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 find the ratio F1/F0, we need to consider the torque applied to the door in both situations.
Torque is the product of the force applied and the distance from the pivot point (hinge) to the point of application of the force. In this case, the torque applied by pushing on the door at a distance x0 from the hinge is given by:
T0 = F0 * x0
Now, when we push on the door at a distance of 2x0 from the hinge (2 times farther), the torque applied would be:
T1 = F1 * 2x0 = 2 * (F1 * x0) = 2T0
Since the angular acceleration α remains the same in both cases, we can equate the torques:
T1 = T0
2T0 = T0
Now, we can simplify the equation:
2 * (F1 * x0) = F0 * x0
2F1 = F0
Divide both sides by F0:
2F1/F0 = F0/F0
2F1/F0 = 1
Therefore, F1/F0 = 1/2
So, the ratio F1/F0 is 1/2. This means you would need to exert half the force when pushing on the door at a distance 2x0 from the hinge compared to pushing at a distance x0.