If you watch little kids try to open doors, they will often just push anywhere on the door, rather than on the edge like adults do. This is because they don't understand that it's easier to open a door if you push on the edge. But how much easier? Consider the following situation: you push on a door perpendicularly at a horizontal distance x0 from the hinge with a force F0, thereby opening the door with some angular acceleration α. Let F1 be the amount of force you'd need to exert to open the door with the same angular acceleration, but pushing perpendicularly at a horizontal distance 2x0 from the hinge. What is F1/F0?
To calculate the amount of force needed to open the door at a distance 2x0 from the hinge, we can use the principle of moments or torque.
Torque is the multiplication of force and the perpendicular distance from the pivot point (hinge) to the line of action of the force. In this case, the torque is responsible for opening the door.
Let's assume the door has a length L and a mass m, and the force F0 is applied perpendicular to the door at a distance x0 from the hinge.
The torque generated by this force is given by:
Torque0 = F0 * x0
Now, let's analyze the situation where the force F1 is applied at a distance 2x0 from the hinge. The torque generated by this force is:
Torque1 = F1 * 2x0
According to the problem statement, both forces F0 and F1 should produce the same angular acceleration α, therefore the torques must be equal:
Torque0 = Torque1
F0 * x0 = F1 * 2x0
Simplifying the equation, we can cancel out x0 from both sides:
F0 = F1 * 2
Therefore, F1/F0 = 1/2
So, the ratio of the amount of force needed to exert to open the door with the same angular acceleration at a distance 2x0 from the hinge, compared to pushing at a distance x0, is 1/2.
In other words, you would need to exert half the force if you push on the edge of the door compared to pushing anywhere else on the door.