Random Thought
If there was no friction, then one could keep a nut on one end of a screw and it would automatically screw/roll-down/tighten itself if kept in right orientation under the the influence of gravity.
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Proof that the acceleration vector always points radially inward for a body in circular motion
A few days ago, me and a friend of mine challenged each other to see if we could mathematically prove that the acceleration vector always points radially inward for a body in circular motion. So here's what I came up with: Proof We know the position vector for a body making an angle \(\theta_0\) with the X-axis is given by: \begin{align*} \vec{r}=r\,(\cos{(\theta_0)}\,\hat{i}+\sin{(\theta_0)}\,\hat{j}) \\ \end{align*} But for a body revolving around the origin, the angle changes at the rate \(\omega\). That is to say: \begin{align*} \omega=\frac{d\theta}{dt} \\ \end{align*} So in general, the angle at any given time \(t\) would be (\(\theta_0+\omega t\)). Which implies the position vector would be: \begin{align*} \vec{r}=r\,(\cos{(\theta_0+\omega t)}\,\hat{i}+\sin{(\theta_0+\omega t)}\,\hat{j}) \\ \end{align*} Now since we have the position as a function of time, we can take the its derivative w.r.t. time to get the velocity function. \begi...