi t + 8\cos 10 \pi t$

where x is in cm and t in second. Find (a) period (b) amplitude and (c) initial phase of motion

7. In what time after its motion begins, will a particle oscillating according to the equation $\fn_cm \large y=7\sin 0.5\pi t$ moves from the mean position to maximum displacement?