A uniform 22 kg door that is 2.5 m high by 0.80 m wide is hung from two hinges that are 20 cm from the top and 20 cm from the bottom. If each hinge supports half the weight of the door, find the magnitude and direction of the horizontal components of the forces exerted by the two hinges on the door. (Take the direction of forcing the door away from the hinges to be positive and the direction of forcing the door toward the hinges to be negative.)

Find the force of upper hinge.
Find the force of lower hinge.

draw the figure, with the hinges, and the center of gravity.

Write sum of moments about the hinges.

Here is the equation from the lower hinge:

.40meter*Weight-2.1meter*Fupperhinge=0
That gives you Fupperhinge. Do the same for the other hing, but I bet a coke it is = - Fupperhinge.

check my math.

2/4=.5

To find the forces exerted by the hinges, we need to consider the torque balance around a point. In this case, we can choose the lower hinge as the point of consideration.

Step 1: Find the weight of the door.
The weight of the door can be found using the formula:
Weight = mass * acceleration due to gravity
Given that the mass of the door is 22 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 22 kg * 9.8 m/s^2 = 215.6 N (Newtons)

Step 2: Calculate the torque due to the weight of the door and the forces exerted by the hinges.
Torque is the rotational force that causes an object to rotate. In this case, we have two forces exerted by the hinges and the weight of the door acting as a torque.

The torque exerted by a force can be calculated using the formula:
Torque = Force * Distance

For the upper hinge:
Let F1 be the force exerted by the upper hinge.
The torque exerted by F1 is: Torque1 = F1 * Distance1 = F1 * 0.20 m

For the lower hinge:
Let F2 be the force exerted by the lower hinge.
The torque exerted by F2 is: Torque2 = F2 * Distance2 = F2 * 0.20 m

Since the door is in equilibrium, the total torque exerted by the two hinges must balance the torque due to the weight:
Torque1 + Torque2 = Weight * Distance

Substituting the given values:
F1 * 0.20 m + F2 * 0.20 m = 215.6 N * 0.20 m

Simplifying the equation:
0.20 m * (F1 + F2) = 43.12 N*m

Step 3: Calculate the forces exerted by the hinges.
Since the hinges each support half the weight of the door, we can use this information to relate the forces:
F1 = 0.5 * Weight
F2 = 0.5 * Weight

Substituting these values into the equation:
0.20 m * (0.5 * Weight + 0.5 * Weight) = 43.12 N*m

Simplifying the equation:
0.20 m * (Weight) = 43.12 N*m

Solving for Weight:
Weight = 43.12 N*m / 0.20 m
Weight = 215.6 N

So, the forces exerted by the hinges can be calculated as follows:
F1 = 0.5 * Weight = 0.5 * 215.6 N = 107.8 N
F2 = 0.5 * Weight = 0.5 * 215.6 N = 107.8 N

Thus, the magnitude of the horizontal component of the force exerted by the upper hinge is 107.8 N in the positive direction, and the magnitude of the horizontal component of the force exerted by the lower hinge is 107.8 N in the negative direction.