Physics Principles and Formula Derivation
Analyzing the forces on a door suspended by two hinges involves applying classical static equilibrium principles in two dimensions. When a rigid body like a door is at rest, two fundamental criteria must be met: the net vector force acting on it must be zero, and the net torque about any axis of rotation must equal zero.
Static Equilibrium Equations
To determine reaction forces, we balance horizontal forces, vertical forces, and rotational moments:
1. Vertical Force Balance ($\sum F_y = 0$):
The total downward force exerted on the door is its weight, given by $W = m \cdot g$. Assuming equal vertical load distribution between both hinges:
$$F_{y1} = F_{y2} = \frac{m \cdot g}{2}$$
2. Rotational Moment Balance ($\sum \tau = 0$):
The center of gravity creates a rotational torque pulling the door downward away from the frame. To prevent rotation, the hinges exert equal and opposite horizontal forces. Taking the moment pivot about the bottom hinge:
$$\tau = (W \cdot d_{cg}) - (F_{x1} \cdot d) = 0 \implies F_{x1} = \frac{m \cdot g \cdot d_{cg}}{d}$$
Due to horizontal force balance ($\sum F_x = 0$), the bottom hinge exerts a force of equal magnitude in the opposite direction ($F_{x2} = -F_{x1}$). Thus, the top hinge pulls outward (tension) while the bottom hinge pushes inward (compression).
3. Total Resultant Reaction Force ($F_{net}$):
The magnitude of total reaction force acting on each hinge is solved using the Pythagorean theorem:
$$F_{resultant} = \sqrt{F_x^2 + F_y^2}$$
How to Use This Calculator
- Input Mass & Gravity: Enter the door's total mass in kilograms and adjust gravitational acceleration if needed.
- Set Dimensions: Enter overall width and height of the door in meters.
- Hinge & CG Distance: Specify vertical separation between top and bottom hinges and horizontal distance from hinges to the center of gravity.
- Compute Results: Click "Calculate Hinge Forces" to inspect component reaction loads instantly above the form.