## Proof 37: Elastic potential energy

In a spring that is elongated/compressed the distance $\Delta L=L_\text{end}-L_\text{relaxed}$, an elastic force (due to Hooke’s law) $F_\text{elastic}=k\Delta L$ is pushing/pulling on the surroundings in…

These proofs are described in a concise and down-to-Earth style for everyone to understand and follow along. Be aware that certain mathematical depths and details aren’t fully covered for the sake of simplicity and the intuitive description. The intend is sometimes rather to give the intuitive idea than the exact derivation.

In a spring that is elongated/compressed the distance $\Delta L=L_\text{end}-L_\text{relaxed}$, an elastic force (due to Hooke’s law) $F_\text{elastic}=k\Delta L$ is pushing/pulling on the surroundings in…

Small falling object due to weight A flower pot falls down from a window sill due to gravity. This i because gravity does work $W_g$…

Linear version of kinetic energy A spaceship in outer space starts its rocket engines to propel itself forwards with a rocket force. That force does…

Proof of the precession angular speed formula $\omega_\text{precession}=\frac{wr}{L_\text{spin}}$ is coming soon.

If we define angular momentum as the cross-product $\vec L=\vec r \times \vec p$, its magnitude is $L=r_\perp p$ (see the Cross-product skill). For a…

The moment conservation law and Newton’s 3rd law as well as the idea of impulse seem equivalent (they all describe impacts/collisions). And in fact the…

We call a collision elastic if the total kinetic energy is conserved (same before as after the collision) and there are no other energies involved.…

Solid sphere (ball) spinning about an axis through its centre-of-mass Let’s cut the ball into many thin slices. Each slice has a small thickness $\mathrm…

Holed cylinder (with a hole in the middle) spinning about its axis With a single point having the moment-of-inertia $I_\text{point}=r^2m$ ($m$ and $r$ are mass…

Plate rotating perpendicularly about its centre-of-mass With a single point having the moment-of-inertia $I_\text{point}=r^2m$ ($m$ and $r$ are mass and distance from the axis-of-rotation), infinitely…

A rod rotating perpendicularly about its centre-of-mass With a single point having the moment-of-inertia $I_\text{point}=r^2m$ ($m$ and $r$ are mass and distance from the axis-of-rotation),…

When rotating about the centre-of-mass, the moment-of-inertia $I_\text{com}$ is often easier to find. When the axis-of-rotation is somewhere else, we’ll below derive a formula that…

An object’s centre-of-mass is a point $\vec r_\text{com}=(x_\text{com},y_\text{com})$ that “represents” the object, taking into account all the “particles” it consists of as well as how…

Get some friends to help you push the stuck carousel with your daughter sitting in it. Each applies a force on her. Friction and other…

$$\vec c=\vec a \times \vec b$$ A cross-product is defined with two features: It’s magnitude $c$ is the perpendicular part of one vector multiplied with…

Angle $\theta$ measured in radians is ‘the number of radius-lengths along the periphery’. Multiply that number with the radius $r$ to get the total length…

We here derive the sine and cosine values of a few chosen angles. Half ($\pi$) and quarter turns ($\frac \pi 2$, $-\frac \pi 2$): Cosine…

We will prove that inwards (radial/centripetal) acceleration is squared speed over radius: $$ a_\perp=\frac{v^2}r $$ When driving around a roundabout (a circle) with constant speed,…

Definitions: A logarithm finds the exponent $a$ that when put on a base $b$ gives the number. In other words, it ‘takes out’ the exponent: $$\log_b(b^a)=a$$…

Requirements: Remember the sign rule: both ‘positive times positive gives positive’ and ‘negative times negative gives positive’. Only ‘negative times positive (or opposite) gives negative’.…

Definitions: The pattern looks as if reducing the exponent means removing an $a$ term: $$\vdots\\a^4=a\cdot a \cdot a \cdot a\\a^3=a\cdot a \cdot a\\ a^2=a\cdot a…

Requirements: The denominator in a fraction tells ‘how many portions we split a value into’. Split it into 2 or 3 or 4 portions: $\frac12$,…

A ‘binomial’ is a word for two added terms, such as: $a+b$. The square of a binomial is naturally $(a+b)^2$. $$(a+b)^2=(a+b)(a+b)=aa+ab+ba+bb=a^2+b^2+2ab\quad_\blacksquare$$ A small mnemonic rhyme…

In motion along a path, $a$ is acceleration, $v_0$ and $v$ are initial and final (current) speed, and $t_0$ and $t$ are initial and final…

The circle equation (see Proof 12) can be rearranged: $$(x-c_1)^2+(y-c_2)^2=r^2\\\frac{(x-c_1)^2}{r^2}+\frac{(y-c_2)^2}{r^2}=1\\\left(\frac{x-c_1}{r}\right)^2+\left(\frac{y-c_2}{r} \right)^2 =1$$ The $x-c_1$ and $y-c_2$ distances are being ‘squeezed’ by $r$, so to say.…

There is a round amphitheatre 1000 metres East and 3500 metres North from your hotel (let’s call those numbers $c_1$ and $c_2$). You go there…

A sphere can be cut into slices. Each slice is almost a disk, or a very flat cylinder, except for the curved edge. If we…

A trapezium consists of a rectangle and two weird triangles at the ends. $$\begin{align}A_\text{trapezium}&= A_\text{rectangle}+ A_{\text{triangle } a}+ A_{\text{triangle } b}\\~&=\underbrace{lw_1}_\text{rectangle}+\underbrace{\frac12 w_al}_{\text{rectangle } a}+\underbrace{\frac12 w_b…

A rhombus actually consists of four right-angled triangles, because the diagonals $d_1$ and $d_2$ are perpendicular. Those triangles are pair-wise equal. The rhombus area is…

The formula for the sum of squares is: $$1^2+2^2+\cdots+n^2=\frac16 n(n+1)(2n+1)$$ It is useful because it converts a row of confusingly many terms (therefore the dots…

The Cheops pyramid in Egypt is constructed of many equally thick layers of box-shaped stones (more or less). Imagine more and more such layers being…

In a room with 3 metres to the ceiling, there are 3 cubic metres above each square metre floor regardless of the shape. The 3…

Imagine a circle with a radius of exactly 1, $r=1$ (called a unit circle). Circle area is $A_\text{circle}=\pi r^2$ (see Proof 4), so such a…

A pizza slice’s edge is the pizza’s radius $r$. The slice is almost a triangle if it wasn’t for the rounding. That rounding has a…

The circumference formula is quick to figure out since by definition: $\text{circumference}=\pi\;\text{diameter}$ (as explained in the Natural constants and pi skill) and since the diameter…

Two brothers share a slice of beer cake. Its face has a rectangle-shape. They cut it from corner to corner to get two equal halves.…

‘A room is $4\,\mathrm m$ long and $2\,\mathrm m$ wide’; a rectangle-shape. For each of the 4 metres there are 2 square metres across. That…