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Mechanical Properties Of Fluids

Question
CBSEENPH11019114
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A particle executing S.H.M. of amplitude 25 cm and time period 3 s. What is the minimum time required for the particle to move between two points 12.5 cm on either side of the mean position?  

Solution
Here comma space Amplitude comma space straight A space equals space 25 space cm semicolon space straight T space equals space 3 space straight s Let space the space particle space be space at space location space minus 12.5 space cm space at space time space straight t subscript 1 space and space plus 12.5 space cm space at space time space straight t subscript 2. space Now comma space usign space the space relation comma space straight x space equals space straight a space cos space open parentheses fraction numerator 2 πt over denominator straight T end fraction space plus space straight phi close parentheses First space condition comma space minus 12.5 space equals space 25 space cos space open parentheses fraction numerator 2 πt subscript 1 over denominator 3 end fraction space plus space straight phi close parentheses space space space space space... space left parenthesis straight i right parenthesis thin space Second space condition comma space 12.5 space equals space 25 space cos space open parentheses fraction numerator 2 πt subscript 2 over denominator 3 end fraction space plus space straight phi close parentheses space space space space space space space space... space left parenthesis ii right parenthesis space From space left parenthesis straight i right parenthesis comma space cos space open parentheses fraction numerator 2 πt subscript 1 over denominator 3 end fraction space plus space straight phi close parentheses equals negative space fraction numerator 12.5 over denominator 25 end fraction space equals space 1 half space equals space cos space fraction numerator 2 straight pi over denominator 3 end fraction therefore space fraction numerator 2 πt subscript 1 over denominator 3 end fraction space plus space straight phi space equals space fraction numerator 2 straight pi over denominator 3 end fraction rightwards double arrow space 2 πt subscript 1 space plus space 3 straight phi space space equals space 2 straight pi space space space space space space space space space space space space space space space space space space space space space... space left parenthesis iii right parenthesis equals negative fraction numerator 12.5 over denominator 25 end fraction space equals space 1 half space equals space cos space straight pi over 3 therefore space fraction numerator 2 πt subscript 2 over denominator 3 end fraction space plus space straight phi space equals space straight pi over 3 rightwards double arrow space space space 2 πt subscript 2 space plus space 3 straight phi space space equals space straight pi space space space space space space space space space space space space space space space space... space left parenthesis iv right parenthesis Subtracting space left parenthesis iv right parenthesis space from space left parenthesis iii right parenthesis comma space we space get space space space space space space space 2 straight pi space left parenthesis straight t subscript 1 space minus space straight t subscript 2 right parenthesis space equals space straight pi space rightwards double arrow space left parenthesis straight t subscript 1 space minus space straight t subscript 2 right parenthesis space equals space fraction numerator straight pi over denominator 2 straight pi end fraction space equals space 1 half space equals space 0.5 space straight s
0.5 seconds is the minimum time required to move between two points. 

Some More Questions From Mechanical Properties of Fluids Chapter

Define frequency.

What is the condition for motion to be simple harmonic motion?

In the displacement equation y = r sin(ωt/ + ϕ) what does ϕ called and what information does it give?

What determines the natural frequency of an oscillator?

Is there any relation between uniform circular motion and simple harmonic motion?

What is periodic motion?

Write down the expression for the instantaneous velocity and acceleration of particle vibrating in simple harmonic motion.

How do you define the mean position of particle executing simple harmonic motion?

Define amplitude.

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