Statics & dynamics · first-year engineering

Engineering Mechanics

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Statics: Forces and Moments

L1
Lesson 1
Reading the forces
newtons, kilograms, pascals, newton-metres
Statics is the study of things that are NOT moving — and the whole subject rests on telling four quantities apart. The unit does that for you, every time, so learn to read it first and think second.
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L2
Lesson 2
Weight and mass
The kilogram-newton confusion, priced
Your mass is how much of you there is.
The formula in this lesson
$W = mg$
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L3
Lesson 3
Resolving on the incline
sin θ down the slope, cos θ into it
Put a crate on a ramp and its weight stops being one number. Gravity still pulls straight down, but the ramp only cares about the two directions it knows: along the slope and into it.
The formulas in this lesson
$N = mg\cos\theta$ , $F_{\parallel} = m g \sin\theta$
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L4
Lesson 4
Friction holds
μₛ before the slide, μₖ during it, tan θ at the tipping point
If your μ ever turns up wearing newtons, a rearrangement upstream is lying to you. The subscripts name a STATE, not a first-and-second: s is static, still at rest, and k is kinetic, already sliding.
The formulas in this lesson
$f_k = \mu_k N$ , $f_{s,\max} = \mu_s N$ , $\mu_s = \tan\theta$
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L5
Lesson 5
The moment of a force
The arm is half the answer
A force applied off a pivot does not just push — it TURNS.
The formula in this lesson
$\tau = r F \sin\theta$
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L6
Lesson 6
Levers and pulleys
mechanical advantage, and what friction takes back
No machine gives you energy. Every one of them trades: less force, more distance, and the product stays put. That trade has a name — mechanical advantage.
The formulas in this lesson
$MA = \dfrac{d_e}{d_l}$ , $F = \dfrac{W}{n}$ , $\eta = \dfrac{W_{out}}{W_{in}}$
Start
L7
Lesson 7
Beam reactions
the near support carries more — and by how much
A simply supported beam sits on two supports and carries a load somewhere between them.
The formulas in this lesson
$R_A = \dfrac{P(L-a)}{L}$ , $M = \dfrac{Pa(L-a)}{L}$
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L8
Lesson 8
Loaded two ways
a load spread along the span, then two loads at once
Real beams rarely carry one tidy point load. They carry floors, snow and their own weight, spread along the whole length, or they carry several loads at once.
The formulas in this lesson
$R = \dfrac{wL}{2}$ , $R_A = \dfrac{P_1 (L - a_1) + P_2 (L - a_2)}{L}$
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Boss lesson
The Free-Body Final
the boss · no calculator · one rig, fully balanced
No calculator — today g = 10 N/kg, the round number every engineer keeps for mental work and site sanity checks, and the values are chosen to fit in your head.
The formula in this lesson
$W = mg$
Start

Dynamics of a Particle

L10
Lesson 10
Newton's second law
F = ma, asked all three ways
Statics asked what holds still. Dynamics asks what happens when the arrows do NOT balance, and the answer is one line: F = ma.
The formula in this lesson
$F = ma$
Start
L11
Lesson 11
Kinematics revisited
uniform acceleration, now with plant numbers
Two relations from school physics come back, and they will not leave again.
The formulas in this lesson
$v = v_0 + at$ , $d = v_0 t + \tfrac{1}{2} a t^2$
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L12
Lesson 12
The no-time equation
When the clock is missing
Sometimes nobody timed anything. A braking test gives you a speed and a distance; a chute gives you a drop and an arrival speed.
The formula in this lesson
$v^2 = v_0^2 + 2ad$
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L13
Lesson 13
Down the incline
a = g sin θ, and what friction takes off it
Sine goes with the side OPPOSITE the angle, cosine with the side that touches it — swap them and every answer in this lesson is wrong.
The formulas in this lesson
$a = g\sin\theta$ , $N = mg\cos\theta$ , $a = g\left(\sin\theta - \mu_k\cos\theta\right)$
Start
L14
Lesson 14
Ropes and tension
The hoist that reads heavy
A rope holding a still load pulls with exactly the weight, mg. Start lifting that load with an upward acceleration and the rope has two jobs at once: beat gravity, AND supply the ma the acceleration demands.
The formulas in this lesson
$T = m\left(g + a\right)$ , $a = \dfrac{\left(m_1 - m_2\right)g}{m_1 + m_2}$
Start
L15
Lesson 15
Drag and terminal speed
The speed where drag wins
And v is the speed through the fluid, in m/s.
The formulas in this lesson
$F_D = \tfrac{1}{2} C_d \rho A v^2$ , $v_t = \sqrt{\dfrac{2mg}{\rho A C_d}}$
Start
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Boss lesson
The Elevator Test
the boss · no calculator · rope to ramp to verdict
One lift, one ramp, and every answer feeds the next: weight, then the rope's win over it, then Newton, then the speed that acceleration buys, then the climb, then the verdict the commissioning sheet actually wants.
The formula in this lesson
$W = mg$
Start

