As you finish lessons and skills through courses and disciplines, you gradually gain more knowledge. It can be a lot and hard to remember and keep track of. Therefore this site stores your newly gained scientific key-points as memories in your profile’s Badge & Memory Inventory.

Memories contain either **properties**, **mechanisms** or **phenomena**. Below is a list of all achievable memories (the list is continuously growing as more courses are made available on the site).

## Time $t$

The distinction between the ‘after’ and the ‘before’. The arrow of time tells that time has a direction (always from ‘before’ to ‘after’).

- Fundamental scalar property
- Symbol: $t$
- SI-unit: second $\mathrm s$

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## Quantity $n$

The number of things; the counting of things.

- Fundamental scalar property
- Symbol: $n$
- SI-unit: (none)

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## Length $L$, area $A$ & volume $V$

Size properties that cover a portion of space in 1, 2 and 3 dimensions. Length is the space straight between two positions. Sweep out the space from each point along a length to have area. Sweep out the space from each point of an area to have volume.

- Scalar properties derived from position
- Symbols: $L$, $A$ and $V$
- SI-units: metre $\mathrm m$, square metre $\mathrm m^2$ and cubic metre $\mathrm m^3$

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## Angle $\theta$

The turning or direction away from a reference direction.

- Scalar property derived from position
- Symbol: $\theta$
- SI-unit: none (or radians). Other relevant unit: degree $^\circ$

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## Position $\vec s$

Points to a location in space. The property of space that defines space.

- Vector property
- Symbol: $\vec s$
- It’s magnitude is typically termed
*distance*or*length* - SI-unit: metre $\mathrm m$

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## Curvature $k$

How a line bends or curves. One-dimensional only. Upwards-curving (“happy smile”) gives a positive curvature; downwards-curving (“sad smile”) a negative curvature.

$$k=\frac 1r$$

$r$ is radius of the imagined circle that the curve is a part of.

- Scalar property
- Symbol: $k$
- SI-unit: Inverse metres $\mathrm{m^{-1}}$

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## Gaussian curvature $G$

How a surface curves in 2 dimensions. If negative, the shape is saddle-like (some directions upwards, others downwards); if positive, all directions are the same way (a ball, a mountain top, a valley); if zero, at least one direction is straight/non-curving. Calculated for a single point.

$$G=k_\text{smallest}\cdot k_\text{largest}$$

The $k$’s are the smallest and largest 1D curvatures (principle curvatures) that pass through the point.

- Scalar property
- Symbol: $G$
- SI-unit: Inverse metres $\mathrm{m^{-2}}$

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## Velocity $\vec v$

How fast you go; how far you move over time in a particular moment.

$$\vec v=\frac{\mathrm d\vec s}{\mathrm dt}$$

$\vec s$ is position and $t$ is time.

- Vector property
- Symbol: $\vec v$
- It’s magnitude is typically named
*speed* - SI-unit: Metres-per-second $\mathrm{m/s}$

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## Acceleration $\vec a$

How much faster you go; how fast you are speeding up; how many extra metres-per-second your velocity increases with each second.

$$\vec a=\frac{\mathrm d\vec v}{\mathrm dt}$$

$\vec v$ is velocity and $t$ is time.

- Vector property
- Symbol: $\vec a$
- SI-unit: Metres-per-second-squared $\mathrm{m/s^2}$

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## Abseleration, absity, absition… Jerk, snap, crackle, pop

A couple of the derivatives and integrals of position $\vec s$ are the following. We do not assign them symbols here as they are rare:

**Abseleration**is the integral of absity: $\int\int\int \vec s\,\mathrm dt \,\mathrm dt \,\mathrm dt$ with units of $\mathrm{m\cdot s^3}$,**absity**the integral of absition: $\int\int \vec s\,\mathrm dt \,\mathrm dt$ with units of $\mathrm{m\cdot s^2}$, and**absition**the integral of position: $\int \vec s\,\mathrm dt$ with units of $\mathrm{m\cdot s}$.

**Jerk**is the derivative of acceleration: $\frac{\mathrm d^3\vec s}{\mathrm dt^3}$ with units of $\mathrm{m/s^3}$,**Snap**the derivative of jerk: $\frac{\mathrm d^4\vec s}{\mathrm dt^4}$ with units of $\mathrm{m/s^4}$,**Crackle**the derivative of snap: $\frac{\mathrm d^5\vec s}{\mathrm dt^5}$ with units of $\mathrm{m/s^5}$, and**Pop**the derivative of crackle: $\frac{\mathrm d^6\vec s}{\mathrm dt^6}$ with units of $\mathrm{m/s^6}$.

