Motion in the Macrocosm and the Microcosm

 

Motion at large scale or universal dimensions versus motion at small or atomic scale.

 

 

Classical Mechanics and Quantum Mechanics

 

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

 

 

Figure 1: From Wikipedia - Danleo~commonswiki assumed (based on copyright claims)

 

 

 

Sir Isaac Newton used "quantitas motus" to describe what we refer to as "momentum"

 

Sir Isaac Newton used the term "Quantitas motus" or "Quantity of motion" as "arising from the velocity and quantity of matter conjointly" to describe the notion we currently refer to as "mometum" (Reference). Please consider also this reference & Aristotelian dynamics for the tem "Impetus" and the greek term "Oρμή" (represented in “hormone”).

 

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

 

 

An interesting tweet from "Collège de France" on Umberto Eco: The quest for a perfect language in the history of european culture > http://bit.ly/1L5Xpz2

 

 

 2: Picture of Umberto Eco

 

 

 

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 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

Linear motion (also called rectilinear motion) is a one-dimensional motion along a straight line.

 

 

> Rotational Motion

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

"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."

 

 

> Displacement

In linear motion, displacement is the distance traveled by the object.

 

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

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

 

> Velocity

Generally, velocity is the displacement in the unit of time.

 

Angulal Velocity

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

http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#avel

In rotational motion, as the displacement is an angle, we use angular velocity represented by ω, which we define as the rate of change of angular velocity or informally as the "angle" that the object "runs" in the unit of time:

 

ω=Δθ/Δt

 

 

Figure 3: Angular velocity (From Wikipedia)

 

https://en.wikipedia.org/wiki/Angular_velocity#Particle_in_three_dimensions

Angular velocity is a vector (or pseudovector) whose magnitude measures the rate at which the radius sweeps out angle, and whose direction shows the principal axis of rotation. Its up-or-down direction is given by the right-hand rule.

 

 

Figure 4: Angular velocity (From Wikipedia) - By dnet based on raster version released under GFDL - self-made by tracing raster version

 

The angular velocity is measured relative to a chosen center point, called the origin (O in the next figure).

The direction axis is the axis of the rotation, the reason being that a rotating object changes direction continuously and it is therefore difficult to track its direction (reference).

 

 

 

Figure 5: Linear velocity in rotational motion (From Wikipedia)

 

The linear velocity v of the particle is split in a radial component (parallel to the radius) VII and a tangential component (perpendicular to the radius or circular or cross-radial)

 Therefore:

 

The angular velocity is linked to the tangential or cross-radial velocity as follows:

  • {\displaystyle \omega ={\frac {d\phi }{dt}}={\frac {v_{\perp }}{r}}.}
 
 

 

> 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 p

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

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

https://www.chegg.com/homework-help/definitions/linear-momentum-2

 

The momentum of a particle, traditionally represented by the letter p, is the product its mass (m) times its velocity (v).

It is a vector quantity, meaning it has both a magnitude and a direction.

 

 

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

Another expression can be derived from the above (https://en.wikipedia.org/wiki/Angular_momentum). Therefore:

 

The angular momentum of a particle with respect to the origin O is a vector quantity defined as the cross product of cross product of the position vector r and the linear momentum p=mv of the particle.

L= r * p (vector symbols over letters omitted)

 

Please note that it was mentioned previously that the angular velocity depends on the tangential component of the linear velocity. Similarly:

L= r * ptangential

 

The angular momentum has meaning only with respect to a specific origin and its direction is always perpendicular to the plane formed by the position and linear momentum vectors r and p.

 

 

Figure 6: Reproduction from a Wikipedia animation "by Yawe - Own work, Public Domain, " included at the Wikipedia page https://en.wikipedia.org/wiki/Angular_momentum

 

 

 

 

 

> Orbital angular momentum and spin angular momentum

 

 

Figure 7 | On the left: Spin angular momentum S due to the rotation of a sphere upon itself. In the middle: Orbital angular momentum L of a sphere due to its rotation around an origin (the angular momentum has meaning only with respect to a specific origin and its direction is always perpendicular to the plane formed by the position and linear momentum vectors r and p.) On the right: Total angular momentum J is the vector sum of L and S. (Figure 3: From Wikipedia (selection) - By Maschen - Own work, CC0, included at https://en.wikipedia.org/wiki/Angular_momentum).

 

The above image from Wikipedia shows on the left an object that rotates upon itself (rotation axis passes from mass center) or "spins" and on the right an object that performs a rotation around an origin at distance r (it is in orbit around an origin). We refer to the first as having "spin angular momentum" S and to the second as having "orbital angular momentum" L. We can add the two vectors L and S to calculate the total angulal momentum J. 

 

 

 

Figure 8: From WikipediaBy Maschen - Own work, Public Domain, -included at https://en.wikipedia.org/wiki/Angular_momentum

 

The above image from Wikipedia also shows the additions of the vectors of the two angular momenta to calculate total angular momentum. The conservation of angular momentum applies to the total angular momentum (not the individual components). Please also refer to https://en.wikipedia.org/wiki/Angular_momentum#Spin,_orbital,_and_total_angular_momentum.

 

 

 

> 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:

  • Moment of force or torque: τ  =r * F.  (It is a first moment)
  • Moment of inertia: I = Σ *m* r2( I = Σ m r 2 ) {\displaystyle (I=\Sigma mr^{2})}. (It is analogous to mass in rotational motion. It is a measure of an object's resistance to changes in its rotation rate. It is a second moment of mass.)
  • Electric dipole moment : The electric dipole moment between a charge of –q and q separated by a distance of d is p = q * d. (It is a first moment)
  • Magnetic (dipole) moment cf. Amperian loop model: (μ = Ι * S)  (It represents the strength and orientation of a magnetic source).

 

> On Dipole Moment

 

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

Dipole moment may refer to:

 

 

 

> Torque (Rotational Force)

 

>> 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 rotates the heavy door on its axis (to open it). You are exerting a torque. 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 radius vector r.

 

Image 9: From Wikipedia - by StradivariusTV - Own work

 

 

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 (the component that is perpendicular to the radius) 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

Figure 10: Reproduction from a Wikipedia animation "by Yawe - Own work, Public Domain, " included at the Wikipedia page https://en.wikipedia.org/wiki/Angular_momentum

 

 

 

>> 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

 

Let us consider a spinning top, the toy in the image below. 

 

 

When we spin a top it rotates (spins) 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 of 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.