Mechanics

Physical quantities fall into two categories: fundamental, defined by convention (mass kg, length m, time s, temperature K, current A, amount of substance mol, luminous intensity cd), and derived, obtained by combining fundamental ones (velocity m/s, force N = kg·m/s², energy J = N·m, power W = J/s).

Dimensional analysis is a powerful tool for checking the formal correctness of a physical formula. The principle is simple: every term in an equation must have the same physical dimensions. You cannot add metres to seconds.

We write dimensions of fundamental quantities as $[L]$ for length, $[T]$ for time, $[M]$ for mass. Then:

• Velocity: $[v] = L/T$ — ratio of a length to a time interval.
• Acceleration: $[a] = L/T^2$ — change in velocity over time.
• Force (Newton's 2nd law): $[F] = [m][a] = M \cdot L/T^2$.
• Work/Energy: $[W] = [F][L] = M L^2 T^{-2}$.

Concrete example: in the pendulum period formula $T = 2\pi\sqrt{L/g}$, check the dimensions: $[L/g] = L/(L/T^2) = T^2$, so $\sqrt{L/g}$ has dimensions $[T]$, which is correct. If we had written $T = 2\pi\sqrt{g/L}$, the dimensions would be $T^{-1}$, not a time — the formula would be wrong before any measurement is taken.

Dimensional analysis cannot determine dimensionless constants (like $2\pi$), but it is the first check to perform on any formula: if the dimensions do not match, the formula is certainly wrong.

To describe the motion of a point particle we first need to know where it is and how it moves. The position vector $\vec{r}(t)$ is the fundamental quantity: it locates the point in space relative to a fixed origin O, and it depends on time because the point moves.

When the point moves from $\vec{r}_1$ to $\vec{r}_2$, we define the displacement $\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$. Crucially, displacement depends only on the initial and final points, not on the path taken. If you hike a winding mountain trail and return to the starting point, $\Delta\vec{r} = 0$ even if you have walked kilometres.

Average velocity is the ratio of displacement to time interval: $\vec{v}_m = \Delta\vec{r}/\Delta t$. It tells us, on average, how fast the position is changing.

Instantaneous velocity is the limit of the average velocity as the time interval tends to zero: $$\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta\vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$$ The derivative of the position vector with respect to time. Geometrically, $\vec{v}$ is always tangent to the trajectory: imagine a car on a curved road — its velocity at every instant points in the exact direction the car is heading at that moment.

Speed (magnitude of velocity) is $|\vec{v}| = ds/dt$, where $s$ is the arc length travelled. The car's speedometer measures exactly $|\vec{v}|$.

Acceleration measures how velocity changes over time. Just as velocity is the derivative of position, acceleration is the derivative of velocity: $$\vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$$
Note: a body can accelerate even if its speed is constant — it suffices that the direction of motion changes. This leads to the decomposition of acceleration into two components:

• Tangential acceleration $a_t = dv/dt$: measures how the magnitude of velocity changes. If you press the accelerator, $a_t$ is positive and the car goes faster; if you brake, $a_t$ is negative. It is parallel to the velocity (same or opposite direction).
• Centripetal acceleration $a_n = v^2/R$: measures how the direction of velocity changes. It is always perpendicular to the velocity and points toward the centre of curvature of the trajectory. On a curve, even at constant speed, you feel pushed outward — that is the effect of centripetal acceleration (which actually points inward).

The total magnitude of acceleration is $|\vec{a}| = \sqrt{a_t^2 + a_n^2}$.

Physical example: a car driving around a curve at a constant 50 km/h has $a_t = 0$ (constant speed) but $a_n = v^2/R \neq 0$ (changing direction). The total acceleration is purely centripetal. Conversely, a car accelerating in a straight line has $a_n = 0$ (straight trajectory) and $a_t \neq 0$.

For rectilinear motion, $R \to \infty$ and therefore $a_n = 0$: the acceleration is purely tangential.

