Finding the contact force in a hinge?

1. May 18, 2012

SopwithCamel

I have been working on a slightly tricky calculus problem and was hoping someone could verify my solution.

1. The problem statement, all variables and given/known data

A 4m long beam whose mass is 5 kg is attached to the wall by a wire and a hinge that allows the beam to rotate. There is a sign (with a mass of 20 kg) attached to the beam. The system is in equilibrium when the beam is at 20° to the horizontal. What is the contact force in the hinge (magnitude and direction)?

2. Relevant equations

3. The attempt at a solution

I first found the tension in the wire by setting the net torque to zero:
0=|r||T|sinθ + |rs||Fs|sinθ - |rb||Fb|sinθ
(where Fs is the force of the sign and Fb is the force of the beam and rs and rb is the distance of the forces from the hinge, respectively)
0=(4)(T)sin135 - (4)(196)sin110 - (2)(49)sin110
Solving for T, I get the tension in the wire = 293.02 N.

Then, to find the force in the hinge, I found two equations by resolving it into it's vertical and horizontal forces:
Horizontal: Tcos25 = Fcosθ ---> Fcosθ = 265.574N
Vertical: 0 = Fsinθ + Tsin25 - 49 - 196 ---> 245 = Fsinθ + 123.574N ---> Fsinθ = 121.161N

From there, I was able to solve... θ = 24.5° and F = 291.9N.

Is this correct? I hope this isn't asking too much... Please tell me if there are any errors in my figuring.

ALSO ... A quick question: When it says the beam is in equilibrium at 20° to the horizontal, does that imply that it is not at equilibrium at any other point? What exactly does it mean in this scenario to be "in equilibrium"?

Thank you!

2. May 19, 2012

vela

Staff Emeritus
This isn't a calculus problem. I've moved the thread to the intro physics forum.