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If fraction numerator sin to the power of 4 space theta over denominator a end fraction plus fraction numerator cos to the power of 4 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction text  then  end text fraction numerator sin to the power of 8 begin display style text end text end style theta over denominator a cubed end fraction plus fraction numerator cos to the power of 8 begin display style space end style theta over denominator b cubed end fraction equals

  1. fraction numerator 1 over denominator left parenthesis a plus b right parenthesis cubed end fraction
  2. fraction numerator 1 over denominator left parenthesis a plus b right parenthesis squared end fraction
  3. a plus b
  4. fraction numerator 1 over denominator a plus b end fraction

The correct answer is: fraction numerator 1 over denominator left parenthesis a plus b right parenthesis cubed end fraction

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A piston fitted in cylindrical pipe is pulled as shown in the figure. A tuning fork is sounded at open end and loudest sound is heard at open length 13cm, 41 cm and 69 cm, the frequency of tuning fork if velocity of sound is 350 m s to the power of negative 1 end exponent is

In a closed organ pipe in which length of air-column can be increased or decreased, the first resonance occurs at lambda divided by 4 and second resonance occurs at 3 lambda divided by 4.
Thus, at first resonance
fraction numerator lambda over denominator 4 end fraction equals 13 blank horizontal ellipsis open parentheses i close parentheses
And a second resonance
fraction numerator 3 lambda over denominator 4 end fraction equals 41 blank horizontal ellipsis open parentheses i i close parentheses


Subtracting Eq.(i) from Eq.(ii), we have
fraction numerator 3 lambda over denominator 4 end fraction minus fraction numerator lambda over denominator 4 end fraction equals 41 minus 13
⟹ fraction numerator lambda over denominator 2 end fraction equals 28
therefore blank lambda equals 56 blank c m
Hence, frequency of tuning fork
v equals fraction numerator v over denominator lambda end fraction equals fraction numerator 350 over denominator 56 cross times 10 to the power of negative 2 end exponent end fraction equals 365 blank H z

A piston fitted in cylindrical pipe is pulled as shown in the figure. A tuning fork is sounded at open end and loudest sound is heard at open length 13cm, 41 cm and 69 cm, the frequency of tuning fork if velocity of sound is 350 m s to the power of negative 1 end exponent is

physics-General
In a closed organ pipe in which length of air-column can be increased or decreased, the first resonance occurs at lambda divided by 4 and second resonance occurs at 3 lambda divided by 4.
Thus, at first resonance
fraction numerator lambda over denominator 4 end fraction equals 13 blank horizontal ellipsis open parentheses i close parentheses
And a second resonance
fraction numerator 3 lambda over denominator 4 end fraction equals 41 blank horizontal ellipsis open parentheses i i close parentheses


Subtracting Eq.(i) from Eq.(ii), we have
fraction numerator 3 lambda over denominator 4 end fraction minus fraction numerator lambda over denominator 4 end fraction equals 41 minus 13
⟹ fraction numerator lambda over denominator 2 end fraction equals 28
therefore blank lambda equals 56 blank c m
Hence, frequency of tuning fork
v equals fraction numerator v over denominator lambda end fraction equals fraction numerator 350 over denominator 56 cross times 10 to the power of negative 2 end exponent end fraction equals 365 blank H z
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fraction numerator d y over denominator d t end fraction equals A cos invisible function application left parenthesis k x minus omega t right parenthesis open square brackets negative omega close square brackets
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And slope at x equals 0 and t equals 0is positive, in figure. Therefore, particle velocity is in negative y-direction.

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fraction numerator d y over denominator d t end fraction equals negative v open parentheses fraction numerator d y over denominator d x end fraction close parentheses
i e blank p a r t i c l e blank v e l o c i t y equals negative left parenthesis w a v e blank s p e e d right parenthesis cross times s l o p e.
And slope at x equals 0 and t equals 0is positive, in figure. Therefore, particle velocity is in negative y-direction.
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