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Mechanical Properties Of Fluids

Question
CBSEENPH11019200

A particle executes simple harmonic motion of amplitude 16cm and time period 2.4s. What is the minimum time taken by the particle to move between two points 8cm on either side of mean position?

Solution

The position of the particle executing simple harmonic motion is given by
space space space space space space space space space space space space y equals A space sin fraction numerator 2 pi over denominator T end fraction t
G i v e n space h e r e comma space

A m p l i t u d e space o f space t h e space p a r t i c l e comma space A equals 16

T i m e space p e r i o d comma space T equals 2.4 space s space

therefore space y equals 16 space sin fraction numerator 2 pi over denominator 2.4 end fraction t

Let the position of particle be +8 cm at tand -8cm at t2. 

therefore space space space 8 equals 16 space sin space fraction numerator 2 straight pi over denominator 2.4 end fraction straight t subscript 1 space

rightwards double arrow space space space space 1 half equals space sin fraction numerator 2 straight pi over denominator 2.4 end fraction straight t subscript 1 space

rightwards double arrow space space space space fraction numerator 2 straight pi over denominator 2.4 end fraction straight t subscript 1 equals straight pi over 6 or fraction numerator 5 straight pi over denominator 6 end fraction

rightwards double arrow space space space space space space straight t subscript 1 equals 0.2 straight s space or space 1.0 straight s

and space minus 8 equals 16 space sin fraction numerator 2 straight pi over denominator 2.4 end fraction straight t subscript 2
rightwards double arrow space space minus 1 half equals sin fraction numerator 2 straight pi over denominator 2.4 end fraction straight t subscript 2 space end subscript

rightwards double arrow space space fraction numerator 2 straight pi over denominator 2.4 end fraction straight t subscript 2 equals fraction numerator 7 straight pi over denominator 6 end fraction or fraction numerator 11 straight pi over denominator 6 end fraction

Therefore the time taken by particle to move between two points 8cm on either side of mean position is,

straight t subscript 2 minus straight t subscript 1 equals 1.2 straight s space or space 0.4 straight s space or space 2 straight s

The minimum time is 0.4s.