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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Simple Harmonic Motion Page 2

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We have,  

d2x

 = 

-k

 x

dt2

m

 

i.e., a =  

-k

 x

m

i.e., a = -w2x

 

 

 

 

 

where,   

k

 = w2

m


The dimensions of both are [MLT
-2]

A solution of differential equation 

d2x

 + w2x = 0

dt2

 

d2x

 + w2x = 0

dt2

 

\ 

d2x

 = -w2x

dt2

 

\ 

dn

 = - w2x

dt

...(1)

\  

d2x

 = 

d

(

dx

)

dt2

dt

dt

 

 

dn

dt

 

Now,  

dn

 = 

dn

  

dx

 = 

dn

 .n

(2)

dt

dx

dt

dx

 

where n =   

dx

 is the velocity of the particle

dt

 

From (1) and (2); n   

dn

 = -w2x

dx


\ ndn = -w2xdx

On integrating,  

n2

 = 

-w2x2

 + C, where C is the constant of integration.

2

2


When
n = 0, x = A

\ C =   

w2A2

2

 

\ 

v2

 = 

-w2x2

 + 

w2A2

2

2

2


\ n = w

This is the expression for the velocity of the particle in terms of its displacement.

Now, v

dx

 = 

 w  

(considering only the positive root)

dt

 

\   

dx

 = wdt

 

On integration, sin-1

(

x

)

 = (wt + a) where a is the constant of integration

A

which depends on the initial conditions.

\ x = A sin (wt + a)
This is the expression for the displacement of the particle.

Alternate Expressions For Velocity and Acceleration.
Since
x = A sin (wt + a)

Velocity, n

dx

 = Aw cos (wt + a)

dt

 

Acceleration, a =  

dn

 = - Aw2 sin (wt + a) = -w2x

dt


Maximum and Minimum Values of
x, n and a

a)

x = A sin (wt + a)

 

 

\ |x max| = A

when sin (wt + a) = 1

 

x min = 0

when sin (wt + a) = 0

 

 

 

b)

n = A w cos (wt + a)

 

 

\ |nmax| = Aw

when cos (wt + a) = 1

 

nmin = 0

when cos (wt + a) = 0

 

Also, n = w

 

 

\ n = nmax = wA

when n = 0

 

nmin = 0

when x = A

c)

a = -Aw2 sin (wt + a)

 

 

\ |amax| = Aw2

when sin (wt + a) = 1

 

and amin = 0

when sin (wt + a) = 0

 

also a = -w2x

 

 

\ |amax| = w2A

when x = A

 

and amin = 0

when x = 0


Expression for Time Period and Frequency

Time period T =  

2p

 , but w2 =  

k

, where k is the force constant and

w

m

m is the mass of the particle.
\ T = 2p

Also, since |a| = w2x

w2

|a|

x

 

 \ w =  


\ w =

T =  

2p

  =  

2p

w

 

Frequency n =  

1

 = 

w

 = 

1

 

T

2p

2p

 

 

 

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