Mr. Rogers' AP Physics C: Newton's Law Problems

Scale  Problems
What will each of the scales shown below read? Assume that the pulleys are massless and frictionless and the ropes and scales massless.
 Case I Case II Case III

 SolutionTo understand the scale reading, we must first draw free body diagrams and compare them for the three different cases.   Case I: From the free body diagram for the scale in Case I  the  upward force's magnitude equals the downward force's magnitude of mg. Here clearly the scale will read mg. Case II: From the free body diagram for the scale in Case II the magnitudes of the two horizontal forces have to be the same for static conditions to exist. Clearly FT2 is equal to mg, so both forces are equal to mg. Hence, the free body diagram is identical to one in diagram shown above except that it is turned horizontally. Therefore  the scale will also read mg.  Case III: A free body diagram of Case III will be identical to Case II, hence the scale reading for Case III will also be mg.
Conclusion and Significance

In all three cases, if the scales were removed and replaced with rope, the tension in the rope would be mg not 2 mg or zero even though a force of mg is applied to both ends of the rope.

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