Physics SIMPLE HARMONIC MOTION

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SIMPLE HARMONIC MOTION

MOTION OF A FOOT OF PERPENDICULAR UPON ANY DIAMETER OF A PARTICLE EXECUTING UNIFORM CIRCULAR MOTION

Consider a particle (reference particle) moving
along a circle of radius a, with uniform angular
speed w . The circle is known as circle of reference.
Let x'x and y'y be any two mutually perpendicular
diameters, which can be taken as X and Y � axes
respectively.

Let M be the foot of perpendicular drawn from the
Particle P upon the diameter x'x. As will be seen
ahead that, as the reference particle executes
uniform circular motion along the circle, the foot of
perpendicular M (or the projection of P upon the
diameter x'x) executes SHM along the diameter x'x.

Let `P_0` be the initial position of the reference
particle and corresponding angle made by the
radius vector joining center o to particle `P_0` with the
axis OY' be f . Further, let in a time t, the radius
vector `OP_0` rotate by an angle q = w t, and reach the
new position OP. M is the projection of P along X'X.

The displacement of M (referred to 'O ' the center of
circle of reference) will be given by

OM = x = a sin ( w t + f ) [From rt. angled D MOP] .....
(a)

The velocity of the projection M, will be the
component of linear velocity of particle p along X -
axis.

`\ v = w a cos ( w t +f) ..... (b)`

Finally, the acceleration (linear) of the projection
M, will be the component of centripetal
acceleration of particle P along X - axis.

`\ f = -w^2 a sin ( w t + f) ..... (c)`

From equation (a) and (c), it is evident that

`f = - w^2x`

Proving the fact that the projection M executes SHM
along the X - axis.

Consider a particle (reference particle) moving
along a circle of radius a, with uniform angular
speed w . The circle is known as circle of reference.
Let x'x and y'y be any two mutually perpendicular
diameters, which can be taken as X and Y � axes
respectively.

Let M be the foot of perpendicular drawn from the
Particle P upon the diameter x'x. As will be seen
ahead that, as the reference particle executes
uniform circular motion along the circle, the foot of
perpendicular M (or the projection of P upon the
diameter x'x) executes SHM along the diameter x'x.

Let `P_0` be the initial position of the reference
particle and corresponding angle made by the
radius vector joining center o to particle `P_0` with the
axis OY' be f . Further, let in a time t, the radius
vector `OP_0` rotate by an angle q = w t, and reach the
new position OP. M is the projection of P along X'X.

The displacement of M (referred to 'O ' the center of
circle of reference) will be given by

OM = x = a sin ( w t + f ) [From rt. angled D MOP] .....
(a)

The velocity of the projection M, will be the
component of linear velocity of particle p along X -
axis.

`\ v = w a cos ( w t +f) ..... (b)`

Finally, the acceleration (linear) of the projection
M, will be the component of centripetal
acceleration of particle P along X - axis.

`\ f = -w^2 a sin ( w t + f) ..... (c)`

From equation (a) and (c), it is evident that

`f = - w^2x`

Proving the fact that the projection M executes SHM
along the X - axis.

NOTE : rhe projection of reference particle P rtpon any diameter will exewte SHM . Significance of w and f

Evidently, the projection M repeats its motion accordingly as the reference particle P
repeats its motion.

\The time duration for one complete oscillation for
M is same as that for one complete revolution of
reference particle P along the circle of reference.

Now, time period `= T = (2 pi)/omega`
( w is the angular speed of P]

The phase of the particle `( w t + f )` is the angle made
by the radius vector OP with the axis OY'. As the
particle P goes on rotating along the circle, the
phase of the projection `M` goes on varying
uniformly from o to `2 p` , then from `2 p` to `4 p` (which
can be treated as `0` to `2 p` ) and again from `4 p` to `6 p`
(which can again be treated as `0` to `2 p` )and so on.
Even though, the term "phase" is Concerned with
particle executing SHM (here projection `M`) but still
it is worthwhile to associate it with the reference
particle. Similarly, the term angular frequency `( w )`
can also be better understood and interpreted in
term of the reference particle.

Therefore, whenever, we come across problems
related to phase difference, it is always preferable
and wise to think in terms of the corresponding
circle of reference and its reference particle. Things
get more easier and more lucid.

Evidently, the projection M repeats its motion accordingly as the reference particle P
repeats its motion.

\The time duration for one complete oscillation for
M is same as that for one complete revolution of
reference particle P along the circle of reference.

Now, time period `= T = (2 pi)/omega`
( w is the angular speed of P]

The phase of the particle `( w t + f )` is the angle made
by the radius vector OP with the axis OY'. As the
particle P goes on rotating along the circle, the
phase of the projection `M` goes on varying
uniformly from o to `2 p` , then from `2 p` to `4 p` (which
can be treated as `0` to `2 p` ) and again from `4 p` to `6 p`
(which can again be treated as `0` to `2 p` )and so on.
Even though, the term "phase" is Concerned with
particle executing SHM (here projection `M`) but still
it is worthwhile to associate it with the reference
particle. Similarly, the term angular frequency `( w )`
can also be better understood and interpreted in
term of the reference particle.

Therefore, whenever, we come across problems
related to phase difference, it is always preferable
and wise to think in terms of the corresponding
circle of reference and its reference particle. Things
get more easier and more lucid.

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