Work, Energy, Momentum

L17
Lesson 17
Work done
The sideways pull that does nothing
A force on its own does nothing measurable.
The formula in this lesson
$W = F d \cos\theta$
Start
L18
Lesson 18
Two energy accounts
½mv² for motion, mgh for height
Energy is the capacity to do work, and mechanics keeps it in two everyday accounts. Both are measured in joules, both can be paid into and drawn out of, and the whole subject is bookkeeping between them.
The formulas in this lesson
$E_k = \tfrac{1}{2} m v^{2}$ , $U = m g h$
Start
L19
Lesson 19
The work–energy theorem
net work IS the change in kinetic energy
Here is the sentence that ties the chapter together: the net work done on a body equals the change in its kinetic energy.
The formula in this lesson
$W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)$
Start
L20
Lesson 20
Springs store it
kx is linear, ½kx² is not — and that gap is the lesson
A spring is the cleanest energy store in mechanics: squeeze it, it pushes back; let it go, it gives the energy back. Two relations govern it and they are NOT the same relation, which is where every mark gets lost.
The formulas in this lesson
$F = kx$ , $U = \tfrac{1}{2} k x^{2}$
Start
L21
Lesson 21
Power delivered
the same job, done faster, costs more machine
Work says how much energy changed hands.
The formulas in this lesson
$P = \dfrac{W}{t}$ , $P = F v$
Start
L22
Lesson 22
Momentum and impulse
p = mv, and the crumple zone that buys time
Energy is not the only thing a moving body carries. Momentum is p = mv, read aloud p equals m v: p is the momentum, m is the mass in kilograms and v is the velocity in metres per second.
The formulas in this lesson
$p = mv$ , $J = F \, \Delta t$
Start
L23
Lesson 23
Collisions
stick together, bounce apart — momentum survives both
Whatever happens in the milliseconds between — noise, heat, bent steel — total momentum is the same before and after, provided nothing outside the pair pushes on them.
The formulas in this lesson
$v = \dfrac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$ , $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$ , $e = \dfrac{v_2 - v_1}{u_1 - u_2}$
Start
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Boss lesson
The Runaway Ramp
the boss · no calculator · energy in, friction out
No calculator — today g = 10 N/kg, the round number every engineer keeps for mental work and site sanity checks, and the values are chosen to fit in your head.
Start

Rotation and Vibration

L25
Lesson 25
Angular kinematics
ω, α and the rim speed they set
Everything you learned about motion along a line has a twin that turns, and the twins share one shape.
The formulas in this lesson
$\omega = \dfrac{\theta}{t}$ , $\alpha = \dfrac{\Delta\omega}{t}$ , $v = \omega r$
Start
L26
Lesson 26
RPM and radians
the 2π/60 bridge, in both directions
Every nameplate, tachometer and datasheet in a plant is written in rpm — revolutions per minute, the symbol n. Every equation in your course is written in rad/s.
The formulas in this lesson
$\omega = \dfrac{2\pi}{T}$ , $v = \dfrac{2\pi r}{T}$
Start
L27
Lesson 27
Torque makes it spin
Where the mass is sitting
Whichever of the three the question withholds is the one you solve for. But I is not simply the mass, and that is the interesting part.
The formulas in this lesson
$I = \tfrac{1}{2} m r^{2}$ , $I = m r^{2}$ , $\tau = I\alpha$
Start
L28
Lesson 28
Round the bend
v²/r and ω²r · the same fact in two dresses
A car at a constant 20 m/s through a curve is accelerating, and the speedometer has nothing to say about it. Acceleration is a change in velocity, and velocity carries a direction.
The formulas in this lesson
$a_c = \dfrac{v^{2}}{r}$ , $F_c = \dfrac{m v^{2}}{r}$ , $a_c = \omega^{2} r$
Start
L29
Lesson 29
Rotational energy and momentum
The analogy, complete
The translation table finishes here.
The formulas in this lesson
$KE_{rot} = \tfrac{1}{2} I \omega^{2}$ , $L = I\omega$
Start
L30
Lesson 30
Power through a shaft
P = τω · and the rpm that must become rad/s
It is the rotational cousin of P = Fv, and a newton-metre per second is exactly one watt, which is the whole reason the relation is this tidy.
Start
L31
Lesson 31
The natural frequency
√(k/m), and the sag that measures it for you
Stiffer rings faster; heavier rings slower.
The formulas in this lesson
$f_n = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}$ , $f_n = \dfrac{1}{2\pi}\sqrt{\dfrac{g}{\delta_{st}}}$
Start
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Boss lesson
The Flywheel Final
the boss · no calculator · rpm to energy to power
That is a 4.5% lie, said out loud, and it is the same move as rounding g to 10 — every line then lands on a whole number and the chain stays in your head.
The formula in this lesson
$\omega = \dfrac{2\pi}{T}$
Start

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