# Why have we taken different areas for both the cases?

My teacher had solved an example question related to stress and strain. The question is as follows:

If a hollow glass sphere of inner radius $$r$$ and thickness $$\Delta r ~ (\Delta r \ll r)$$ is filled with some gas, then find the maximum pressure the gas can exert before the sphere breaks. Given: the breaking stress for the glass sphere is $$S_b$$

Below I have attached a picture of the solution that the teacher had provided to us.

Now, I get that the sphere would break when the stress being applied by the gas on the sphere would be equal to the breaking stress, but then why has the teacher taken two different areas? for one side, its $$\pi r^2$$ and for the other side its $$2\pi r \Delta r$$. From what I understand stress is internal force per unit area being applied by an elastic object to resist deformation, so shouldn't the area be the same? The solution is pretty straightforward.

First of all you need to know what stress is. Stress is nothing but the force across a certain area. We can express the stress $$\sigma$$ (your teacher named it $$S$$) as the quotient of force $$F$$ and area $$A$$

$$\begin{equation} \sigma = \frac{F}{A} \quad . \end{equation}$$

If the force produced by the gas ($$F_g$$) exceeds the force that holds the sphere together ($$F_b$$), the sphere breaks. We can write down the force balance equation while making use of the force-stress relation from above for the breaking force $$F_b$$.

\begin{align} F_g &= F_b \\ F_g &= \sigma_b \, A_b \end{align}

Lastly, we have to identify the force produced by the gas which can be expressed in terms of the gas pressure $$\rho$$ ($$F_g = \rho A_r$$) and the area that is related to the breaking stress. The gas pressure acts on the inside of the glass sphere where the surface area is given as: $$A_r = \pi r^2$$.

If the glass sphere breaks, the wall shatters. This is why the breaking stress is related to the area of the glass sphere wall. We can compute the area of this annulus by substracting the area of the bigger circle from the smaller one.

\begin{align} A_b &= \pi [(r + \Delta r)^2 - r^2] \\ &= \pi [r^2 + 2r \Delta r + \Delta r^2 - r^2] \\ &\approx \pi 2r \Delta r \end{align}

The last line follows from $$\Delta r \ll r$$ which lets us omit the even smaller term $$\Delta r^2$$.

Putting everything together, leaves us with the desired result

\begin{align} F_g &= \sigma_b \, A_b \\ \rho A_r &= \sigma_b A_b \\ \rho \pi r^2 &= \sigma_b \pi 2r \Delta r \\ \rho &= \frac{\sigma_b 2 \Delta r}{r} \quad . \end{align}

The gas pushes on all the inner surface of the half spheres, so the blue arrows show the force due to the gas trying to 'break' the sphere.

For example the gas might push the left half sphere to the left with $$100N$$ and the right half sphere to the right with $$100N$$.

The material of the sphere that can resist the breaking is shown by red shading. This opposing force is from the stress in the material times the red shaded area, if the $$\Delta r$$ were bigger, then the material would be under less stress.