Dynamics and Conservation Laws

A body with zero net force remains at rest or continues in uniform linear motion (ULM). This is Newton's First Law of Inertia: a body does not change its state of motion unless compelled to by a net external force.

Inertial mass is the property of matter that resists changes in velocity — a heavier object requires a greater force to achieve the same acceleration.

An inertial frame is a reference frame in which the First Law is valid: a body subject to no forces moves in a straight line at constant speed. Any frame in uniform translation relative to an inertial frame is itself inertial. The Earth is a good approximation of an inertial frame for most laboratory purposes (the corrections due to its rotation are negligible except for large-scale atmospheric phenomena).

The net (resultant) force acting on a body equals the product of its mass and its acceleration: $\vec{F}_{net} = m\vec{a}$. This is the quantitative heart of Newtonian mechanics.

Inertial mass $m$ measures the resistance of a body to changes in its state of motion. A larger mass requires a proportionally larger force to produce the same acceleration.

In practice the law is applied component by component: $\sum F_x = ma_x$, $\sum F_y = ma_y$. A free-body diagram is drawn first to identify all forces; the net force in each direction is then set equal to $ma$ in that direction. If the net force is zero the body is in dynamic equilibrium (First Law).

Forces always come in pairs: if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. The two forces are equal in magnitude and opposite in direction.

A crucial point: the action and reaction forces act on different bodies — they can never cancel each other in a single free-body diagram. For example, when you push against a wall the wall pushes back on you with the same force; these two forces act on different objects (wall vs. you) and do not cancel.

The Third Law is the microscopic origin of the conservation of momentum: for an isolated system the internal action–reaction pairs sum to zero, so the total momentum is constant.

Weight $P = mg$ always acts vertically downward toward the Earth's centre. On an inclined plane at angle $\theta$ it is convenient to decompose weight along the plane: $P_{\parallel} = mg\sin\theta$ (tends to slide the body down) and perpendicular to the plane: $P_{\perp} = mg\cos\theta$ (balanced by the normal force).

The normal force $N$ is the contact force perpendicular to the surface; on a horizontal surface $N = mg$, on an incline $N = mg\cos\theta$.

Friction opposes relative motion or its tendency:

• Static friction $f_s \leq \mu_s N$: prevents motion; it takes any value from zero up to its maximum $\mu_s N$.
• Kinetic (dynamic) friction $f_k = \mu_k N$: constant opposing force once sliding has begun; always $\mu_k < \mu_s$.

The dimensionless coefficients $\mu_s$ and $\mu_k$ depend on the materials in contact, not on the contact area or the speed of sliding.

The work done by a force $\vec{F}$ along a displacement $d\vec{s}$ is $dW = \vec{F}\cdot d\vec{s} = F\,ds\cos\theta$, where $\theta$ is the angle between force and displacement. Only the component of force along the displacement does work; a force perpendicular to the motion (e.g. the normal force, centripetal force) does zero work.

The Work–Energy Theorem states that the net work done on a body equals its change in kinetic energy: $W_{net} = \Delta KE$. This is a scalar relation derived directly from Newton's Second Law integrated over the path.

A force is conservative if the work it does is independent of the path taken between two points (equivalently, the work around any closed path is zero). Gravity and the spring force are conservative; friction is not. For conservative forces a potential energy function $U$ exists such that $W = -\Delta U$.

Power $P = dW/dt = \vec{F}\cdot\vec{v}$ is the rate at which work is done [W = J/s].

When only conservative forces act, the total mechanical energy is constant: $E_{mech} = KE + PE = \mathrm{const}$. This is the law of conservation of mechanical energy. Energy is continuously exchanged between kinetic and potential forms, but their sum never changes.

Two important potential energy functions:

• Gravitational PE (near Earth's surface): $U_g = mgh$, measured from a chosen reference height $h = 0$.
• Elastic PE (spring): $U_e = \tfrac{1}{2}kx^2$, where $x$ is the compression or extension from the natural length and $k$ is the spring constant [N/m].

Hooke's law: the spring force is $F = -kx$ (restoring: always directed back toward equilibrium). The minus sign means the force opposes the displacement.

When non-conservative forces (such as friction) are present, the work they do equals the change in mechanical energy: $W_{nc} = \Delta E_{mech}$. Friction converts mechanical energy irreversibly into heat, so $E_{mech}$ decreases.