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## Force $\vec F$

How hard something is hitting/impacting; the cause of changes in motion; the cause of acceleration.

- Vector property
- Symbol: $\vec F$
- SI-unit: Newton $\mathrm{[N]=[kg\cdot m/s^2]}$

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## Mass $m$

How “tough” something is to move; resistance against changes in linear motion; resistance against acceleration.

Mass is linear inertia.

- Scalar property
- Symbol: $m$
- SI-unit: Kilogram $\mathrm{kg}$

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## Friction coefficients $\mu_k$, $\mu_s$

The two touching surfaces’ resistance against being separated.

- Scalar property
- Symbol: $\mu$
- SI-unit: (none)

For kinetic friction models, its symbolised $\mu_k$, for static friction models $\mu_s$.

## Spring constant $k$

The resistance against elastic elongation or compression of a material in one dimension.

- Scalar property
- Symbol: $k$
- SI-unit: Newton-per-metre $\mathrm{N/m}$

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## Angular position $\vec \theta$

Where you are in a turn. Location as an angle.

- Vector property
- Symbol: $\vec \theta$
- SI-unit:
*none*(radians) - Other usual units: degree $^\circ$

## Angular velocity $\vec \omega$

How fast something is turning.

$$\vec \alpha=\frac{\mathrm d\vec \omega}{\mathrm dt}$$

$\vec omega$ is angular velocity and $t$ is time.

- Vector property
- Symbol: $\vec \omega$
- SI-unit: $\mathrm{s^{-1}}$

## Angular acceleration $\vec \alpha$

How fast a turning changes. How much faster/slower somethings speeds up/down.

$$\vec \omega=\frac{\mathrm d\vec \theta}{\mathrm dt}$$

$\vec \theta$ is angular position and $t$ is time.

- Vector property
- Symbol: $\vec \alpha$
- SI-unit: $\mathrm{s^{-2}}$

## Torque $\vec \tau$

How hard something is being turned; the cause of changes in angular motion; the cause of angular acceleration.

$$\vec \tau=\vec r\times \vec F$$

$\vec r$ is the position vector from the axis-of-rotation and $\vec F$ the force. Numerically, only the perpendicular components are multiplied: $$\tau=r\,F_\perp=r_\perp\,F$$

- Vector property
- Symbol: $\vec \tau$
- SI-unit: Newton-metre $\mathrm{[N\cdot m]=[kg\cdot m^2/s^2]}$

## Moment-of-inertia $I$

How “tough” it is to turn something (to slow down or speed up that turning). Rotational equivalent to mass.

$$I=\sum mr^2$$

$m$ is mass and $r$ distance from the axis-of-rotation to each particle of the rotating object. $I$ is different about different axes.

- Scalar property
- Symbol: $I$
- SI-unit: Kilogram-metres-squared $\mathrm{kg\cdot m^2}$

Moments-of-inertia about various axes are found in Resource: Moments-of-inertia.

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## Momentum $\vec p$

The “toughness” of catching/stopping something in motion or throwing something/speeding something up from rest.

$$\vec p=m\vec v$$

where $m$ is mass and $\vec v$ impact velocity.

- Vector property
- Symbol: $\vec p$
- SI-unit: kilogram-metres-per-second $\mathrm{kg\cdot m/s}$

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## Impulse $\vec J$

The “hardness” of an impact/collision.

$$\vec J=\vec F\Delta t=\Delta \vec p$$

where $\vec F$ is impact force (constant during the impact), $\Delta \vec p$ total transferred momentum, and $\Delta t$ impact duration. For non-constant impact force, we’ll instead sum up the impulse during all instantaneous moments with an integral: $\vec J=\int \vec F\,\mathrm dt$.

- Vector property
- Symbol: $\vec F$
- SI-unit: Kilogram-metre-per-second $\mathrm{kg\cdot m/s}$

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## Angular momentum $\vec L$

The “toughness” of stopping something heavy which is spinning, or starting a spinning motion of something from rest.

$$\vec L=I\vec \omega$$

where $I$ is moment-of-inertia and $\vec \omega$ angular speed at impact. For separated objects/particles rotating, the angular momentum that each contributes with is:

$$\vec L=\vec r \times \vec p$$

where $\vec r$ is the object’s distance from the centre-of-rotation and $\vec p$ is it’s momentum.

