A student tries to open a door at a distance of 1cm from the hinge, at a distance of 10cm from the hinge,....?

Question

and at a distance of 50cm from the hinge.

A. Why is a larger force required at 1cm than at 50cm?

B. How much more force is required at 1cm than at 50cm?

Answer 1

A)Torque is required to open the door.

Torque = Force * arm-length

Here arm-length is the distance of the point where force is applied to the axis of rotation, i.e. hinge.

Or, force = torque/arm-length

As you can see from the above equation, if arm-length is smaller then required force is more to generate a given torque.

In first case, arm length is less (as 10cm < 50cm).

Therefore, more force is required in first case.

B) 50 times more force is required.

Answer 2

The most basic answer, as far as I see it, is leverage. The closer you are to the hinge to try to move the mass and weight of the door, the more force you need to overcome that weight and mass.  The further out you go then the more leverage you have. The easier to move the door.

   On how much more force that is required, I guess that depends on the weight, mass, friction of the hinge, and air resistance on the door. I'm sure there is some mathematical formula to apply to this.