Write the necessary equations of equilibrium of the object shown in the figure below. Take the origin of the torque equation about an axis perpendicular to the page through the point O. (Let counterclockwise torque be positive and let forces to the right and up be positive. Use the following as necessary: the forces Rx, Ry, Fx, Fy, and Fg; the length ℓ of the object; and the angle 𝜃 that the object makes with the horizontal.)A rod of length ℓ makes an acute angle 𝜃 clockwise from the horizontal. A point at the left end of the rod is labeled O. Five forces act on the rod as follows.A force with magnitude Rx begins at the left end of the rod and points horizontally left.A force with magnitude Ry begins at the left end of the rod and points vertically up.A force vector Fg begins at the center of the rod and points vertically down.A force with magnitude Fx begins at the right end of the rod and points horizontally right.A force with magnitude Fy begins at the right end of the rod and points vertically up.ΣFx = = 0ΣFy = = 0Σ𝜏 = 0 = 0

The equilibrium of an object requires that both the net force and the net torque on the object be zero.

The net force in the x-direction (horizontal) is the sum of all the forces in the x-direction. According to the problem, Rx is to the left (negative) and Fx is to the right (positive). So, the sum of the forces in the x-direction (ΣFx) is:

ΣFx = Fx - Rx = 0

The net force in the y-direction (vertical) is the sum of all the forces in the y-direction. According to the problem, Ry and Fy are upwards (positive) and Fg is downwards (negative). So, the sum of the forces in the y-direction (ΣFy) is:

ΣFy = Ry + Fy - Fg = 0

The net torque (Σ𝜏) is the sum of all the torques about the point O. The forces Rx and Ry pass through point O, so they do not contribute to the torque. The forces Fx, Fy, and Fg create torques about point O. The torques due to Fx and Fy are counterclockwise (positive) and the torque due to Fg is clockwise (negative). The distances from O to the points of application of these forces are ℓ cos(𝜃), ℓ sin(𝜃), and ℓ/2, respectively. So, the sum of the torques (Σ𝜏) is:

Σ𝜏 = Fx * ℓ cos(𝜃) + Fy * ℓ sin(𝜃) - Fg * ℓ/2 = 0

These are the necessary equations of equilibrium for the object.