It is harder to move a door if you lean against it (along the plane of the door) toward the hinge than if you lean against the door perpendicular to its plane. Why is this so?

Torque is the measure of the force that can cause an object to rotate about an axis. Mathematically, torque (τ) is calculated as the product of the force (F) and the moment arm (r), where the moment arm is the perpendicular distance between the axis of rotation and the line of action of the force. In our case, the axis of rotation is the hinge and the force is the leaning action. So, torque can be expressed as τ = rF.

Now we will compare the torque when leaning against the door along the plane (Scenario 1) and when leaning against the door perpendicular to its plane (Scenario 2). Scenario 1: When leaning along the plane of the door towards the hinge, the moment arm is very small as the force is being applied close to the hinge, i.e., the axis of rotation. As a result, the torque generated in this case would be low (τ1 = r1F1, with r1 being small). Scenario 2: When leaning against the door perpendicular to its plane, the moment arm is equal to the distance between the hinge and the point where the force is applied (usually the doorknob or the edge of the door). As a result, the torque generated in this case would be relatively higher (τ2 = r2F2, with r2 being significantly larger than r1).

Since torque is the force causing the rotation, a higher torque will result in easier rotation of the door. In Scenario 2, when leaning against the door perpendicular to its plane, the torque is higher due to the larger moment arm. Thus, it is easier to move the door in this situation compared to Scenario 1, when leaning along the plane of the door towards the hinge, where the torque generated is lower due to a smaller moment arm.