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Relation Between Volumetric Strain, Lateral Strain And Poisson’s Ratio - Practice Questions & MCQ

Edited By admin | Updated on Sep 25, 2023 25:23 PM | #NEET

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  • 11 Questions around this concept.

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A bottle has an opening of radius a and length b.  A cork of length b and radius (a + a) where (a<<a) is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is B and frictional coefficient between the bottle and cork is µ then the force needed to push the cork into  the bottle is :

 

Concepts Covered - 1

Relation Between Volumetric Strain, Lateral Strain and Poisson’s Ratio

Let us long rod have a length L and radius 'r', then volume of this rod = \pi r^2L . . . . . . . . (1)

Now, Differentiating both the sides of (1), we get

                                                             d V=\pi r^{2} d L+\pi 2 r L d r

Now, dividing both the sides by volume of rod , i.e., \pi r^2L, we get , 

                                                            \frac{d V}{V}=\frac{\pi r^{2} d L}{\pi r^{2} L}+\frac{\pi 2 r L d r}{\pi r^{2} L}=\frac{d L}{L}+2 \frac{d r}{r} . . . . (2)

So we can say that,

Volumetric strain = Longitudinal strain + 2(Lateral strain)

Also, equation(2) can be written as, 

                                                     \Rightarrow \frac{d V}{V}=\frac{d L}{L}-2 \sigma \frac{d L}{L}=(1-2 \sigma) \frac{d L}{L}

                              This is because , \left[\mathrm \ \sigma=\frac{-d r / r}{d L / L} \Rightarrow \frac{d r}{r}=- \sigma \frac{d L}{L}\right]

 

Special case -

  •  When \sigma = 0.5, then dV = 0. It means that the substance is incompressible, so there is no change in volume.

 

  • If a material having \sigma = 0, it means lateral strain is zero. So, when a substance is stretched its length increases without any decrease in diameter. For example - cork has \sigma = 0. Also, in this case change in volume is maximum.

 

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Relation Between Volumetric Strain, Lateral Strain and Poisson’s Ratio

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Relation Between Volumetric Strain, Lateral Strain and Poisson’s Ratio

Physics Part II Textbook for Class XI

Page No. : 238

Line : 4

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