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Where am I wrong in proving that force perpendicular to motion increases speed and kinetic energy?

First, I am using $z$ for the vertical direction. The eq. of motion are: $$\frac{d^2\vec r}{dt^2} = \frac 1 m \vec F = -g\hat z$$ or: $$\ddot x = 0$$ $$\ddot y = 0$$ $$\ddot z = -g$$ which is ...
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The conservative force

While @basics has the math right, I understand your question is about the physical interptetation. To answer this, we must understand what a force field is, since the definition of the rotation ...
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The conservative force

An example might be the best way to see it. Set up some electrodes that create a horizontal electric field. Make it so the field strength is proportional to height. This is a non-conservative field. ...
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The conservative force

Remark the curl it's the measure of the rotation of the vector field around a specific point Before answering your question, I'd replace (local) "rotation" with (local, or elementary) &...
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What are the forces that do work when a spring between two masses pulls inwards?

There is no kinetic energy change in the second system (1,2, spring). So there is no net work done, so no need to ask which external forces that do the work. The constituent particles of the system do ...
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