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Just want to know why the door turns less (less torque) at the point closer to the hinge (point A) than at the outer point at the handle (point B) when the same force is applied. If same force is not doing same work then where the energy is lost or transferred at point A (law of conservation of energy)?


When the forces are perpendicular to the arm as shown in your diagram then the work done is simply


Where $Fr$ is the torque and $r$ is the distance from the hinge to the point of application of the force $F$.

Although the displacement angle θ is the same for the forces at both A and B, The distance that the force at B travels along the arc is greater than the distance that the the force at A has to travel for the same angle displacement θ. For this reason when the forces are the same, the force at B does more work than that at A.

If the torque at A and B is the same, the forces at A and B do the same work, but less force is needed at B than A to do the same work because of the longer moment arm, $r$. The advantage of applying the same torque at B is it requires less force (effort) to do the same work, providing a mechanical advantage like a lever. Of course the force has to be applied over a greater arc length. Nothings for free.

Hope this helps.

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