A rod of length 4.00 m with negligible mass is hinged to a wall. A rope attached

Question

A rod of length 4.00 m with negligible mass is hinged to a wall. A rope attached to the end of the rod runs up to the wall at an angle of exactly 45°, helping support the rod, while a sign of weight 40.0 N is hanging by two ropes attached to the bottom of the rod. The ropes make an angle of exactly 30° with the rod as shown in the figure below. Another sign with a weight of 20.8 N is attached to the top of the rod with its center of mass at the midpoint of the rod. The entire system is in equilibrium. Find the magnitude of the vertical and horizontal components of the force that the hinge must exert on the rod to keep the system in equilibrium.

Explanation / Answer

here,

length of rod , l = 4 m

theta = 45 degree

phi = 30 degree

let the tension in the rope be T

let the normal reaction be Nx and Ny

taking moment of force at hinge

t*sin(45) * 4 - 2*20.8 - 20*sin(30)*2 - 20 * sin(30)*4 = 0

t = 143.68 N

and

Nx = t*cos(45)

Nx = 101.6 N

Ny = t*sin(45) - 40 - 20.8 = 0

Ny = 40.8 N

the vertical and horizontal component of reaction are 40.8 N and 101.6 N

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