Material shear and normal stress question

[Redoctober]

Homework Statement

Everything needed is written in the question ( Open the attached image )
Nyhow the required is to find the value of P.

Homework Equations

$$Shear stress = \frac{ \Delta V }{\Delta A}$$ eq 1
$$normal stress = \frac{ \Delta N }{\Delta A}$$ eq 2

The Attempt at a Solution

For the coned part,

I know that

$$r = (h + 25)/1000 meters$$ so is substute in

$$A = 2 \pi r h$$ to get

$$A = 2 \pi \frac{(h^2 + 25h)}{1000}$$ therefore taking the derivative DA/DH give

$$dA = \frac{4 \pi h + 50 \pi}{1000} .dh$$ substute to derivative of eq 2 and then integrate wiith respect to h for the boundries 0 < h < 50*10^-3 m and then multiplying by sin (45) to get sum of vertical force.

For the cylindrical part,

Just multiply A = 2*pie*r*30*10^(-3) to shear stress to get the force.
suming up the two force should equal P. however i got it wrong :/ !

So any ideas of how to solve this beast ??

 

[KataKoniK]

Thank you for your question. It seems like you are on the right track with your approach. However, there are a few things that need to be clarified in order to solve this problem correctly.

Firstly, the shear stress equation you have written is incorrect. The correct equation is:

Shear stress = \frac{F}{A}

Where F is the force and A is the area on which the force is acting. This means that in order to find the force, you need to multiply the shear stress by the area, not the other way around.

Secondly, in order to find the value of P, you need to find the sum of the vertical forces acting on the coned and cylindrical parts. This means that you need to integrate the normal stress equation (eq 2) with respect to h, not the shear stress equation.

Finally, when integrating with respect to h, you need to use the correct bounds for the coned and cylindrical parts. For the coned part, the bounds should be 0 < h < 50*10^-3 m, as you have correctly stated. However, for the cylindrical part, the bounds should be 50*10^-3 m < h < 80*10^-3 m, as the cylindrical part starts at h = 50*10^-3 m and ends at h = 80*10^-3 m.

I hope this helps you to solve the problem correctly. Good luck!