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?

Question

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?

Answer 1

To determine the ratio F1/F0, we can apply the principles of torque and angular acceleration.

When opening a door, the torque applied is directly proportional to the force applied and the perpendicular distance from the hinge where the force is applied. Mathematically, the torque (τ) is given by the equation τ = F * d, where F is the force and d is the distance from the hinge.

In this scenario, we can consider the initial torque (τ0) when the force F0 is applied at the distance x0, and the final torque (τ1) when the force F1 is applied at the distance 2x0.

The torque is also related to the angular acceleration (α) and the moment of inertia (I) of the door. The moment of inertia represents how resistant an object is to changes in its rotational motion. For a door, the moment of inertia depends on its mass distribution and shape.

The relationship between torque (τ), angular acceleration (α), and moment of inertia (I) is given by the equation τ = I * α.

Since we're interested in the ratio F1/F0 for the same angular acceleration, we can equate τ0 to τ1:

F0 * x0 = F1 * 2x0

We can cancel out the distance x0 from both sides, resulting in:

F0 = 2 * F1

Dividing both sides by F0, we get:

F1/F0 = 1/2

Therefore, the ratio F1/F0 is 1/2. This means that if you double the perpendicular distance from the hinge at which you apply the same force, you would need half the force to achieve the same angular acceleration.