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- Homework Statement
- This US Physics Olympiad problem 2018A Question

A uniform stick of mass ##m## is originally on a horizontal surface. One end is attached to a vertical rope,

which pulls up with a constant tension force ##F## so that the center of the mass of the stick moves upward

with acceleration ##a < g##. The normal force ##N## of the ground on the other end of the stick shortly after

the right end of the stick leaves the surface satisfies ##mg > N > mg/2##

- Relevant Equations
- ##F_{net} = ma ##

##\tau_{net} = I \alpha##

The question is not directly related to the problem itself, but an odd discovery that when using left end of the stick as axis of rotation versus using the CM as axis of rotation, two different normal force expressions are found. Note that the solutions are for the initial moment when the stick is still horizontal. I am looking for suggestions to reconcile the two different answers, thanks!

The link to the problem set is here: https://www.aapt.org/physicsteam/2019/upload/Fma-2018-A.pdf It's question 9.

Using left end as axis of rotation:

$$ma=F+N-mg$$

$$\alpha * L/2= a$$

$$Fl-mgl/2=\alpha * \frac{1}{3} ml^2=\frac{2}{3} mal$$

$$F=\frac{2}{3}ma+mg/2=mg+ma-N$$

$$N=\frac{1}{3} ma+mg/2$$

Using CM as axis of rotation:

$$Fl/2-Nl/2=\frac{1}{12} ml^2 \alpha$$

$$a=\alpha l/2$$

$$F-N=ma/6$$

$$F+N-mg=ma$$

$$F=ma+mg-N = ma/6 + N$$

$$2N = \frac{5}{6}ma+mg$$

$$N=\frac{5}{12} ma + mg/2$$

The difference between the two normal force is ##1/12 ma##.

The link to the problem set is here: https://www.aapt.org/physicsteam/2019/upload/Fma-2018-A.pdf It's question 9.

Using left end as axis of rotation:

$$ma=F+N-mg$$

$$\alpha * L/2= a$$

$$Fl-mgl/2=\alpha * \frac{1}{3} ml^2=\frac{2}{3} mal$$

$$F=\frac{2}{3}ma+mg/2=mg+ma-N$$

$$N=\frac{1}{3} ma+mg/2$$

Using CM as axis of rotation:

$$Fl/2-Nl/2=\frac{1}{12} ml^2 \alpha$$

$$a=\alpha l/2$$

$$F-N=ma/6$$

$$F+N-mg=ma$$

$$F=ma+mg-N = ma/6 + N$$

$$2N = \frac{5}{6}ma+mg$$

$$N=\frac{5}{12} ma + mg/2$$

The difference between the two normal force is ##1/12 ma##.