Motion in the Macrocosm and the Microcosm

(large/universe or small/atomic scale)

Classical Mechanics and Quantum Mechanics

 

 

What concept/notion would you invent to describe the following image?

 

Image: Danleo~commonswiki assumed (based on copyright claims)

 

 

 

Sir Isaac Newton used "quantitas motus":

quantity of motion", as "arising from the velocity and quantity of matter conjointly", as mentioned in the wikipedia link (and section "History of concept")

 

 

Cf. https://en.wikipedia.org/wiki/Theory_of_impetus & Aristotelian dynamics

Impetus, Momentum, greek “Oρμή» (represented in “hormone”)

 

(Note: It could be argued that the notion of mass is not present in the image, but still the notion of momentum could be conveyed)

 

 

An interesting tweet:

 

College of France - Umberto Eco - The search of a perfect language in the history of the european culture

Collège de France ‏@cdf1530  Feb 22

#UmbertoEco – La quête d'une langue parfaite dans l'histoire de la culture européenne >> http://bit.ly/1L5Xpz2

 

 

 

 

 

 

Motion in the Macrocosm - Classical Mechanics

 

https://en.wikipedia.org/wiki/Motion_(physics)

Motions of all large scale and familiar objects in the universe (such as projectilesplanetscells, and humans) are described by classical mechanics.

 

Whereas the motion of very small atomic and sub-atomic objects is described by quantum mechanics.

 

http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html

Motion is described in terms of position / displacement (x), time (t), velocity (v), and acceleration (a).

 

 

 

Introduction to Linear Motion and Rotational Motion

 

> Linear Motion 

https://en.wikipedia.org/wiki/Linear_motion

 

> Rotational Motion

https://en.wikipedia.org/wiki/Rotation

Excerpt: "A rotation is a circular movement of an object around a center (or point) of rotation. A three-dimensional object always rotates around an imaginary line called a rotation axis. If the axis passes through the body's center of mass, the body is said to rotate upon itself, or spin. A rotation about an external point, e.g. the Earth about the Sun, is called a revolution or orbital revolution, typically when it is produced by gravity."

(End of excerpt).

 

In rotational motion, the displacement is an angle and we therefore talk about “angular displacement”.

 

For this reason, we use angular velocity (rate of change of angular displacement) defined  as ω=Δθ/Δt.

 

We also use angular acceleration termed α (rate of change of angular velocity).

 

 

> On momentum

https://en.wikipedia.org/wiki/Motion_(physics)

Momentum is a quantity which is used for measuring the motion of an object. An object's momentum is directly related to the object's mass and velocity.

 

Linear momentum

https://en.wikipedia.org/wiki/Momentum

Momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities.

The momentum of a particle is traditionally represented by the letter p. It is the product of two quantities, the mass (represented by the letter m) and velocity (v):[1]

 

 

The rotational analog of mass for linear motion, is rotational inertia termed moment of inertia. One has to define both the mass and its distance from the rotation axis. For a point mass the moment of inertia is the mass times the square of perpendicular distance to the rotation axis, I = m * r2

The moment of inertia has the same direction as the rotation axis.

 

http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html#am

The angular momentum of a rigid object is defined as the product of the moment of inertia and the angular velocity.

L= I * ω

The above relationship can be transformed (https://en.wikipedia.org/wiki/Angular_momentum) as follows:

L= r * p

This is the cross product of the position vector r and the linear momentum p=mv of the particle.

 

 

 

> On Moment

What is moment in physics?

https://en.wikipedia.org/wiki/Moment_(physics)

In physics, moment is a combination of a physical quantity and a distance.

For example, a moment of force is the product of a force and its distance from an axis, which causes rotation about that axis.

 

Principle derived from Archimedes. “Archimedes noted that the amount of force applied to the object, the moment of force, is defined as M = rF, where F is the applied force, and r is the distance from the applied force to object.”

 

 

Examples

 

 

 

 

> Rotational Motion and Torque (<to twist) : A rotation action, a turning action

 

>> Pushing a knob on a heavy door

A knob is located as far away as possible from the axis that holds the door. By pushing the knob, you exert a force that creates a torque and rotates the heavy door on its axis (to open it). What happens if you try to push the door at a point closer to the axis? More effort will be needed.

 

Imagine that the door is represented in the wikipedia figure below by the green arrow and that you exert the force F in purple on a specific point "at the end of the arrow" termed A.

