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 determine how much harder you would have had to push the door to open it at the same speed by pushing near the hinges, we need to understand the concept of torque.
Torque, in this context, is the measure of the force that causes an object to rotate around an axis. When you push a door, you are applying torque to make it rotate around its hinges. The torque is directly proportional to the force applied and the distance from the axis (hinges) at which the force is applied.
Mathematically, the torque (T) can be calculated as the product of the force (F) and the perpendicular distance from the axis (r) at which the force is applied: T = F * r.
In the given scenario, pushing near the edge that is farthest from the hinges (40 inches wide), the distance from the axis (r) is 40 inches. Let's say the force you applied is denoted as F1.
Now, if you had instead pushed the door at a point only 10 inches in from the hinges, the distance from the axis (r) would reduce to 10 inches. Let's denote the force required to open the door at the same speed as F2.
As mentioned earlier, torque is proportional to the force and the distance from the axis. In this case, the torque of the force applied near the edge would be the same as the torque of the force applied near the hinges.
So, we can equate the two torques:
T1 = T2
F1 * r1 = F2 * r2
Since the speed of opening the door is the same, the torques are equal. Now, let's solve for F2, i.e., the force required to open the door when pushing near the hinges.
F2 = (F1 * r1) / r2
Given that r1 = 40 inches and r2 = 10 inches, we can substitute these values into the equation:
F2 = (F1 * 40) / 10
F2 = 4F1
Therefore, to open the door at the same speed by pushing near the hinges (10 inches from the hinges), you would have to push with four times the force you would exert pushing near the edge farthest from the hinges.
In essence, the farther away from the hinges you push, the easier it is to open the door because the lever arm (distance from the axis) is greater, reducing the force required.