# Internal stresses of accelerating body Refer to pics above. I have an accelerating body that has a fin extruded from the side. I want to analyze the stresses at the base of the fin that result from high accelerations. So, the body is accelerating at $a$. I begin by taking the sum of the moments at the cut (base of fin) and get the following: $$-M-F_{air}\frac{w}{2}=m_{fin}a\frac{w}{2}$$ where $M$ is the moment, $F_{air}$ is the total air drag force distributed over the entire face. I want to solve for the moment, $M$ as a function of d so that I can find the bending stresses and determine the thickness $d$ that prevent material failure. However, when I solve this equation and analyze the bending stress with: $$\sigma = \frac{Md}{2I_{zz}}$$ I get strange answers.. I must be doing something wrong.

It would be helpful if someone can walk me through this problem or suggest another way to analyze stresses as a result of shock / high accelerations

## 2 Answers

What do you mean by "strange answers"? Please be more specific.

If there is constant acceleration in empty space, and the force is applied to the main body, then there is a constant inertial force on the fin, causing a torque.

If the object is moving in air, drag force is proportional to $v^2$. If the body is accelerating then $v$ will be constantly increasing, so the drag force will also be constantly increasing.

I think you need to be looking at the constant velocity case when there is a constant drag force. This could be when the body reaches terminal velocity - ie the propulsive force equals the drag force.

• Strange answers as in unrealistically small d values. The air resistance force may be negligible, but i figured The acceleration would result in some sort of inertial stresses. So in a vacuum are you saying the body would not experience higher internal stresses at the base of the fin, even if the accelerating force is a point force at the center of the larger cube volume? I may have the wrong approach here, but id expect for there to be higher stresses at the base of the fin because when I picture this body accelerating I imagine the fin deflecting backwards – Mike James Johnson Apr 7 '17 at 21:18
• If the main body rests on the ground with the fin unsupported, there is a torque acting on the fin. If an applied (point) force on the main body (larger cube) causes an acceleration in empty space, there is likewise a torque on the fin. But if the body moves with constant velocity in empty space there is no such torque. If the body moves with constant velocity through air, due to point force on the larger cube, there is a torque on the fin due to air drag. If the body is accelerating, drag increases and at some point is no longer "negligible". – sammy gerbil Apr 7 '17 at 22:36
• I don't know what you mean by "higher" internal stresses. Higher than what? I think the difficulty here is that you are not being clear what the situation is : what forces are acting, and at what points are they applied? – sammy gerbil Apr 7 '17 at 22:40
• I understand what you are saying. If i grab my notebook by the right edge and accelerate it forward with my arm, the left edge deflects backwards. Does this happen ONLY because of air drag? If we repeat this experiment in a vacuum, and I accelerate my notebook forward while gripping the right edge, will the left edge still deflect due to internal inertial forces? In my mind, I think it would - am i wrong? Those are the stresses I want to measure. We can ignore Air drag for now. Will it still deflect/create internal stress at the base of the due to accelerating the larger cube? – Mike James Johnson Apr 7 '17 at 22:45
• Yes the left edge deflects due to inertial forces, because you are applying force only on the right edge. When applying the force yourself, you are unlikely to reach speeds at which air drag is significant. – sammy gerbil Apr 7 '17 at 23:00

Dear Sir: I am not a physicist so my comment may be misguided however the problem as I see it is that your problem does not account for the mass of the fin compared to the mass of the block.

Let's say the acceleration on the block is a constant $g$ or $10\ \mathrm{m/s^2}$. The block is being pushed to the back of the block is going to want to crunch forward into the front of the block. If the block were a hollow cube made of the plastic of a soda bottle for instance filled with air the constant movement forward would compress the air and the block from front to back would be shorter.

The fin however is attached to the front of the block. It is not being pressed forward at all since the acceleration or engine if you will is at the back of the block. So as the front of the block goes forward the fin is pulled forward via its rigid attachment to the front. Thus there will be a stress on the line where the fin attaches to the block as the fin tends to try to keep up (not a good metaphor). So what is the mass of the fin. This is important because Force equals mass time acceleration $F=ma$. In our example we set acceleration to $g$ or approximately $10\ \mathrm{m/s^2}$ so if we know the mass we have the force. Then you can determine the length of the fin attached to the block as well as the thickness. You then have to calculate the stress over the length since a larger fin up and down with the same mass would spread that force over a larger area.

To me this is what I think you are trying to solve. The physicists would have to develop the formula. Just my thoughts.

• His formula is only looking at the fin with the forces on it already resolved. This doesn't seem relevant. – JMac Sep 28 '17 at 23:02