Can someone please help me with this one? Thanks! :)?

Question

A 1000 N uniform boom is supported by a cable perpendicular to the boom, as in Figure P8.26. The boom is hinged at the bottom, and a 1900 N weight hangs from its top.
(see image link below)
http://www.webassign.net/sercp/p8-26alt.gif
(a) Find the tension in the supporting cable
(b) Find the components of the reaction force exerted on the boom by the hinge.
-to the right_________N
-upward ___________ N

Answer 1

The author was kind to set the angle of the cable such that the angle it subtends with the boom is exactly 90 degrees.
Using counter clockwise as positive
The torques are
Tension (T)
T*3*l/4

The boom
-COS(65)*1000*l/2

The suspended weight

-cos(65)*1900*l

Sum these and set =0 since the boom is static

T*3*l/4=
cos(65)*1000*l/2+
cos(65)*1900*l

Note that l divides out

T=(4/3)*1000*cos(65)*(.5+1.9)
T=1352 N

The reaction forces are found by first looking at the vertical component, which is the sum of vertical weights

1000+1900-1352*sin(25)

Vertical=2328 N upward

Horizontal is the sum of hortizontal forces

cos(25)*1352

Horizontal =1225 N to the right

I like to check my work by computing the resultant force vector magnitude two ways

First, sum the forces parallel to the boom

sin(65)*(1000)+sin(65)*1900
=2628 N
This is the resultant force vector

Then using Pythagorean, compute the resultant from the components

R=sqrt(2328^2+1225^2)
=2630

Close enough

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