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Mechanical Properties Of Fluids

Question
CBSEENPH11020264

A pendulum made of a uniform wire of cross-sectional area A has time period T. When an additional mass M is added to its Bob, the time period changes TM. If Young's modulus of the material of the wire is Y, then 1/Y is equal to (g = gravitational acceleration)

  • open square brackets open parentheses straight T subscript straight M over straight T close parentheses squared minus 1 close square brackets straight A over Mg
  • open square brackets open parentheses straight T subscript straight M over straight T close parentheses squared minus 1 close square brackets Mg over straight A
  • open square brackets 1 minus space open parentheses straight T subscript straight M over straight T close parentheses squared close square brackets straight A over Mg
  • open square brackets 1 minus space open parentheses straight T subscript straight M over straight T close parentheses squared close square brackets Mg over straight A

Solution

A.

open square brackets open parentheses straight T subscript straight M over straight T close parentheses squared minus 1 close square brackets straight A over Mg

Time period, straight T space equals space 2 straight pi square root of straight L over straight g end root
When additional mass M is added to its Bob
straight T subscript straight M space equals space 2 straight pi square root of fraction numerator straight L plus increment straight L over denominator straight g end fraction end root comma where Δ is increase in length
As we know that,
straight Y space equals space fraction numerator Mg divided by straight A over denominator increment straight L divided by straight L end fraction space equals space fraction numerator MgL over denominator straight A increment straight L end fraction
rightwards double arrow increment straight L space equals space MgL over AY
straight T subscript straight M space equals space MgL over AY
straight T subscript straight M space equals space 2 straight pi square root of fraction numerator straight L plus begin display style MgL over AY end style over denominator straight g end fraction end root
rightwards double arrow open parentheses straight T subscript straight M over straight T close parentheses squared space equals space 1 plus Mg over AY
or space Mg over AY equals space 1 minus open parentheses straight T subscript straight M over straight T close parentheses squared
1 over straight Y space equals space straight A over Mg open square brackets open parentheses straight T subscript straight M over straight T close parentheses squared minus 1 close square brackets