A Question regarding stress/cross sectional area

[FaroukYasser]

Homework Statement

A cylindrical rod has a spherical bubble in it. as illustrated in figure 4.2 (in the attachments)
The rod has a cross sectional area of 3.2 x 10^-6 m^2 and is stretched by forces of magnitude 1.9 x 10^3 N.
The maximum Stress that the cylinder can take is 9.5 x 10^8 Pascals.

Homework Equations

Stress = Force/ Cross sectional area

The Attempt at a Solution

Basically what I did was say:
Maximum stress = Force applied/Area
9.5 x 10^8 = 1.9 x 10^3 / Minimum area
Minimum area = 2 x 10^-6 m^2
So the maximum Cross sectional area of a rod is 2 x 10^-6 m^2

The right answer was on the other hand to do all this + subtract new area from old area so it becomes (3.2 - 2) x 10^-6 = 1.2 x 10^-6
This is the part I don't understand. Shouldn't the maximum cross section of the ball be the area of the new cylinder so that if it is more then the cylinder breaks?

 

[Chestermiller]

Hi Farouk,

I'd like to help with this, but I don't see the actual question that is being asked. What are they asking you to find?

Chet

 

[FaroukYasser]

Sorry forgot to include it . It was: what is the maximum cross section of the ball.
Its really vague what is asked here :/

 

[Chestermiller]

The bubble can't support any stress, so all the stress must be carried by the 2x10^-6 m² of rod material surrounding the bubble. That leaves 1.2x10^-6 m² cross sectional area available for the bubble.

Chet

 

[HalfLostAndSearching]

I understand your confusion with the solution provided. The key concept to understand here is that the maximum stress that the cylinder can take is determined by the material properties of the cylinder, not the size or shape of the bubble inside it. The maximum stress value given (9.5 x 10^8 Pascals) is a property of the material and indicates the point at which the material will fail or break.

In this case, the solution is taking into account the fact that the overall cross sectional area of the cylinder is reduced due to the presence of the bubble. So, instead of simply calculating the minimum area based on the force applied, the solution is also taking into consideration the reduced area of the cylinder due to the bubble.

To put it simply, the maximum stress that the cylinder can take is determined by its material properties, not the size or shape of the bubble inside it. Therefore, the solution provided is correct and takes into account the reduced cross sectional area of the cylinder due to the bubble.