How Do You Calculate the Tension in String A?

[DameLight]

Solved Thank You : )
1. Homework Statement
A rigid, uniform, horizontal bar of mass m₁ and length L is supported by two identical massless strings. (Figure 1) Both strings are vertical. String A is attached at a distance d < L/2 from the left end of the bar and is connected to the ceiling; string B is attached to the left end of the bar and is connected to the floor. A small block of mass m₂ is supported against gravity by the bar at a distance x from the left end of the bar, as shown in the figure.

Throughout this problem positive torque is that which spins an object counterclockwise. Use g for the magnitude of the acceleration due to gravity.

m₁
L
d_A < L/2
m₂
d_m2 = x
g

Find T_A, the tension in string A.

Homework Equations

Στ = Iα = F × r
I_bar = 1/12 m₁L²

The Attempt at a Solution

the rod isn't moving so
Στ = 0

If I make the point of rotation the location of string B
Στ = 0 = - τ_bar - τ_m2 + τ_A

τ_bar = F_g × L/2 = m₁g(L/2)
τ_m2 = F_g × x = m₂gx

τ_A = T × d_A = Td_A(sin90)
= τ_bar + τ_m2
= m₁g(L/2) + m₂gx

d_A = this is a known variable so
T_Ad = m₁g(L/2) + m₂gx
T_A = [m₁g(L/2) + m₂gx]/d

Conclusion:
Reason for error- the answer did not require the cross product to have sinθ

 

[theodoros.mihos]

All forces are 90 degrees cross to lengths. $\sin(90)=1$ for all. Your calculation is correct. The rope of T according to B point must be opposite to total masses rope $\tau_{m_2}+\tau_{bar}$.