Civil and Environmental Engineering center of mass problem

[Sandstormer]

Homework Statement

The uniform rod is bent into the shape of a parabola and has a weight per unit length of 9 lb/ft. Determine the reactions at the fixed support A

Homework Equations

Equation of the rod is: y^2=3x
From A the rod goes 3m up and 3m to the right

The Attempt at a Solution

I know that you have a length dL, which is dL = sqrt((dx/dy)+1) dy. Since the equation of the rod is given by y^2=3x, I solved it for x, giving y^2/3=x. Then i found dx/dy by deriving it, getting 2y/3=dx/dy. Now I substitute it in my dL equation, and now I have sqrt
dL=((2y/3)^2+1) dy. After simplifying this more, I end up with dL = 1/3sqrt(4y^2+9) dy. And now the weight of this small length dW is equal to the weight/unit length times dL. This means that dW = 9(1/3sqrt(4y^2+9) = 3sqrt(4y^2+9). To find the weight of this rod, I now have to integrate dW. The only problem is, I got stuck! I tried to integrate by substitution, but that didn't work. Any help is greatly aprreciated.

 

[KataKoniK]

I would like to offer some suggestions for solving this problem. First, it's important to understand the physical meaning of the problem. The uniform rod is bent into a parabolic shape, which means it is subjected to a bending moment and a shear force. The weight per unit length of the rod also adds to the load on the structure.

To determine the reactions at the fixed support A, we need to consider the equilibrium of forces and moments. The weight of the rod exerts a downward force along its entire length, while the fixed support at A exerts an upward reaction force to counteract this weight. Additionally, there will be a reaction moment at A to counteract the bending moment caused by the weight of the rod.

To solve for these reactions, we can use the equations of equilibrium: ΣFy = 0 and ΣM = 0. Since the rod is symmetric, we can consider only half of the rod and then multiply our results by 2 to account for the other half.

Starting with the moment equation, we can choose a point to take moments about. Let's choose point A as our reference point. This means that the reaction moment at A will be zero, since it is the reference point. The only other moment acting on the rod is the bending moment caused by the weight of the rod, which we can calculate using the equation M = Wx, where W is the weight per unit length and x is the distance from the reference point.

Next, we can use the force equation to solve for the reaction force at A. Since there is no horizontal movement at A, the sum of the horizontal forces must be zero. This means that the reaction force at A must be equal to the horizontal component of the weight of the rod.

Using the equation of the parabola given in the problem, y^2 = 3x, we can solve for the horizontal distance x at the point where the rod meets the support A. This will give us the distance from the reference point to the point where the reaction force is acting.

Finally, we can plug in our values for the weight per unit length, the distance x, and the length of the rod to solve for the reactions at the fixed support A.

I hope this helps in solving the problem. Remember to always consider the physical meaning of the problem and use the equations of equilibrium to solve for unknown forces and moments. Good