Find the position of the center of mass of the bar, x, measured from the bar's left end.?

Question

A nonuniform, horizontal bar of mass m is supported by two massless wires against gravity. The left wire makes an angle phi_1 with the horizontal, and the right wire makes an angle phi_2. The bar has length L.

http://session.masteringphysics.com/problemAsset/1011052/17/MTS_st_24.jpg

Express the center of mass in terms of L, phi_1, and phi_2.

Answer 1

Let F₁  and F₂ be the forces exerted by lift and right wire on the bar. Both forces act upward in direction of the wires. So you can split the forces into x- and y-component.

F₁x = -F₁ · sin(φ₁)

F₁y = F₁ · cos(φ₁)

F₂x = F₂ · sin(φ₂)

F₂y = F₂ · cos(φ₂)

As always for such a plane system in equilibrium you can  balances horizontal forces(x) , vertical forces (y) and the torques about an arbitrary fulcrum.

Set up balance of forces acting on the bar in x-direction

F₁x + F₂x = 0

<=>

-F₁ · sin(φ₁) +  F₂ · sin(φ₂) = 0

<=>

F₁/F₂ = sin(φ₂)/sin(φ₁)

From the balance in y-direction you could calculate the magnitude of F₁ and F₂ but this is not the focus, so skip this step.

As fulcrum for torque balance choose the point directly above the center of mass at the upper edge of the bar. By this choice there are no torques exerted by the F₁x and F₂x nor the weight of the bar.

- F₁y · x + F₂y · (L-x) = 0

=>

x = L · F₂y / (F₁y  + F₂y)

<=>

x = L  / ( (F₁y/F₂y )  + 1)

express forces in terms of magnitude an angle

x = L  / ( (F₁/F₂)·(cos(φ₁)/cos(φ₂))  + 1)

use ratio derive from x-direction balance

x = L  / ( (sin(φ₂)/sin(φ₁))·(cos(φ₁)/cos(φ₂))  + 1)

<=>

x = L  / ( (tan(φ₂)/tan(φ₁) + 1)