Recent content by wrobel

I get it. That’s not really my area — I mostly just solve problems. But this forum has experts on the otherworldly essence of definitions. I’m sure they’ll be delighted to help.

One just need to understand how the very concept of the moment of inertia arises. Assume a rigid body rotates about a fixed point that coincides with the origin ##O## of an inertial coordinate frame. The rigid body consists of particles of masses ##m_1,\ldots,m_N## with position vectors...

Just found a really good lecture on tensors. It starts with the absolute basics and goes all the way up to universal mapping:

Suppose two particles of masses ##m_1## and ##m_2## moving along a line collide perfectly elastically. By definition, this means that if the velocities of the particles before the collision were ##\boldsymbol v_1^-,\boldsymbol v_2^-##, then their velocities after the collision ##\boldsymbol...

Two unknowns: the force ##P## and system acceleration ##a## and two equations: projections of Newton's second law onto the horizontal axis for each body.

Basically, a good problem can be made out of this. We just need to define the law of motion along the meridian. A person walking along the meridian is acted upon by two forces: gravity ##\boldsymbol G## and the Earth's reaction force ##\boldsymbol R##. The left-hand side of the equation...

I do not know what you understand by 'meaning'. You started by objecting to my phrase 'A definition is a definition.' Perhaps you see in a definition something more than just a definition. That is your point, not mine. I formulate theorems; I do not search for a sacred meaning in definitions as...

So it's all the same thing. Until you write a formula that connects kinetic energy with some other quantity, the formula ##T=mv^2/2## makes no more sense than that gibberish you came up with above.

Explain this to me with an example. In #15, I wrote down a formula derived from Newton's second law. This formula is useful for solving problems. Beyond that (other than obtaining useful formulas), what reasons do you see to define kinetic energy and work in this specific way and no other?

That seemed like an objection, but I didn't quite catch it. Could you please clarify your point?

Most pointless debates arise precisely from searching for a 'meaning behind it'. In fact, there are definitions and theorems that connect them, for example: $$T\mid_{t=t_2}-T\mid_{t=t_1}=\int_{t_1}^{t_2}(\boldsymbol F,\boldsymbol v)dt,$$ $$ T:=\frac{m\vert{}\boldsymbol v\vert{}^2}{2}$$

There’s a great concept called 'definition.' A definition is a definition. The definition of work done is $$\int_{t_1}^{t_2}(\boldsymbol F,\boldsymbol{\dot r})dt$$

There is nothing strange about this. For example, gravity also tries to pull a simple pendulum off its circular path, but that doesn't mean we can't calculate its work.

Nice bait. I have a feeling the OP won't show up again.

This problem can, of course, be solved by elementary means. A bicycle consists of 4 rigid bodies (the frame, two wheels, and the pedals). For each body, one needs to set up a system of equations of motion accounting for reaction forces—there may be some indeterminate constraints somewhere—add...