Force and potential energy of a mail bag

[physstudent1]

Homework Statement

A 115 kg mail bag hangs by a vertical rope 3.8m long. A postal worker then displaces the bag to a position of 2.2m sideways from its original position, always keeping the rope taut.

What horizontal force is necessary to hold the bag in the new position?

Homework Equations

The Attempt at a Solution

Ok, so I set up the new position of the bag and used Pythagoreans theorem to find that the bag is now .7m higher than it was where it started. I called the original position my reference line so it has no potential energy there. At a height of .7 it has .7*9.8*120
In my book it says that F(in x direction) = -(change in potential)/(change in x)

so I thought dividing this potential energy I found by the change in the x direction (2.2) would give me the Force in the horizontal?

 

[rl.bhat]

Draw the diagram as explained in the problem. Draw three vectors. One tension, second horizontal force and third weight vertically downwards. Draw a right angled triange such that length of the rope is hypotenuse, horizontal displacement is horizontal side. According to your calculation the third side becomes 3.1m. Applying the law of trianglesof vectors we can wright F/2.2 = Mg/3.1. Now find F. You can also solve this problem by applying law of moments.

 

[ogb p]

Your approach is on the right track, but there are a few things to consider. First, when calculating potential energy, it is important to use the height above the ground, rather than the displacement from the original position. So in this case, the potential energy at the new position would be (2.2 + 3.8) * 9.8 * 115 = 8,626 J.

Second, the formula you mentioned (F = -ΔU/Δx) is actually for the force due to a change in potential energy, not just the potential energy itself. So in this case, you would need to divide the change in potential energy (8,626 J) by the change in x (2.2 m) to get the force in the horizontal direction, which is 3,921 N.

However, this only gives the minimum force needed to hold the bag in the new position. In reality, the postal worker would also need to exert some force in the vertical direction to keep the bag from swinging or falling. This would depend on the angle at which the rope is held and the tension in the rope. So the actual force needed to hold the bag in the new position would be greater than 3,921 N.