- Vector property
- Symbol: $\vec L$
- SI-unit: Kilogram-times-metres-squared-per-second $\mathrm{kg\cdot m^2/s}$

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## Kinetic energy $K$

Energy carried with with motion (translational and rotational).

$$K_\text{trans}=\frac 12 mv^2\qquad\qquad K_\text{rot}=\frac12 I\omega ^2$$

$m$ is mass, $v$ speed, $I$ the moment-of-inertia and $\omega$ angular speed.

- Scalar property
- Symbol: $K$, $KE$, $E_k$
- SI-unit: Joule $\mathrm{[J]=[kg\cdot m^2/s^2]}$

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## Gravitational potential energy $U_g$

Energy stored due to separation in location between objects that attract each other via gravity.

$$U_g=mgh$$

$m$ is mass, $g$ the gravitational acceleration and $h$ the height (distance) between the attracting objects.

- Scalar property
- Symbol: $U_g$, $PE$, $E_\text{pot,g}$, $E_\text{grav}$
- SI-unit: Joule $\mathrm{[J]=[kg\cdot m^2/s^2]}$

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## Elastic potential energy $U_\text{elastic}$

Energy stored in a taught (elongated or compressed) spring, rubber band or other elastic material which via an elastic force wants to return to its original size.

$$U_\text{elastic}=\frac 12k\Delta L^2$$

$k$ is the spring constant and $\Delta L$ the difference in length (from relaxed length).

- Scalar property
- Symbol: $U_\text{elastic}$, $E_\text{pot,sp}$, $U_\text{spring}$
- SI-unit: Joule $\mathrm{[J]=[kg\cdot m^2/s^2]}$

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## Work $W$

Energy that is transferred mechanically (over a distance)

$$W=\vec F\cdot \Delta \vec s$$

$\vec F$ is force that acts while a displacement of $\Delta \vec s$ takes place.

- Scalar property
- Symbol: $W$
- SI-unit: Joule $\mathrm{[J]=[kg\cdot m^2/s^2]}$

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## Heat $Q$

Energy being transferred thermodynamically (from a cold to a warm place).

- Scalar property
- Symbol: $Q$
- SI-unit: Joule $\mathrm{[J]=[kg\cdot m^2/s^2]}$

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## Power $P$

How fast energy is transferred

$$P=\frac{\mathrm dE}{\mathrm dt}$$

$E$ is energy being transferred (work $W$ or heat $Q$) over time $t$.

- Scalar property
- Symbol: $P$
- SI-unit: Joule-per-second $\mathrm{[J/s]=[kg\cdot m^2/s^3]}$

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## Curvature conservation law

The Gaussian curvature $G$ of a 2D surface stays constant before and after any deformation of non-flexible materials.

$$G_\text{before}=G_\text{after}$$

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## Motion equations

The four motion equations combine the properties (position/length $s$, speed $v$, initial speed $v_0$, acceleration $a$, time $t$) of motion along a path/direction.

$$v=v_0+at\\

s=s_0+v_0t+\frac12 at^2\\

v^2=v_0^2+2a(s-s_0)\\

s=s_0+\frac12 (v+v_0)t$$

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## Turning & circular motion

Turning requires sideways acceleration, perpendicular to the velocity’s direction. This acceleration is called *centripetal *or *radial acceleration*; typically symbolised $a_\perp$, $a_{cen}$ or $a_{rad}$.

Keeping $a_\perp$ constant causes circular motion (as driving in a roundabout). If the speed $v$ is constant (no parallel acceleration), the circular motion is called *uniform* and the following expression holds true:

$$a_\perp=\frac{v^2}r$$

$r$ is the radius of the circular path.

## Newton’s 1st law

An object does not accelerate if not acted upon by a net force / resulting force $\sum \vec F$. Can be considered a special case of Newton’s 2nd law. Can be considered defining the concept of inertial reference frames.

$$\sum \vec F=0$$

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## Newton’s 2nd law

Acceleration is caused by a net/resulting force, $\sum \vec F$; and vice versa, a net force causes acceleration $\vec a$. This acceleration is “dampened” or resisted by the mass $m$.

$$\sum \vec F=m\vec a$$

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## Angular-motion equations

The four motion equations combine the properties (angular position/length $\theta$, angular speed $ \omega $, initial angular speed $ \omega_0$, angular acceleration $\alpha$, time $t$) of angular motion in a curved turn.

