Solved Examples on Deformable Bodies and Elastic Deformation:-

Question 1:-

The tension in a string holding a solid block below the surface of a liquid (of density greater than the solid) is T₀ when the containing vessel in the below figure is at rest.

Show that the tension T, when the vessel has an upward vertical acceleration a, is given by T₀(1+a/g).

Concept:-

The various forces acting on the block when it is inside the water are the weight, the buoyant force and the tension in the string. When the vessel is at rest, there is no net force acting on the block. The sum of the three forces is equal to zero when the vessel is at rest.

When the vessel moves upward with an acceleration, the buoyant force and the tension in the string will be differ. There is a net force acting vessel. Thus, the sum of the three forces is equal to the net force acting on the block.

The weight of the block having mass m which is acting downward is

W = mg

Here,

Acceleration due to gravity at the point of observation is g

The magnitude of the buoyant which acts upwards is

F_b = Vρg

Here,

Volume of the water displaced by the block is V

Density of the water is ρ

The various forces acting on the block which is placed inside the vessel is shown below:

As the vessel is rest, the three forces are in equilibrium. Thus, the sum of the three forces is equal to zero.

Thus,

∑F = 0

This makes

W-F_b-T₀ = 0

Here,

Tension in the string when the vessel is at rest is T₀.

Insert the values of the various terms involved in the above equation gives

F_b-W-T₀ = 0

Vρg-mg - T₀ = 0

T₀ = Vρg-mg

This represents the tension in the string when the vessel is at rest.

The buoyant forces acting on the block is

F_b,a = Vρ(g+a)

Thus, the net forces is given as

∑F = ma

It is equal to the sum of the buoyant force, the weight of the block and tension (T).

Thus,

F_b,a-W-T = ma

Substitute the values of F_b,a  and W  in the above equation gives

F_b,a-W-T = ma

Vρ(a+g) – mg - T = ma

T = Vρa+ Vρg -mg-ma

T = Vρg - mg + Vρa -ma

  = T₀ + a (Vρ-m)

Dividing both numerator and denominator in the second part in the right hand side of the equation by g gives

T = T₀+[(a/g) (Vρg – mg)]

  = T₀+[(a/g) T₀]

  = T₀[1+ (a/g)] 

This represents the tension in the string when the vessel is accelerating upward.

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Question 2:-

Using a soap solution for which the surface tension is 0.025N/m, a child blows a soap bubble of radius 1.40 cm. How much energy is expended in stretching the soap surface?

Concept:-

Surface tension is the surface potential energy per unit area of surface.

The surface area of the soap bubble blown by the child having radius r is

A = 4πr²

The surface tension of the soap bubble is

γ = U/2A

The factor 2 is due to the presence of two surfaces of the soap bubble.

Substitute A =4πr² in the above equation, the surface tension becomes

γ = U/2A

  = U/[2(4πr²)]

  = U/8πr²

Solution:-

From the above equation, the energy stored in the surface of the soap bubble is

U = 8γπr²     

This much of energy is used in stretching the soap surface.

To obtain required energy, substitute 1.40 cm for the radius r of the soap bubble, 0.025 N/m for the surface tension of the soap bubble γ in the equation U = 8γπr²

U = 8γπr²

  = 8(0.025 N/m) (3.14) (1.40 cm)² (10^-2 m/1 cm)²

 = (1.2308810^-4 N.m) (1 J/1 N.m)

= 1.2308810^-4 J

Rounding off to three significant figures, the energy expend in stretching the soap surface is 1.23×10^-4 J.

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Question 3:-

In 1654 Otto von Guericke, Burgermeister of Magdeburg and inventor of the air pump, gave a demonstration before the Imperial Diet in which two teams of horses could not pull apart two evacuated brass hemispheres. (a) Show that the force F required to pull apart the hemispheres is F = πR²Δp, where R is the (outside) radius of the hemispheres and Δp is the difference in pressure outside and inside the sphere in the below figure. (b) Taking R equal to 0.305 m and the inside pressure as 0.100 atm, what force would the team of horses have had to exert to pull apart the hemispheres? (c) Why were two teams of horses used? Would not one team prove the point just as well?

Concept:-

The two brass hemispheres with an open flat can be replaced with two hemispheres with a closed flat end.

The force required to pull apart the hemispheres is equal to pressure times the surface area.

(a) It is given that the pressure difference between outside and inside the sphere is Δp. The radius of the circle formed by the half hemispheres which face each other is R.

Now, the surface area of the hemisphere is

A = πR²

The force required to pull the apart the brass hemisphere is

F = ΔpA

  =Δp(πR²)

  = πR² Δp

Therefore, the force required to pull apart the two hemispheres is πR²Δp.

Thus, the pressure difference between the inside and outside of the hemispheres is

Δp = 1.00 atm – 0.11 atm

     = 0.89 atm

Substitute 0.305 m for R  and 0.89 atm   for Δp  in the equation F = πR²Δp,

F = πR²Δp

   = (3.14) (0.305 m)² (1.01×10⁵ Pa/1 atm) (1 kg/m.s²/1 Pa)

  = (2.5997×10⁴ kg.m/s²) (1 N/1 kg.m/s²)

 = 2.5997×10⁴ N

Rounding off to three significant figures, the force required to pull apart the two hemispheres by the team of horses is 2.60×10⁴ N.