Rongzhou: /* Gravity: any two mass attracts */

2026-09-24T15:13:07Z

Gravity: any two mass attracts

← Older revision		Revision as of 15:13, 24 September 2026
Line 142:		Line 142:

	The potential energy as a result of gravitational attraction is known as [[gravitational potential energy]].		The potential energy as a result of gravitational attraction is known as [[gravitational potential energy]].

			<blockquote>
			See also: [[Two body problem]]
			</blockquote>

Rongzhou: /* Gravity: any two mass attracts */

2026-09-24T13:34:58Z

Gravity: any two mass attracts

← Older revision		Revision as of 13:34, 24 September 2026
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	! Symbol !! Meaning		! Symbol !! Meaning
	\|-		\|-
	\| $\mathbf r_A$ \|\| position of ~~a tiny~~ piece of body $A$.		\| $\mathbf r_A$ \|\| position vector of an infinitesimal piece of body $A$.
	\|-		\|-
	\| $\mathbf r_B$ \|\| position of ~~a tiny~~ piece of body $B$.		\| $\mathbf r_B$ \|\| position of an infinitesimal piece of body $B$.
	\|-		\|-
	\| $\rho_A(\mathbf r_A), \rho_B(\mathbf r_B)$ \|\| density at those points.		\| $\rho_A(\mathbf r_A), \rho_B(\mathbf r_B)$ \|\| density at those points.
	\|-		\|-
	\| $dV_A, dV_B$ \|\| ~~tiny~~ volumes.		\| $dV_A=dr_A^3, dV_B=dr_B^3$ \|\| infinitesimal volumes.
	\|-		\|-
	\| $dm_A=\rho_A dV_A$, $dm_B=\rho_B dV_B$ \|\| ~~tiny~~ masses.		\| $dm_A=\rho_A dV_A$, $dm_B=\rho_B dV_B$ \|\| infinitesimal masses.
	\|-		\|-
	\| $\lvert \mathbf r_B-\mathbf r_A\rvert$ \|\| distance between the two ~~tiny~~ pieces.		\| $\lvert \mathbf r_B-\mathbf r_A\rvert$ \|\| distance between the two infinitesimal pieces.
	\|-		\|-
	\| $\dfrac{\mathbf r_B-\mathbf r_A}{\lvert \mathbf r_B-\mathbf r_A\rvert^3}$ \|\| unit direction from $A$-piece to $B$-piece, divided by distance squared.		\| $\dfrac{\mathbf r_B-\mathbf r_A}{\lvert \mathbf r_B-\mathbf r_A\rvert^3}$ \|\| unit direction from $A$-piece to $B$-piece, divided by distance squared.
	\|}		\|}

			The potential energy as a result of gravitational attraction is known as [[gravitational potential energy]].

Rongzhou at 12:48, 23 September 2026

2026-09-23T12:48:26Z

← Older revision		Revision as of 12:48, 23 September 2026
Line 122:		Line 122:

	{{#content:Q1778}}		{{#content:Q1778}}

			{\| class="wikitable"
			! Symbol !! Meaning
			\|-
			\| $\mathbf r_A$ \|\| position of a tiny piece of body $A$.
			\|-
			\| $\mathbf r_B$ \|\| position of a tiny piece of body $B$.
			\|-
			\| $\rho_A(\mathbf r_A), \rho_B(\mathbf r_B)$ \|\| density at those points.
			\|-
			\| $dV_A, dV_B$ \|\| tiny volumes.
			\|-
			\| $dm_A=\rho_A dV_A$, $dm_B=\rho_B dV_B$ \|\| tiny masses.
			\|-
			\| $\lvert \mathbf r_B-\mathbf r_A\rvert$ \|\| distance between the two tiny pieces.
			\|-
			\| $\dfrac{\mathbf r_B-\mathbf r_A}{\lvert \mathbf r_B-\mathbf r_A\rvert^3}$ \|\| unit direction from $A$-piece to $B$-piece, divided by distance squared.
			\|}

Rongzhou: Created page with "Classical mechanics is one framework to the study of mechanics. == Validity == == Vector description == Concise description of mechanical systems can be provided with vectors. === Reference frame === A reference frame is defined by an origin $O$, and the three unit vectors for each dimension of space: $\overrightarrow{x}$, $\overrightarrow{y}$, and $\overrightarrow{z}$. It can be noted as: $$R(O;\overrightarrow{x},\overrightarrow{y},\overrightarrow{z})$$..."

2026-09-23T06:47:23Z

Created page with "Classical mechanics is one framework to the study of mechanics. == Validity == == Vector description == Concise description of mechanical systems can be provided with vectors. === Reference frame === A reference frame is defined by an origin $O$, and the three unit vectors for each dimension of space: $\overrightarrow{x}$, $\overrightarrow{y}$, and $\overrightarrow{z}$. It can be noted as: $$R(O;\overrightarrow{x},\overrightarrow{y},\overrightarrow{z})$$..."

New page

Classical mechanics is one framework to the study of [[mechanics]].

== Validity ==

== Vector description ==

Concise description of mechanical systems can be provided with [[vector]]s.

