3D Statics Equilibrium - Dot Everything with Vector P?

[absolutezer0es]

Homework Statement

Bead B has negligible weight and slides without friction on rigid fixed bar AC. An elastic cord BD has spring constant k = 3 N∕mm and 20 mm unstretched length, and bead B has a force of magnitude P in direction BC. If bead B is positioned halfway between points A and C, determine the value of P needed for equilibrium, and the reaction between bead B and rod AC.

Homework Equations

See the equations below.

The Attempt at a Solution

I'll summarize all the steps I've taken.

1) I found position vectors BC and BD.

2) Found unit vectors BC and BD.

3) Calculated vector P = P(u_BC).

4) Calculated F_BD using F=kδ.

5) Found force vector F_BD using vector F_BD = F_BD(u_BD).

Now from here is where I believe I'm having trouble. The sum of all the forces is equal to 0, and my FBD includes vector P, vector F_BD pointing towards D, and vector R, which is perpendicular to vector P at point B. I know:

P + F_BD + R = 0 [these are all vectors]

My strategy was to dot each of those terms with vector P, yielding:

(P⋅P) + (F_BD⋅P) + (R⋅P) = 0

Giving:

(P⋅P) + (F_BD⋅P) = 0, since the third term above is 0 (because they are at right angles).

Is this sound mathematics after my step 5, or should I approach this another way? I even tried dotting each vector with its appropriate unit vector (P with u_AB, F_BD with u_BC, and R with u_BC), but neither approach is getting me correct answers.

I'm actually closer with my initial approaching (dotting everything with P) - I think.

Ideas?

 

[haruspex]

Dotting with P has the benefit of eliminating R, but at the cost of introducing P.F_BD. Can you think of a vector to take the dot product with that gets rid of R without introducing that complication?

 

[absolutezer0es]

The only way I can think of to eliminate R by dotting the entire equation with a vector is something that is perpendicular to it.

In that case, would vector u_BC work?

My confusion then is with vector P. We've already defined vector P as vector P = P(u_BC). Would the equation then become:

(P⋅u_BC⋅u_BC)+(F_BD⋅u_BC) = 0, where P and F_BD are forces?

Another source of confusion - the second term above is still a vector. How do we reconcile that?

 

[haruspex]

The only way I can think of to eliminate R by dotting the entire equation with a vector is something that is perpendicular to it.

In that case, would vector u_BC work?

My confusion then is with vector P. We've already defined vector P as vector P = P(u_BC). Would the equation then become:

(P⋅u_BC⋅u_BC)+(F_BD⋅u_BC) = 0, where P and F_BD are forces?

Another source of confusion - the second term above is still a vector. How do we reconcile that?

There seems to be some notation confusion with P. You are using it both as a vector and as the magnitude of that vector. In your last equation above, it is the magnitude, no? The equation sums two terms, each being a dot product and hence a scalar.