[hamdani yusuf (OP)]

Current SI system makes them dimensionally equivalent.

Nothing to do with SI, which simply names the quantities. Energy and torque are indeed dimensionally equivalent but because they are different entities (energy is a scalar, torque is a vector) , SI gives them different names and different units.

SI is really nothing more than a universal multilingual dictionary. You might be less confused if you used the German word for torque: Drehkraft.

It happens because SI declared radian is dimensionless, which makes it possible to appear and disappear from equations without consequences.

[hamdani yusuf (OP)]

The formula v = ω . r  implies  r = v / ω   

Numerically, yes. Logically, no.
Remember the convention:  effect = f(cause).
when dealing with torque, r is not an effect

if F = m . a, then a = F / m
Would you say that the last formula above is not logically correct?

Here's another example.
V = I . R implies that I = V / R

[hamdani yusuf (OP)]

You only need to determine the behavior of rotational axis while being influenced by force.

which you have still failed to do.

You can start with simple examples. Someone else has made a video about elliptical orbit, so I don't have to.

How Newton deduced gravity obeys inverse square law | Hidden gems #2

Both Newtonian gravity and Hooke's law admit elliptical orbits in 3D (or 2D, same thing since all solutions are planar), but is it a coincidence? Newton said no, but this connection was only discovered accidentally when he was trying to deduce why gravity obeys inverse square law from astronomical observations at the time. In fact, if you know the history between Newton and Hooke, you will find the close connection between these two forces ironic!

This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.

I'm making a video about elliptical trajectory with constant tangential speed. But it takes a significant amount of my time, which is currently limited by 8 to 5 job. So, please be patient.

[alancalverd]

Is it still a Newton meter?

No. We measure energy in joules. |Newton|.|meter| (scalar, energy) is not the same as newton x meter (vector, torque), so 1 |Newton|.|meter|  is called a joule.

What if the angle between force and displacement is 45 degrees?

The product of the force component in the direction of movement x distance moved in that direction is energy, measured in joules. The product of force x distance perpendicular to the line of action of the force is torque, measured in newton meters.

[alancalverd]

if F = m . a, then a = F / m
Would you say that the last formula above is not logically correct?

No, the second formula is conventionally correct: acceleration is the result (effect) of applying a force to a mass (cause). That was Newton's logical deduction from observation. ("Every body remains in a state of rest or uniform motion unless acted upon by a force....")

Here's another example.
V = I . R implies that I = V / R

Again, you have got the logic backwards. A current (effect) flows when you apply a potential across a conductor (cause).  A conductor does not spontaneously generate a potential difference between its ends.

[alancalverd]

Both Newtonian gravity and Hooke's law admit elliptical orbits in 3D (or 2D, same thing since all solutions are planar), but is it a coincidence?

Newton: F α 1/r²

Hooke; F α r

AFAIK, r ≠ 1/r² unless r = 1, which is clearly not the case for an ellipse!

Newton: F α m

Hooke: F is independent of m

[hamdani yusuf (OP)]

if F = m . a, then a = F / m
Would you say that the last formula above is not logically correct?

No, the second formula is conventionally correct: acceleration is the result (effect) of applying a force to a mass (cause). That was Newton's logical deduction from observation. ("Every body remains in a state of rest or uniform motion unless acted upon by a force....")

Here's another example.
V = I . R implies that I = V / R

Again, you have got the logic backwards. A current (effect) flows when you apply a potential across a conductor (cause).  A conductor does not spontaneously generate a potential difference between its ends.

Here's the equivalent in rotating systems.
"Every body remains in a state of rest or uniform rotation unless acted upon by a torque...."

[alancalverd]

True. But not every torque produces rotation.

Here's another example of two quantities being dimensionally equivalent but  entirely unrelated:

The amount of coal burned to keep a steam locomotive running is measured in pounds per mile (ML^-1).

The weight of nylon thread woven into a parachute is measured in pounds per mile (ML^-1) 

Both very important quantities in their own context but with no logical connection..

[hamdani yusuf (OP)]

True. But not every torque produces rotation.

Here's another example of two quantities being dimensionally equivalent but  entirely unrelated:

The amount of coal burned to keep a steam locomotive running is measured in pounds per mile (ML^-1).

The weight of nylon thread woven into a parachute is measured in pounds per mile (ML^-1) 

Both very important quantities in their own context but with no logical connection..

If a torque doesn't produce angular acceleration, it must have been counterbalanced by other torque(s). There's no other way to explain it.

[alancalverd]

Indeed. And when you release the brake, the car/bucket/whatever accelerates. Which is why you need  a quantity called torque which has nothing to do with displacement, because cars, wells, mechanical stopwatches.....are not psychic! The accelerating torque must have been present whilst the brake was on and nothing moved.