$$\omega= \omega_0+\alpha t\\

\theta=\theta_0+v_0t+\frac12 \alpha t^2\\

\omega^2= \omega_0^2+2\alpha (\theta-\theta_0)\\

\theta=\theta_0+\frac12 ( \omega + \omega_0)t$$

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## Parallel-axis theorem

To find the moment-of-inertia $I$ of an object about any axis, find the moment-of-inertia about a parallel axis through the centre-of-mass $I_\mathrm{com}$ and add a term that covers the distance $d$ from new axis-of-rotation to centre-of-mass as well as the total mass $m$ of the object:

$$I= I_\mathrm{com} +dm^2$$

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## Newton’s 2nd law in rotation

Angular acceleration is caused by a net/resulting torque, $\sum \vec \tau$; and vice versa, a net torque causes angular acceleration $\vec \alpha$. This angular acceleration is “dampened” or resisted by the moment-of-inertia $I$.

$$\sum \vec \tau=I\vec \alpha $$

## Newton’s 3rd law

When exerting a force on something, that something always exerts an equal but opposite force on you. Typically called an action/reaction force-pair.

$$\vec F_\text{A on B}=-\vec F_\text{B on A}$$

## “Speed difference” conservation

In elastic collisions, the difference in speed between the two colliding objects/particles stays the same before and after the collision.

$$\Delta v_\text{before}=\Delta v_\text{after}$$

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## Momentum conservation

The total momentum $\vec p$ before and after any collision always stays the same.

$$\sum \vec p_\text{before}=\sum \vec p_\text{after}$$

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## Angular momentum conservation

The total angular momentum $\vec L$ before and after any rotational collision (such as two spinning cogwheels suddenly interlocking) always stays the same.

$$\sum \vec L_\text{before}=\sum \vec L_\text{after}$$

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## Potential energy

Stored energy that can be retrieved when released (a taut spring, a high-up object on Earth, two separated charges etc.) is called *potential *$U$ and exists due to a conservative force $\vec F$ which tries to minimise this energy.

$$\vec F=-\nabla U$$

#### 2 Required Steps

- Complete the Gradient skill
- Complete the Potential energy vs force skill

## Work-energy theorem

When there’s no energy waste (no work nor heat transferred out of the system), the sum of all work $W_\text{all}$ done by all forces causes the change in kinetic energy $K$.

$$W_\text{all}=\Delta K$$

Here, the work $W_\text{all}$ contains that of both non-conservative and conservative forces (the latter will usually be excluded from $W$ and instead added as potential energies $U$).

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## Energy conservation law (1st law of thermodynamics)

Energy cannot disappear into thin air, not appear out of the blue. With a total energy amount $E_1$ at any moment, energy can be added/removed as work $W$ or heat $Q$, giving the new total energy amount $E_2$ at a later moment.

$$E_1+W+Q=E_2$$

*Sign convention*: $W$ is work done on the system and $Q$ heat added to the system.

## Shape

The arrangement of points or “positions” in a continuous manner to form figures or objects. Shapes geometrically appear in 1, 2 and 3 dimensions and may include points (like corners), lines (like edges), areas (like surfaces) and volumes.

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## Gravitational acceleration $g$

All objects fall equally fast (with the same acceleration) due the gravity, regardless of their mass (ideally; in a vacuum). The acceleration everything falls with is called the **gravitational acceleration** $g$. On Earth’s surface, it has a value around:

$$g\approx 9.8\,\mathrm{m/s^2}$$

The value varies with around $\pm 0.02\,\mathrm{m/s^2}$ depending on location.

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## Centrifugal effect

The effect of an object being flung outwards (or forwards/backwards etc.) because the surroundings accelerate away from it (by turning, speeding, braking etc.).

Often gives the illusion that the object itself is accelerating rather than the surroundings. Is sometimes called *centrifugal force* (although no new force is involved).

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## Kinetic friction $\vec f_k$ (Amontons’ law)

Kinetic friction $\vec f_k$ applies solely when there is sliding between two surfaces. It is proportional to the normal force $n$ (the load) at not-too-large normal forces per area (does not depend on contact area or sliding speed).

$$f_k=\mu_k n$$

This is a macro-scale friction model. At larger normal forces per area, other kinetic friction models take over.

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## Static friction $\vec f_s$

Static friction $f_s$ applies solely when two touching surfaces do *not* slide over one another. It has an upper limit:

$$f_s\leq \mu_s n$$

$\mu_s$ is static friction coefficient, $n$ is normal force (the load).

