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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Simple Harmonic Motion Page 1

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Periodic Motion

 


Any motion that repeats itself at regular interval of time is called periodic motion. Periodic motion can be described in terms of harmonic function, (sine and cosine functions) and hence is also called harmonic motion.

Examples: (1)

Rotation of the earth

(2)

Motion of the bob of a simple pendulum

(3)

Motion of the hands of a clock


 

Oscillatory or Vibratory motion

 


If a particle performing periodic motion moves back and forth over the same path, its motion is called oscillatory or vibratory.

Examples: (1)

Motion of the bob of a simple pendulum

(2)

Motion of a mass attached to a spring

(3)

Motion of the prongs of an excited tuning fork

Definition of Linear Simple Harmonic Motion

 


Linear simple harmonic motion is defined as the linear periodic oscillatory motion of a body under the influence of a restoring force which is always directed towards a fixed point in its path (equilibrium position) and is of magnitude directly proportional to the magnitude of the displacement of the body from the fixed position.

Hence, restoring force,

F -x
F =
-kx


where, k is a proportionality constant called the force constant. The negative sign indicates that the direction of the force is always opposite to the displacement.

Terms Used in the Study of S.H.M.


One Oscillation: One oscillation is defined as a complete to and fro motion of the oscillating body.

Period (T):
The period of S.H.M. is defined as the time taken by the body to complete one oscillation.
Unit of Period: Second in both SI and CGS systems.

Frequency (
n): The frequency of S.H.M. is defined as the number of oscillations completed by a body in one second.

Unit of Frequency: Hertz in both S.I. and C.G.S. systems.
In time T seconds, a body completes one oscillation.

\ Number of oscillations in one second = 

1

 = frequency, n

T

 

\  

1

 = n

T


Amplitude (A): The maximum displacement of a body in S.H.M. from its mean position is called the amplitude of S.H.M.

Unit of Amplitude: metre in the SI system, centimetre in the CGS system.

Path Length or Range: The total length of the path over which the particle performing SHM moves is called the path length or range of SHM.

Path Length = 2A, where A is the amplitude.

Unit of Path Length: metre in the SI system, centimetre in the CGS system.

Uniform Circular Motion (U.C.M.), Simple Harmonic Motion (S.H.M.), Phase of S.H.M.

 


S.H.M. as a projection of U.C.M.


Consider a particle performing U.C.M. with an angular velocity
w along the circumference of circle of radius A.

At
t = 0, the particle is at P.
At
t = t, the particle is at P'.
Let the angle made by OP with the Y
- axis be a m POP' = wt.
AB is any diameter of the circle.
Let particle M be another particle such that its position always coincides with the foot of the perpendicular from P to AB. Thus M is the projection of particle P on the diameter AB.

It is clear that as P moves along the circumference of the circle, M moves to and fro along diameter AB.

When particle P completes one revolution, particle M completes one oscillation.

In
DOP'M', P'OM' = [90 - (wt + a)]
OM'= x = OP' cos [90 - (wt + a)]
\ x = A sin (wt + a)
x is the displacement of particle M from O.
Velocity is the time rate of change of displacement.

\ Velocity n =  

dx

 = 

d

 [A sin (wt + a)]

dt

dt

 

i.e. n =  

A w cos w /t


Acceleration is the time rate of change of velocity.

\ acceleration a =  

dn

 = 

d

 [A w cos (wt + a)]

dt

dt

 

 

= - Aw2 sin (wt + a)


since
x = A sin (wt + a), a = -w2 x
w is a constant.
Hence, from the above relation,
a -x

The acceleration of the particle M is directly proportional to its displacement and oppositely directed to the displacement

Hence, particle M performs linear S.H.M. M is a projection of a particle P that performs U.C.M.
Thus, projection of U.C.M. is a linear simple harmonic motion.

Differential Equation of S.H.M.

The restoring force acting on a body performing linear S.H.M. is given by the relationship F =
- kx, where, k is the force constant and x is the displacement.

From
Newton's second law of motion,
F =
ma, where, m is the mass of the particle and a is its acceleration.

Equating the above equations,
ma = - kx

\ a =   

-k

 .x

m

 

But a =  

dn

 =

d

(

t

dx

)=

dt

d2x

dt2

dt

 

\ 

d2x

 = 

-k

 .x

dt2

m

 

or 

d2x

 + 

k

 x = O

dt2

m


This expression is called the second order differential equation of linear S.H.M.

 

 

 

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