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Question

A non-isotropic solid metal cube has coefficient of linear expansion as 5×105/C along the x-axis and 5×106/C along y-axis and z-axis. If the coefficient of volumetric expansion of the solid is C×106/C then the value of C is

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Solution

  • Let the side of the cube taken as l1,l2,l3 in x, y, z directions.
  • Hence the volumeV=l1×l2×l3 .....(1)
  • Let the change in volume beΔv
  • The coefficient of volume expansion can be calculated as:

Volume expansion coefficient=ΔvVΔt

  • ​Take log of (1) on both sides

lnV=lnl1+lnl2+lnl3

  • Differentiating the equation above.

ΔvV=Δl1l1+Δl2l2+Δl3l3

  • Multilply the eqaution by 1Δt
  • 1VΔvΔt=1l1Δl1Δt+1l2Δl2Δt+1l3Δl3Δt
  • Given that 1l1Δl1Δt=5×105/ oC, 1l2Δl2Δt=5×106/ oC
  • Then,​

    1VΔvΔt=5×105+5×106

    =6×106

    6×106=C×106

    C=6



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