Shows a metal advertising sign of weight ww suspended from the end of a horizontal rod of negligible mass and length LL. The end of the rod with the sign is supported by a cable at an angle θθ from the horizontal, and the other end is supported by a hinge at point PP. Using the idea that there is zero net torque about the end of the rod with the attached sign, find the vertical component of the force F⃗ hingeF→hinge exerted by the hinge. Express your answer in terms of some or all of the variables www, LLL, θθtheta.

Fv = 0

Explanation:

The pic shown can help us to understand the question.

If there is zero net torque about the end of the rod with the attached sign, we apply

∑ Fx = 0  (→+)

- Tx + Px = 0

⇒ - T*Cos θ + Px = 0

⇒  Px = T*Cos θ  (i)

where T is the force exerted by the cable and Px is the horizontal component of the force exerted by the hinge. Then

∑ Fy = 0  (↑+)

Ty - w = 0

⇒  T*Sin θ - w = 0  ⇒  T = w/Sin θ

Plugging this value into equation (i)

Px = (w/Sin θ)*Cos θ

⇒  Px = w*Cot θ = P

where w is the weight of the metal advertising sign.

Since the hinge only exerts an horizontal force (Px), the vertical component  is zero.