 

Image: By StradivariusTV - Own work, CC BY-SA 3.0

 

 

You analyze the force in two components. As the door is fixed on its axis the FII  (parallel) component cannot have any productive effect (it cannot pull the door out of its axis). Only the FI tangential component (that is tangent to the circular path) can move/accelerate the door along its path.

 

In this way we have analysed the motion in two linear components.

We can relate the FI to the tangential acceleration of the point with Newton’s second law by writing:

FI=m *aI

Where aI is the linear acceleration tangent to the path of the point

 

How is this specific linear component related to angular acceleration?

It is proven that aI=a *r

 

The torque acting on the point is

τ= FI * r =  m *ai * r= m * (a *r) *r= m * r2 *a = (m * r2 ) *a

 

The quantity m * r2 is called rotational inertia, I.

 

Rotational inertia is considered to be the rotational analog for mass. We could say that in rotation it is not only the mass that is important but also the distance of the mass from the axis (cf. knob on heavy door).

 

Therefore, the torque is:

τ  = I * a

 

 

>> Turning a wheel

 

Here is an example of turning a wheel presented with screen captures that have been copied from this video:

https://www.youtube.com/watch?v=ty9QSiVC2g0

 

Torque is "Force applied" times the "distance or radius away from the rotating axis".

 

 

 

 

"Force" points downwards.

Determine the direction of the "torque".

It is 90 degrees to the force and points outwards towards the reader.

 

 

 

 

The more you apply the "torque", the more you increase the "angular momentum" of the wheel (the red arrow becomes stronger and stronger").

 

 

 

 

 

Rotating bodies and Precession

 

Spin a top (toy in image below). The top spins/rotates around its (rotational) axis.

 

 

After a while, it becomes instable, it wobbles. The rotation axis is no longer vertical but it tilts; there is an angle with the vertical axis. We say that the object "precesses".

 

Precession: Change in the orientation in the rotating axis of a rotational body

 

Can you imagine what kind of shape the axis creates in space?

Stereometry enhances imagination!

It is a cone.

 

Note that as mentioned in wikipedia there is "gradual shift in the orientation of Earth's axis of rotation, which, similar to a wobbling top, traces out a pair of cones joined at their apices in a cycle of approximately 26,000 years.[1] "

 

Image source

 

 

 

 

> What is the cause of precession for a top?

From https://en.wikipedia.org/wiki/Precession

"The torque caused by the normal force (–Fg) and the weight of the top causes a change in the angular momentum L in the direction of that torque. This causes the top to precess."

 

http://hyperphysics.phy-astr.gsu.edu/hbase/top.html

A rapidly spinning top will precess in a direction determined by the torque exerted by its weight.

 

 

 

 

> Earth Precession and Equinox March 20th 4:30 UTC/GMT 

 

By I, Dennis Nilsson, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=3262268

 

"Everything you need to know: Vernal (spring) equinox 2016"

http://earthsky.org/?p=68679

 

 

 

>> "Great tilt gave Mars a new face"

 

Link to Press Release in English

Excerpt: "It wasn't the rotation axis of Mars that shifted (a process known as variation of obliquity) but rather the outer layers (mantle and crust) that rotated with respect to the inner core (...)."

 

#Mars : un grand basculement a refaçonné sa surface bit.ly/1TovnlG c/ @INSU_CNRS pic.twitter.com/S3VqMYcIvW

Extrait: "Ce n'est pas l'axe de rotation de Mars qui a bougé (phénomène que l'on appelle variation de l'obliquité) mais les parties externes (manteau, croûte) qui ont tourné par rapport au noyau interne (...)"

 

 

> An example of precession for a wheel

[Gyroscopic Precession (torque-induced Precession)]

 

Source 1:

http://hyperphysics.phy-astr.gsu.edu/hbase/rotv2.html#rvec4

 

 

Source 2 - presented with screen captures below:

https://www.youtube.com/watch?v=ty9QSiVC2g0

 

Let us examine what will happen if we have the wheel in the following setting and we let it go.

 

Weight points down. This force is applied at a distance x (blue arrow) from its pivoting point (attached to the string).

It will do a swing (around that point).

 

 

 

But what if it is turning and we let it go?

 

"Angular momentum" in red.

What is the direction of the "torque"?

Indicated by the person's finger.

What will be the result?

 

 

Torque will shift angular momentum.