Solve Statics Problem: Find Theta Angle Balance

[Ortix]

I basically have to solve the system below for which angle theta it's in balance. Meaning all forces equal out to 0. The answer is in terms of the variables.

It's been a while since I've done statics like this but it has come back to bite me in the *** again.

I started out by moments about A:
2L*cos(θ)*Cx - P*L*sin(θ)

But now what? If I do sum of forces in X I get that Ax = Cx and then I get stuck with the sum of forces in Y

Do I have to make a cut somewhere? Not sure about to go about solving with the spring.

The answer is:
cos(θ) = 0.5*(1+(P/2kL))

Any ideas?

 

[mfb]

At C, there is an unknown horizontal force C<->wall, a vertical force (depends on the position of C only via the spring) and an unknown force along BC. You can relate theta to the position of C, and that allows to determine the two unknown forces.
Consider the equlibrium at B to relate the force in the lower arm to the force in the upper arm. That might be sufficient to solve the problem, otherwise consider forces at A.

 

[Ortix]

with the unknown force along BC do you mean the normal force inside the beam?

 

[haruspex]

with the unknown force along BC do you mean the normal force inside the beam?

The compression force in the beam. Note that because of the free joints at each end, there can be no torque passed through the beam.

 

[paranormal]

I would approach this problem by first identifying all the forces acting on the system and their respective directions. From there, I would apply the principles of equilibrium, which state that the sum of all forces in the x and y directions must equal zero, and the sum of all moments about any point must also equal zero. In this case, the point A seems like a good choice for calculating moments.

To solve the problem, I would start by setting up the equations for the sum of forces and moments in terms of the variables given. This would involve using trigonometric functions to represent the forces in their respective directions. From there, I would manipulate the equations to solve for the unknown variable, in this case, theta.

I would also consider making a cut in the system to isolate the spring and analyze it separately. This would involve using Hooke's Law to relate the force exerted by the spring to its displacement.

Once I have the equations set up, I would use algebraic manipulation and possibly calculus techniques to solve for theta. I would also check my answer by plugging it back into the original equations to ensure that all forces and moments are balanced.

In conclusion, solving this statics problem requires a thorough understanding of the principles of equilibrium and the ability to manipulate equations and solve for unknown variables. By carefully analyzing the system and applying these principles, the desired answer can be obtained.