Question

This is a pretty basic doubt.
We say dU (potential energy) = - Force. dx
Where it's a dot product. From there, dU = -F.dx *cos(theta) { theta is the angle between F and dx}
However, we write F as -dU/dx. Where did the cos(theta) in the dot product vanish?

Answer 1

\(\theta\) vanishes only when \(\theta\)=0
That is cos\(\theta\)=1
that is to say when force and displacement are in same direction, cos\(\theta\) vanishes.

Answer 2

There is a little bit of ambiguity in your question. Writing
\[\frac{dU}{dx} = -F \rightarrow dU=-F\cdot dx \]
implies that you're working in a one-dimensional system, where vectors are unnecessary. If you're working in multiple spatial dimensions, it would be more appropriate to write
\[\vec{\nabla}U = -\vec{F} \rightarrow dU = -\vec{F}\cdot d\vec{r} = -F\cdot dr \cdot \cos(\theta)\]
where theta is the angle between the displacement dr and the force F.

Answer 3

You as using the same notation for F, magnitude of force and Fx, component of force along the x-axis.

\(\vec F=-\vec{\nabla}U\) means

\(F_x=-\Large\frac{\partial U}{\partial x}\)
\(F_y=-\Large\frac{\partial U}{\partial y}\)
\(F_z=-\Large\frac{\partial U}{\partial z}\)

Answer 4

Oh right.
It;s the component. That clears it.
THanks a lot. :)