Recent content by kuruman

I messed up the definition of ##\sin\theta## because I copied and pasted from @brotherbobby's expression without changing ##s-x## to ##x## in the numerator. The original equation is inconsistent with the text. See edit note and added figure in post #4. Thanks for the catch.

Read about quarks here.

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Say I have vector ##\mathbf C## in a given 2D coordinate system specified by unit vectors ##\mathbf {\hat e}_1## and ##\mathbf {\hat e}_2.## In that system I write $$\mathbf C=C_1~\mathbf {\hat e}_1+C_2~\mathbf {\hat e}_2.$$In the above equation, ##C_1~\mathbf {\hat e}_1## and ##C_2~\mathbf...

I think your problem was that you measured the "jump in" distance from the starting point. If ##x_{opt}## is the optimum distance from C, the time will be a minimum no matter how far from C the man starts running as long as the starting point is at ##s>x_{opt}##. So it makes more sense to...

You don't have to repeat, I am with you 100%. I was expressing my doubts (in counterpoint to my "as it should be") about the respectability of NASA whose $125 million satellite to Mars crashed in 1999 because someone apparently used numbers without paying close attention to the units.

Yes it's interesting and as it should be. And then there is this (see below) from the people who landed people on the Moon almost 55 years ago. Resistance is futile.

It is reasonable and appropriate to say that. We say "a vector is the sum of its components". To translate this statement from English into mathematese, we write the equation$$\mathbf F=F_x~\mathbf {\hat x}+F_y~\mathbf {\hat y}.$$ However, when we ask "What is the x-component of vector...

If you are going to quote the Ampere-Maxwell law, quote it correctly. The displacement current term (Maxwell's correction) is ##\mathbf J_d=\dfrac{\partial \mathbf D}{\partial t}=\dfrac{\partial (\epsilon_0 \mathbf E)}{\partial t}##. The ##\epsilon_0## in the numerator cancels the one in the...

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Let's do it formally and top down. I will use boldface for vectors. The net force is always the sum of the restoring and damping force vectors, $$\begin{align}\mathbf F_{\text{net}}=\mathbf F_{\text{restoring}}+\mathbf F_{\text{damping}}~.\end{align}$$The restoring-force vector is always...

It occurred to me that I should complete the picture by adding FBDs of the two cases above when the velocity of the mass is reversed (see below). The reader can readily verify that the equation $$F_{net}=-kx-bv$$ gives the correct direction of the restoring and damping forces.

I think part of the problem with writing Newton's second law in this case is that the velocity, which is not normally part of an FBD, is crucial here for determining the direction of the damping force. The way to sort this out is to draw the FBD at the moment when the velocity is in the...

I became interested in plotting the trajectory of the skater so this is what I did. We have seen elsewhere(1) that when a body is displaced by ##\mathbf s## under constant acceleration ##\mathbf a## the expression $$\mathbf a\times \mathbf s=\mathbf v\times \mathbf u$$ relates the cross product...

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