Who understands this tension problem?

A holiday decoration consists of two shiny glass spheres with masses 0.0240 kg and 0.0360 kg suspended from a uniform rod with mass 0.120 kg and length 1.00 m. The rod is suspended from the ceiling by a vertical cord at each end, so that it is horizontal. Calculate the tension in each of the chords A through F.

Here's the figure.

I need help on Tensions E and F. The solution manual doesn't explain sufficiently. I'm confused by the equations of equilibrium that have to be set up; explain if you can.


Starting with the lower mass (.036), the only forces acting on it are the tension in cord A (Ta) and it's weight (mg) so

Ta = mg = 0.036(kg) * 9.8 (m/s^2) = 0.353 N

For the other mass, the forces are Ta (down), Tb (up), and weight (down)

Tb = Ta + mg = 0.353 + 0.024*9.8 = 0.588 N

Consider the point where B, C, and D intersect. The forces acting on this point must sum to zero.


Tc * cos(53.1) = Td * cos(36.9)

and Vertical:

Tc * sin(53.1) + Td * sin(36.9) = Tb ...(1)


Tc = Td * (cos(36.9)/cos(53.1)) ...(2)

substitute this in (1) and solve for Td

Td = Tb / [sin(36.9) + sin(53.1)*cos(36.9)/cos(53.1)] = 0.353 N

using this value of Td in (2):

Tc = 0.470 N

To get Te and Tf, pick a couple of points on the rod and sum the torques. They must add up to zero. If you choose the point where cord E joins the rod...

(clockwise torque = counterclockwise torque)

(torque due to D + torque due to C + torque due to mass of rod) = torque due to Tf

0.2*Td*sin(36.9) + 0.8*Tc*sin(53.1) + 0.5*0.120*9.8 = 1.0*Tf

Tf = 0.931N

and where F joins the rod...

1.0*Te = 0.2*Tc*sin(53.1) + 0.8*Td*sin(36.9) + 0.5*0.120*9.8

Te = 0.833N

Note that the sum of the tensions in E and F should equal the sum of the three weights:

0.931 + 0.833 = 9.8 * (0.036 + 0.024 + 0.120)

1.764 = 1.764