A door, essentially a uniform rectangle of height 2.00 m, width 0.78 m, and weight 136.0 N, is supported at one edge by two hinges, one 18.1 cm above the bottom of the door and one 181.9 cm above the bottom of the door. Calculate the horizontal components of the forces on the two hinges.
lower hinge magnitude:
upper hinge magnitude:
To calculate the horizontal components of the forces on the two hinges, we will use the concept of torque.
Torque is the tendency of a force to cause rotation about an axis. In this case, we can consider the axis of rotation to be at the hinges. To maintain equilibrium, the torques acting on the door must sum to zero.
Let's label the forces on the hinges as F1 and F2, with F1 acting at the lower hinge and F2 acting at the upper hinge. We are looking for the horizontal components of these forces.
The torque due to F1 will depend on its distance from the axis of rotation (the lower hinge), which is 18.1 cm (convert to meters by dividing by 100, so 0.181 m). Similarly, the torque due to F2 will depend on its distance from the axis of rotation (the upper hinge), which is 181.9 cm (convert to meters by dividing by 100, so 1.819 m).
Since the weights acts vertically downwards, it will not contribute to the torque.
The equation for torque is given by:
Torque = Force x Distance.
For the door to be in equilibrium, the sum of the torques must be zero.
Sum of Torques = Torque due to F1 + Torque due to F2 = 0.
Therefore,
F1 x 0.181 m + F2 x 1.819 m = 0.
Simplifying the equation, we have:
F1 = - F2 x (1.819 m / 0.181 m).
Now, we need to consider the vertical components of the forces to find their magnitudes.
The weight of the door acts vertically downwards and can be calculated using the formula:
Weight = mass x acceleration due to gravity.
Given that the weight is 136.0 N, we can find the magnitude of the weight. Divide it by the acceleration due to gravity to get the mass of the door.
The acceleration due to gravity is approximately 9.8 m/s^2.
Weight = mass x 9.8 m/s^2.
136.0 N = mass x 9.8 m/s^2.
From here, solve for the mass of the door.
Once we have the mass of the door, we can determine the magnitude of each force using the formula:
Force = mass x acceleration.
The acceleration is the acceleration due to gravity, which is 9.8 m/s^2.
Given that the forces act vertically upwards at the hinges, the magnitudes of F1 and F2 will be equal to the magnitude of the weight of the door.
Once we have the magnitudes of F1 and F2, we can substitute them into the equation for the horizontal components of the forces to find the values of F1 and F2.