Question

A farmer making grape juice fills a glass bottle to the brim and caps it tightly. The juice expands more than the glass when it warms up, in such a way that the volume increases by 0.2% (that is, ΔV / V0 = 2×10−3 ) relative to the space available. Calculate the magnitude of the normal force exerted by the juice per square centimeter if its bulk modulus is 1.8×109 N/m2 , assuming the bottle does not break.

Answer 1

The magnitude of force per
`cm^(2)` is
`3.6* 10^(6)\ N/m^(2)`

Solution:

As per the question:

Increase in the volume,
`(\Delta V)/(V_(o)) = 2* 10^(- 3)`

Bulk Modulus, B =
`1.8* 10^(9)\ N/m^(2)`

Now, to calculate the Normal Force's magnitude, F exerted by the juice:

Volume change of an elastic substance on the application of a force is given by:

`(\Delta V)/(V_(o)) = (1)/(B)*((F)/(Area,\ A))`

`(F)/(Area,\ A) = B((\Delta V)/(V_(o)))`

Now, putting suitable values in the above eqn:

`(F)/(Area,\ A) = 1.8* 10^(9)* 2* 10^(- 3) = 3.6* 10^(6)\ N/m^(2)`

Here, F is the force exerted on the juice by the container per
`cm^(2)`, there an equal reaction force per
`cm^(2)` will be exerted by the juice on the container.

Answer 2

Explanation: Bulk Modulus (K) =
`(Volumetric Stress)/(Volumetric Strain)`

This is the same as saying it equal to the change in pressure with respect to the change in volume divided by initial volume.

Bulk modulus elasticity may also be expressed in terms of pressure and density:

Remember: Strain is a unit-less quantity.

`K=((p_1 - p_0))/(((\rho_1- \rho_0))/(\rho_0))`

Here,

• ρ0 and ρ1 are the initial and final density values.

• p₀ & p₁ are the initial and final pressure values.

Given data:

bulk modulus,
`k= 1.8 *10^(9) N.m^(-2)`

volumetric strain,
`\epsilon _(v)=2*10^(-3)`

To find: Force exerted by the juice per square centimeter,
`\sigma _(v)`

Solution:
`\sigma _(v)=\epsilon _(v)* K`

⇒
`\sigma _(v)= 2*10^(-3)*1.8 *10^(9)`

`\sigma _(v)= 3.6*10^(6) N.m^(-2) =3.6*10^(2)N.cm^(-2)`