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Oscillation (S.H.M)

It’s a Periodic Motion

It is a motion due to vibration

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Oscillatory Motion

It is type of motion in which a body moves to and fro,

tracing the same path again and again, in equal

intervals of time.

What is Simple Harmonic Motion (S.H.M.)?

Simple harmonic motion is periodic motion

produced by a restoring force that is directly

proportional to the displacement and oppositely

directed.

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type of S.H.M.

1) If the object is moving along a straight line path,

it is called ‘linear simple harmonic motion’

(L.S.H.M.)

In S.H.M., the force causing the motion is directly

proportional to the displacement of the particle from

the mean position and directed opposite to it. If x is

the displacement of the particle from the mean

position, and f is the force acting on it, then

f α -x negative sign indicates direction of force

opposite to that of displacement

f = -kx k is called force per unit displacement or

force constant.

The units of k are N/m in M.K.S. and dyne/cm in

C.G.S. Its dimensions are [M1L0T -2]

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S.H.M. is projection of U.C.M. on any diameter

∠ DOP0 = α

∠ P0OP1 = ωt

At this instant, its projection moves from O to M, such

that distance OM is x.

In right angled triangle OMP1,

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∴ x = a sin (ωt + α)

This is the equation of displacement of the particle

performing S.H.M. from mean position, in terms

of maximum displacement a, time t and initial

phase α.

The time derivative of this displacement is velocity v

∴ v = = aω cos(ωt + α)

Time derivative of this velocity is acceleration.

∴ accln = = - aω2 sin (ωt + α)

But, a sin (ωt + α) = x,

∴ accln = - ω2 x

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The negative sigh indicates that the acceleration is

always opposite to the displacement. When the

displacement is away from the mean position, the

acceleration is towards the mean position and vice

versa. Also, its magnitude is directly proportional to

the displacement. Hence, S.H.M. is also defined as,

‘ the type of linear periodic motion, in which the

force (and acceleration) is always directed

towards the mean position and is of the

magnitude directly proportional to displacement

of the particle from the mean position.’

Q.1    In the equation F=-Kx, representing a S.H.M., the force constant K does not depends upon

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          (a.) elasticity of the system  

         (b) inertia of the system

          (c) extension or displacement of the system  

(d.) velocity of the system

Q.2    The suspended mass makes 30 complete

oscillations in 15 s. What is the period and frequency

of the motion?

a) 2s,0.5 Hz b) 0.5s, 2Hz c) 2s,2Hz d)0.5s,0.5Hz

Q.3    A 4-kg mass suspended from a spring

produces a displacement of 20 cm. What is the spring

constant

a) 196 N/m b) 500 N/m c) 100 N/m d) 80N/m

Answers :-

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1. (d.) velocity of the system

2. (b) 0.5s, 2Hz

3. a) 196 N/m

Differential Equation of S.H.M.

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If x is the displacement of the particle performing

S.H.M.,

accln = ,

force = m

But f = - kx

∴ m = - kx

∴ m + kx = 0 ... (1)

∴ + But,

∴ + ω2x = 0 ... (2)

These two equations are called differential equations

of S.H.M.

According to second equation,

Force = mass × acceleration

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∴ f = - mω2x but, f = - kx also

∴ - kx = - mω2x

∴ k = mω2

∴ k/m = ω2

Formula for velocity

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= - ω2x

Separating the variables,

v dv = - ω2x dx

integrating both sides,

∴ = - ω2

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Where C is constant of integration.

at x = a, v = 0

∴ 0 =

∴ C =

Formula for displacement

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But,

Integrating both sides,

∴ sin-1

where α is constant of integration. It is the initial

phase of motion.

Special Cases

1. At t = 0, x = 0

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0 = a sin α

∴ sin α = 0

∴ α = 0 Thus, when the body starts moving from

the mean position, the initial phase is zero.

2. At t = 0, x = a

a = a sin α

∴ sin α = 1

∴ α = π / 2 Thus, when the body starts moving

from the extreme position, the initial phase angle

is π / 2

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