External forces and other influences below this limit will not cause sliding, because $f_s$ will adjust to exactly balance them out.

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## Weigth $\vec w$

The attraction between two objects with mass is a gravitational force. When one object is enormous (like a planet) compared to the other (like a human-scale object), this attraction is typically called weight $\vec w$.

$$w=mg$$

$m$ is the mass of the smaller object, $g$ the gravitational acceleration towards the larger object at the location of the smaller object.

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## Normal force $\vec n$

The “holding-back” or “holding-up” force from surfaces against external influences (such as the wall’s force on your shoulder when leaning on it). Always perpendicular to the surface. Will always adjust exactly to the external force to balance it out.

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## Tension $\vec T$

The force in ropes, string, cables and the like. For non-stretchable ropes, tension is the same at every point along the rope.

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## Elastic force $\vec F_\text{elastic}$ (Hooke’s law)

A spring or other elastic object will want to return to it’s original/natural length when elongated or compressed. The larger the elongation/compression $\Delta L$, the larger the opposite $\vec F_\text{elastic}$.

$$F_\text{elastic}=k\Delta L$$

$k$ is stiffness/spring constant. The right-hand-side is added a negative sign, if $F_\text{elastic}$ and $\Delta L$ are considered with direction.

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## Pulleys

With pulleys, a load can be shared between the lifter and the ceiling or other hooked point. The same load is then lifted with less force by the lifter. The load is also lifted slower, though.

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## Geometric bonds

Geometric bonds tie the linear motion (position/length $s$, speed $v$ and acceleration $a$) of a point at the periphery with the rotational motion (angular position/length $\theta$, angular speed $\omega$ and angular acceleration $\alpha$) of the object, e.g. in the case of a rolling wheel.

$$s=r\theta\\

v=r\omega\\

a=r\alpha$$

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## Centre-of-mass

Shortened $\mathrm{com}$, the centre-of-mass is the point about which an object naturally rotates (e.g. when falling). There is equal “mass influence” on the rotational motion, meaning equal moment-of-inertia, on either side of the point.

$$\vec r_\mathrm{com}=\frac{\sum m\vec r}{\sum m}$$

$\vec r$ and $m$ are position and and mass of each point in the rotating object.

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## Rolling

Rolling (without slipping) can have static friction but no kinetic friction at the contact with the surface. This static friction is unaffected by the rolling direction. While rolling, if no external forces act no energy is spent and the wheel can ideally continue rolling at constant speed forever without ever stopping.

## Elastic & inelastic collisions

Collisions where the (kinetic) energy is kept within the system are called *elastic* (billiard balls colliding), otherwise *inelastic* (cars crashing where energy is spent to deform the material, produce heating etc.)

All collisions/impacts can be categories as one of these two.

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## Mechanical pain $\Delta \vec a_\text{body parts}$

Human beings feel it when one part of their body accelerates differently than another part, since pain receptors detect stretches of molecular bonds within our tissue. Large acceleration differences might cause bonds to break, which our brain via pain receptors interprets as the pain we feel.

## Precession

A falling object that rotates will fall “sideways” rather than straight down in the direction of gravity. This causes a sideways motion that becomes a sideways (horizontal) rotation, called **precession**. The precession angular speed is:

$$\omega_\text{precession}=\frac{wr}{L_\text{spin}}$$

where $w$ is the weight, $r$ the precession circle’s radius and $L_\text{spin} the angular momentum due to the object’s spin about its own tilted axis.

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## Rotational stability (gyroscope)

A spinning object shows gyroscopic behaviour or “rotational stability” due to precession if it spins fast and is heavy (large moment-of-inertia). It’s axis-of-rotation will not change. Useful in e.g. navigation on airplanes and space equipment.

This phenomenon is a consequence of the angular momentum conservation law.

## Energy

Energy $E$ can be thought of as “stored motion”, meaning the “potential” for motion which will create a force that in turn will cause motion if released.

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## Equilibrium

A object/system is in equilibrium when it *stays* stationary after being made stationary, meaning it has no tendency to move, twist or turn.

$$\sum \vec F=0\quad\text{and}\quad\sum \vec \tau=0$$

$\vec F$ is force and $\tau$ torque on the object/system.

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## No-energy rolling

No energy is expended during (pure, ideal) rolling, since not kinetic friction, which would do work, but stationary friction is present, which without local displacement does no work.