The Marina trench is located in the Pacific Ocean, and at one place it is nearly eleven km beneath the surface of the water. The water pressure at the bottom of the trench is about 1.1 × 108 Pa. A steel ball of initial volume 0.32 m3 is dropped into the ocean and falls to the bottom of the trench. What is the change in the volume of the ball when it reaches the bottom?
Here the solid Steel Ball placed in the water under high pressure is compressed uniformly at all Points. The force applied by the water acts in a perpendicular direction at each point on the surface of sphere This leads to decrease in its volume. The body develops internal restoring forces that are equal and opposite to the forces applied to it by water. The internal restoring force per unit is known as hydraulic stress and in magnitude is equal to the hydraulic pressure, since stress is restoring force per unit area. The strain produced by the pressure is called volume strain and is defined as the ratio of change in volume to the original volume. Since the strain is a ratio of change in dimension to the original dimension
The Force on Sphere, Restoring force by it and volume before and after change due to pressure has been shown in the following figures
Here the volume of ball will change and will decrease due to external pressure of water as there is always change in dimensions of a material upon application of an external force which depends upon the elastic property of that element for volumetric change we consider bulk modulus, now ball is made up of steel we have bulk modulus for steel is
B = 1.6 × 1011 N/m2
The force on the ball will be due to pressure on the surface of sphere and on every point on surface of sphere pressure will be
P = 1.1 × 108 Pa
Now the original volume of sphere without shrinking due to external pressure is
V = 0.32 m3
Not let us say change in volume of the ball be ΔV
So the strain on ball or the ratio of change in volume to original volume is given by ΔV/V
Now as explained earlier the external hydraulic pressure on the Ball is same as the stress on its surface
Now bulk modulus is the ratio of stress to strain so bulk modulus is given by
B = -P/(ΔV)/V
Where B is the bulk modulus, P is the excess pressure or the pressure on the surface of body due to restoring forces which are actually the volumetric stress, ΔV is the change in volume of the body initially having original volume V
So re-arranging we get
B = -(P × V)/(ΔV)
Or we can say change in volume
ΔV = -(P × V)/B
Putting values of P, V, and B we get
1Pa = 1 N/m2
Minus sign suggest that volume has decreased as compared to original So we get change in volume of sphere is 2.2 × 10-4 m3