=== Reference frame ===

A reference frame is defined by an origin $O$, and the three unit vectors for each dimension of space: $\overrightarrow{x}$, $\overrightarrow{y}$, and $\overrightarrow{z}$.

It can be noted as:

$$R(O;\overrightarrow{x},\overrightarrow{y},\overrightarrow{z})$$

Conventionally, we choose unit vectors perpendicular to each other ($\overrightarrow{x}\bot\overrightarrow{y}\bot\overrightarrow{z}$), so each spatial direction is independent of another.

=== Objects in the reference frame ===

==== Motion: Position, velocity, accelaration ====

For an object (S) centered at point $P$, its '''position''' relative to reference frame $R$ can be described by its position vector $\overrightarrow{OP}=(\overrightarrow{x_P},\overrightarrow{y_P},\overrightarrow{z_P})$.

Then velocity, defined as the instantaneous change in position with respect to time, is then:

$$\overrightarrow{v_S}=\frac{d \overrightarrow{OP}}{dt}=(\overrightarrow{v_x},\overrightarrow{v_y},\overrightarrow{v_z})$$

Finally, acceleration, defined as the instantaneous change in velocity with respect to time, is then:

$$\overrightarrow{a_S}=\frac{d \overrightarrow{v_S}}{dt}=(\overrightarrow{a_x},\overrightarrow{a_y},\overrightarrow{a_z})$$

==== Mechanical contact: Force and moments ====

Any ''force'' $\overrightarrow{F}$ on object $(S)$ can be decomposed into component forces in each of the three independent spatial direction:

$$\overrightarrow{F}= \overrightarrow{F_x}+\overrightarrow{F_y}+\overrightarrow{F_z}$$

Therefore, it can be described as a vector:

$$\overrightarrow{F}= (F_x,F_y,F_z)$$

where $$F_x=\left\| \overrightarrow{F_x} \right\|$$ and so on.

As a result, if the force $\overrightarrow{F}$ is applied to point $A$ the ''moment'' $M(\overrightarrow{F}\to S)_B$ caused by such force at point $B$ is:

$$\overrightarrow{M}(\overrightarrow{F}\to S)_B=\overrightarrow{BA}\wedge \overrightarrow{F}$$

It can also be decomposed into components along three independent spatial directions:

$$\overrightarrow{M}=(M_x,M_y,M_z)$$

== Key principles ==

=== Newton's 1st law ===

Newton's 1st law describes the phenomenon '''inertia''': when the overall net external force is zero, the velocity of an object is maintained.<ref>{{#cite:Q1761}}</ref>

In vector notation, it can be written as:

{{#content:Q1766}}

The state that $\sum \overrightarrow{F}_{ext}=0$ is known '''mechanical equilibrium'''.

=== Newton's 2nd law ===

Newton's 2nd law describes the relationship between force, mass, and acceleration: the acceleration is proportional to force and the inverse of mass.<ref>{{#cite:Q1761}}</ref>

In vector notation, it can be written as:

{{#content:Q1767}}

=== Newton's 3rd law ===

Newton's 3rd law describes the reciprocity of force: when object $(S1)$ applies a force on object $(S2)$, $(S2)$ must apply a force opposite to $(S1)$ and equal in magnitude.<ref>{{#cite:Q1761}}</ref>

In vector notation, it can be written as:

{{#content:Q1768}}

=== Conservation of momentum ===

For any closed system, the momentum, defined as the product of mass and velocity (m\overrightarrow{v}), is conserved before and after any mechanical interaction within the system (including collision).<ref>{{#cite:Q1761}}</ref>

In vector notation, for a closed system of $n$ objects, the principle can be expressed as:

{{#content:Q1769}}

where $C$ is a constant.

Any change to momentum must be a direct result of an external force $\overrightarrow{F_{ext}}$ acting on a member of the system for a duration $t$. This change in momentum, known as ''impulse'', satisfies the following relationship:

{{#content:Q1770}}

=== Any motion carries kinetic energy ===

The kinetic energy of an object of mass $m$ and velocity $\overrightarrow{v}$ is:

{{#content:Q1771}}

Any change in kinetic energy, is the result of an external force acting over a given distance:<ref>{{#cite:Q1761}}</ref>

{{#content:Q1773}}

=== Conservation of energy ===

In a closed system, energy is conserved. It can be transformed from one form to another, but not created or destroyed:<ref>{{#cite:Q1761}}</ref>

{{#content:Q1772}}

=== Gravity: any two mass attracts ===

Two point masses at points $A$ and $B$ exerts a gravitational force on each other:<ref>{{#cite:Q1761}}</ref>

{{#content:Q1777}}

According to [[shell theorem]], spherically symmetric shapes can be approximated by point masses.

For real-world objects, whose shape may be irregular, if the distance between the object is much larger than the size of a given object ($\left\| \overrightarrow{AB} \right\|\gg R_A$), that object may be approximated as a point mass.

Otherwise, for arbitrary real-world objects:

{{#content:Q1778}}

Classical mechanics - Revision history

Rongzhou: /* Gravity: any two mass attracts */

Rongzhou: /* Gravity: any two mass attracts */

Rongzhou at 12:48, 23 September 2026