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Chapter 15: Problem 74

The difference in air pressure between the inside and outside of a ball is a constant \(\Delta p .\) Show by direct integration that the net pressure force on one hemisphere is \(\pi R^{2} \Delta p,\) with \(R\) the ball's radius.

Short Answer

Expert verified
The net pressure force on one hemisphere of the ball is \(\pi R^2 \Delta p\).

Step by step solution

01

Define the element of the sphere

In order to find the net pressure force on one hemisphere of a ball, it is important to first distinguish a small surface area element of the sphere, say \(dA\), over which the force exerted can be calculated. Consider an element of area at the top of the sphere subtending a small solid angle \(d \Omega\). The surface area \(dA\) of this element is \(R^2 d \Omega\).
02

Calculate the force on the surface

Next, calculate the pressure force acting on the small area of the element. Since the entire external force is perpendicular to the surface, it is given by \(dF = \Delta p \, dA = \Delta p \, R^2 d \Omega\).
03

Calculate the net force

To obtain the total pressure force exerted on the hemisphere, it is necessary to add up (integrate) all the small forces over the entire hemisphere. Thus, the net force \(F\) is given by \(F = \int dF = \int \Delta p \, R^2 d \Omega\).
04

Integrate over the hemisphere

The integral of \(d \Omega\) over a hemisphere gives 2\(\pi\) (solid angle is \(2\pi\) steradians for a hemisphere). Therefore, substituting into the formula, the net force becomes \(F = \Delta p \, R^2 \int d \Omega = \Delta p \, R^2 \cdot 2 \pi = 2 \pi R^2 \Delta p \). However, the exercise requires the answer in terms of a single hemisphere, thus the net force on one hemisphere is \(\frac{F}{2} = \pi R^2 \Delta p\).

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