Question

In the Guericke experiment to show the atmospheric pressure, two copper hemispheres were tightly fitted to each other to form a hollow sphere and the air from the sphere was pumped out to create vacuum inside. If the radius is of each hemisphere is \[R\] and the atmospheric pressure is \[P\], then the minimum force required (when the two hemispheres are pulled apart by the same force) to separate the hemisphere is
A. \[2\pi {R^2}P\]
B. \[4\pi {R^2}P\]
C. \[\pi {R^2}P\]
D. \[\dfrac{{\pi {R^2}P}}{2}\]

Answer 1

Hint: Use the formula for pressure acting on an object. This formula gives the relation between the pressure acting on an object, the force acting on the object and the area on which the force is acting. Check what is the total pressure acting on the hollow spherical shape formed by the two hemispheres and the area on which the force is to be applied to separate the two hemispheres from each other. Use the formula for pressure and determine the necessary force.

Formula used:
The pressure \[P\] acting on an object is given by
\[P = \dfrac{F}{A}\] …… (1)
Here, \[F\] is the force acting on the object and \[A\] is the area on which the force is acting.

Complete step by step answer:
We have given that the two hemispheres of radius \[R\] are fitted into each other to form a hollow sphere of radius \[R\] and the air inside them is evacuated. Hence, there is no air or medium inside the hollow sphere formed by the two hemispheres. Therefore, the pressure inside the hollow sphere is zero. So, we can conclude that the only pressure acting on the hollow sphere is the atmospheric pressure.

We have given that the atmospheric pressure is \[P\]. When the two hemispheres are pulled apart, the force is to be applied on the circular portion which the two hemispheres form at their contact surface. Hence, the area on which the force is to be applied to separate the two hemispheres is \[\pi {R^2}\]. Rearrange equation (1) for the force required to separate the two hemispheres.
\[F = PA\]
Substitute \[\pi {R^2}\] for \[A\] in the above equation.
\[F = P\pi {R^2}\]
\[ \therefore F = \pi {R^2}P\]
Therefore, the force required to separate the two hemispheres is \[\pi {R^2}P\].

Hence, the correct option is C.

Note:The students should be careful while determining the area on which the force is acting. The atmospheric pressure is acting on the whole outer surface of the hollow sphere formed by the two hemispheres but the force required to separate two hemispheres from each other is only acting on the circular portion where the two hemispheres are in contact with each other and not on the whole outer surface of the hollow sphere.