You need to open a heavy door that's 40 inches wide. Instinctively, you push near the edge that's the farthest from the hinges. If instead, you had pushed at a point only 10 inches in from the hinges, how much harder would you have had to push to open the door at the same speed? explain your answer.

To answer this question, we need to consider the concept of torque. Torque is a rotational force used to rotate an object around a fixed point, in this case, the hinges of the door. The torque required to open a door depends on two factors: the force applied (pushing) and the distance from the axis of rotation (hinges).

In the given scenario, instinctively pushing near the edge that's farthest from the hinges (let's call it point A) requires less force to rotate the door because it creates more torque due to the larger lever arm (distance from the axis of rotation). On the other hand, if you had pushed at a point only 10 inches in from the hinges (let's call it point B), you would have had to exert more force to achieve the same speed of door opening.

To understand why, let's analyze the situation mathematically using the equation for torque:

Torque (τ) = Force (F) × Lever Arm (L)

Where:
τ = Torque (measured in Newton-meters or Nm)
F = Applied Force (measured in Newtons or N)
L = Lever Arm (measured in meters or m)

In this case, we can assume that the force applied at point A is F1, and the force applied at point B is F2. The lever arm at point A is L1 and at point B is L2.

We know that the torque at point A is equal to the torque at point B since the door is opening at the same speed. Therefore:

τA = τB

And since torque is calculated using the equation mentioned earlier, we can rewrite the equation as:

F1 × L1 = F2 × L2

Now, let's plug in the given values. The door's width is 40 inches, so the distance from the hinges to point A (L1) would be 40 inches. The distance from the hinges to point B (L2) would be 10 inches.

Using the given information, L1 = 40 inches (or approximately 1.016 meters) and L2 = 10 inches (or approximately 0.254 meters). Since we are comparing the forces, we can replace F1 with 1 and calculate the required force at point B (F2).

1 × 1.016 = F2 × 0.254

After solving the equation, we find that F2 is approximately 4 Newtons. Therefore, if you had pushed at a point 10 inches in from the hinges, you would have had to apply a force of 4 Newtons to open the door at the same speed.

In comparison, when instinctively pushing near the edge of the door, the force required would have been less due to the longer lever arm, resulting in a smaller torque requirement.