Need help verifying symbolic calculations for an FBD statics problem

[rayjbryant]

Homework Statement

A uniform slender beam of mass M has its center of gravity as shown below. The corner on which it rests is a knife; hence the reaction N is perpendicular to the beam. The vertical wall on the left is smooth. What is the value of the angle θ in terms of l and a for equilibrium?

Relevant Equations

Equilibrium equations for any generic statics problem. Sum of moments = 0 and so on.

So basically, I got close to a zero for my solution to this problem. I'm guessing based on the posted solutions that I wasn't able to get partial credit
due to the fact that my coordinate axis was aligned with the slender bar and not in its usual perpendicular position.

This resulted in different equilibrium equations, and I'm guessing I made an error somewhere or my assumptions resulted in a wrong final answer. Rather than follow along with my steps, the TA marked everything off because my work leading up to the final answer didn't match his. I guess I'm
curious where exactly I went wrong with my solution. I'm also curious as to how in the official solutions, we're able to completely ignore the affect of the wall. Even though its frictionless, the path of the slender beam is an arc that would go through the wall. So there must be some kind of "point force" at the wall, which is what I included in my calculations.

Thanks,
Raymond Bryant

 

[Master1022]

Hi,

I'm also curious as to how in the official solutions, we're able to completely ignore the affect of the wall.

We are taking moments at point $O$ and thus all forces there can be ignored as the distance between the line of action and the pivot is $0$.

Why is there no vertical wall force for the vertical force equation?
The wall is frictionless. The wall only provides a normal contact force $R$ in this instance. Friction would create a vertical component $\leq \mu R$. There is no other vertical force to consider.

I'm guessing I made an error somewhere or my assumptions resulted in a wrong final answer.

For your second moments equation (I think it is about $N$), I think your term with $F_0$ is incorrect. Have you treated it as a vertical force? If so, that is probably what led your solution to the wrong answer. As mentioned above it should be horizontal and thus the contribution is $F_0 a tan(\theta)$ if I am not mistaken... By including this vertical force (which shouldn't be there), the problem becomes more complicated. As shown in the solutions, this problem only requires two different equations (to find two ratios of $M/N$ and set them equal to one another). I would just use the moments about O and the vertical equation to simplify the problem.

Even though its frictionless, the path of the slender beam is an arc that would go through the wall. So there must be some kind of "point force" at the wall, which is what I included in my calculations.

I don't see where this "point force" is coming from and I don't think it is needed for this problem. In general for these basic statics problems, contact forces are normal (other components can arise if we have friction).

I hope this is of some help. In short, I think your inclusion of this extra force is what caused issues in your solution. If not, let me know and I can explain further.

 

[rayjbryant]

Thanks for the reply,

The force acts perpendicular to the slender rod because tangents and radii are always perpendicular. The rod wants to move in a circular arc around point N, and therefore is resisted by a tangent force at point O. The force at the wall is vertical if the coordinate axis is aligned with the slender beam. (This is the logic I used)

If there is indeed only a horizontal component of the force at the wall, I'm curious as to why the opposite and equal reaction at the rod's end at point O would be horizontal instead of a force with both x and y components if our coordinate system is globally aligned. I figured they told us to assume a frictionless wall so that we don't have to worry about another x-component force acting downward at that point.

 

[Lnewqban]

Please, see:
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-friction/

 

[rayjbryant]

I read the article over, I believe the wedge example somehow pertains to my issue, but I'm not sure I get it yet.

 

[Lnewqban]

I mainly posted for clarification about the direction of the reactive force from the wall.
Its direction should be perpendicular to the plane of the wall.
Just imagine that there is a microscopic wheel by the end of the beam in contact with the wall.

If we could tilt the wall in any direction, the reactive force could have no other direction than perpendicular to the tilted wall and going through the axis of that little wheel.

 

[rayjbryant]

Okay, I get it now, thank you very much.