Two friends are carrying a crate of mass 200 kg up a flight of stairs. The crate

Question

Two friends are carrying a crate of mass 200 kg up a flight of stairs. The crate has length 1.25 m and height 0.500 m, and its center of gravity is at its center. The stairs make a 45 degree angle with respect to the floor. The crate also is carried at an angle 45 degrees, so that its bottom side is parallel to the slope of the stairs. The force each person applies is vertical.

Part A

What is the magnitude of the upper person's force?

Part B

What is the magnitude of the force applied by the bottom person?

Explanation / Answer

The way to start problems of this variety is to draw a free body diagram. Because
this is a symmetric body, you can depict it as a line segment labeled as being of
length (call this D) 1.25 m angled at 45 degrees. The weight of the object is M g where M=200 kg
draw this as an arrow pointing down at the exact geometric center of the line
segment.

We make the assumption that the object is simply supported so there is an
arrow pointing up at each end of the line segment.


Now, you have two basic physical principles of static equilibrium that are your tools.
These are:
1) the sum of the moments is zero

2) the sum of the forces is zero

Call the forces at the ends L and R for left and right, and I will assume that the object is sloping
left to right. And the weight W = mg

Forces = 0 = L + R - W

Do the sum of the moments about the right end. D is the total length of the rod
Moments =0 = R Dcos(45) - W D/2(cos(45))

so R = W/2 meaning L = W/2

Is that really right? Consider the sum of the moments about the point D/2

moments = 0 = R D/2 - L D/2

so R = L

This is counter intuitive because we all think that the guy on the bottom is going
to have a heavier load. The real life issue is one of being able to apply the vertical
forces. If each end had a handle so that there would be no slipping and a vertical
force could be applied, then in perfect static equilibrium it makes no difference
if the load is on an inclined plane, or if it were horizontal. If the object starts slipping
then we are dealing with a different problem.

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