The net force in the radial direction must be zero to balance the system. This means that the sum of the forces in the x and y directions must be zero. We can write the equations as follows:
ΣFx = ma_r = 0
ΣFy = ma_θ = 0
where a_r and a_θ are the radial and tangential accelerations, respectively. The tangential acceleration is zero because the system is in equilibrium.
Let M be the total mass of the system. Then, the magnitude of mass a can be found using the equation:
Ma_r = Mb(a+b)sinθ
where θ is the angle between the radii of masses b and a. Since the system is balanced, we have:
Ma_r = Mb(a+b)sinθ = 0
Since Mb ≠ 0 and sinθ ≠ 0, we must have a = -b. This means that mass a must be 7 kg.
Next, we can find the magnitude of mass c using the equation:
Mc(a+c)sin(90°-θ) = Mb(b+c)sinθ
Substituting the values, we get:
Mc(a+c) = Mb(b+c)cosθ
Mc(a+c) = 7(b+c)cosθ
Similarly, we can find the magnitude of mass d using the equation:
Md(a+d)sin(θ-240°) = Mb(b+d)sinθ
Substituting the values, we get:
Md(a+d) = Mb(b+d)cos(θ-240°)
Md(a+d) = 7(b+d)cos(θ-240°)
Finally, to find the angular position of mass a, we can use the equation:
ΣFy = Ma_θ + Mb(b+a)cosθ + Mc(c+a)cos(90°-θ) + Md(d+a)cos(θ-240°) = 0
Substituting the values, we get:
7a + 14cosθ + 7c - 7dcosθ = 0
a + 2cosθ + c - dcosθ = 0
This equation can be solved numerically to find the value of θ.