Uniform linear motion (ULM) is the simplest kind of motion: velocity is constant in both magnitude and direction. Consequently, acceleration is zero ($a = 0$).

The law of motion is linear: $$x(t) = x_0 + v \cdot t$$ where $x_0$ is the initial position (at $t=0$) and $v$ is the constant velocity. The $x(t)$ graph is a straight line whose slope equals the velocity. The $v(t)$ graph is a horizontal line (velocity unchanged). The $a(t)$ graph is zero everywhere.

Physical meaning: in ULM the net force on the body is zero (Newton's first law). A car travelling at 130 km/h on a straight highway, neither accelerating nor braking, is in ULM — the engine provides just enough power to overcome friction and air resistance, but the resultant force is zero, so the velocity does not change.

Displacement over an interval $\Delta t$ is simply $\Delta x = v \cdot \Delta t$: the constant velocity multiplied by the elapsed time. If a train travels at 80 m/s for 30 seconds, it covers exactly $80 \cdot 30 = 2400$ metres.

ULM is also the motion of an inertial reference frame: an observer in ULM cannot distinguish their own motion from being at rest (Galilean principle of relativity).

In UAM the acceleration is constant in both magnitude and direction, so the velocity changes linearly in time and the position changes quadratically. The three kinematic equations are obtained by integrating $a = \mathrm{const}$ once (for $v$) and twice (for $x$), or by eliminating $t$ between them.

Free fall is UAM with constant downward acceleration $a = -g = -9.81\,\mathrm{m/s^2}$ (taking upward as positive). Air resistance is neglected. All objects fall with the same acceleration regardless of mass — Galileo's result.

Projectile motion is the superposition of two independent motions: horizontal ULM (constant $v_x = v_0\cos\theta$, zero horizontal force) and vertical UAM (constant downward $g$, zero initial horizontal force). The trajectory is a parabola.

Key results: the time of flight is determined solely by the vertical motion; the range $R$ depends on $v_0^2\sin(2\theta)/g$. Since $\sin(2\theta)$ is maximised at $2\theta = 90°$, the optimal launch angle for maximum range on flat ground is $\theta = 45°$. At the apex the vertical velocity is zero, so only the horizontal component $v_{0x}$ remains.

Uniform circular motion (UCM) is the motion of a point travelling along a circle at constant speed. Note: even though $|\vec{v}|$ is constant, the direction of velocity changes continuously, so there is acceleration.

Angular quantities: rather than describing motion with Cartesian coordinates, it is convenient to use angular coordinates:

• Angular position $\theta(t)$ [rad] — the angle swept by the radius vector.
• Angular velocity $\omega = d\theta/dt$ [rad/s] — constant in UCM.
• Period $T = 2\pi/\omega$ [s] — time for one complete revolution.
• Frequency $f = 1/T = \omega/(2\pi)$ [Hz] — revolutions per second.

Relation between linear and angular quantities: $$v = \omega\,r \qquad a_c = \frac{v^2}{r} = \omega^2 r$$ The centripetal acceleration $a_c$ always points toward the centre of the circle: it is what "bends" the trajectory. Without it, the body would continue in a straight line (Newton's first law).

Non-uniform circular motion (NUCM): when the angular velocity is not constant, we introduce the angular acceleration $\alpha = d\omega/dt$ [rad/s²], analogous to linear acceleration. The kinematic equations are formally identical to those of UAM: $$\omega(t) = \omega_0 + \alpha t \qquad \theta(t) = \theta_0 + \omega_0 t + \tfrac{1}{2}\alpha t^2$$
Real-world examples:

• A geostationary satellite completes one revolution in 24 hours ($\omega = 7.27\times10^{-5}\,\mathrm{rad/s}$).
• A rotating rigid disk: all points have the same $\omega$, but linear speed $v = \omega r$ is larger for points farther from the centre.
• A laboratory centrifuge uses centripetal acceleration to separate components of different densities.