Fundamental Concepts in Classical Mechanics Physics

Posted on Feb 23, 2026 in Physics

Physical Quantities and Measurement

Physical Quantity: A property of a body or phenomenon that can be measured and expressed by a number and a unit (e.g., 5 m, 10 s, 3 kg). It allows us to describe physical laws quantitatively. A physical quantity is defined either by specifying how it is measured or by stating how it is calculated from other measurable quantities.

Unit of Measurement: A standard reference used for comparing quantities of the same kind. Example: meter (m) for length, second (s) for time.

  • Direct Measurement: The quantity is measured directly using an instrument. Example: measuring length with a ruler.
  • Indirect Measurement: The quantity is obtained from other measurements using a mathematical relationship. Example: calculating density using $\rho = m/V$.

Measurement Errors: The difference between the measured value and the true value.

  • Systematic Errors: Occur always with the same value when we use the instrument in the same way (e.g., calibration error).
  • Random Errors: Vary unpredictably (e.g., reading mistakes). They can be reduced by repeating measurements, calibrating instruments, and taking averages.

The SI System and Units

SI System: It is based on seven base units, from which all other units are derived. These units are independent and form the foundation of all measurements:

  • Length (m)
  • Mass (kg)
  • Time (s)
  • Electric Current (A)
  • Thermodynamic Temperature (K)
  • Amount of Substance (mol)
  • Luminous Intensity (cd)

Derived Units: Formed by combining base units according to physical laws.

  • Force $\rightarrow$ Newton (N) = $\text{kg}\cdot\text{m}/\text{s}^2$
  • Energy $\rightarrow$ Joule (J) = $\text{kg}\cdot\text{m}^2/\text{s}^2$
  • Power $\rightarrow$ Watt (W) = $\text{kg}\cdot\text{m}^2/\text{s}^3$

Supplementary Units: Used in conjunction with base units to form derived units (for angles):

  • Plane Angle $\rightarrow$ Radian (rad)
  • Solid Angle $\rightarrow$ Steradian (sr)

Measuring Instruments: Tools used to compare physical quantities with a reference standard.

  • Time: Stopwatch, quartz clock, atomic clock
  • Length: Ruler, laser meter
  • Mass: Balance, scale
  • Temperature: Thermometer
  • Electric Current: Ammeter

Kinematics: Describing Motion

Rectilinear Motion Types

  • Rectilinear Uniform Motion: Motion in a straight line with constant velocity. The body covers equal distances in equal time intervals. Formula: $v = s/t = \text{const}$.
  • Non-uniform Rectilinear Motion: Motion in a straight line with changing velocity (acceleration $\neq 0$). The object covers unequal distances in equal times. Formula (for constant acceleration): $v = v_0 + at$; $s = v_0 t + \frac{1}{2} a t^2$.

Complex Motion: A motion that combines two or more independent motions (e.g., linear + vertical). Examples: A projectile has horizontal uniform motion + vertical accelerated motion; a boat crossing a river with current flow.

Kinematic Variables

  • Distance ($s$): The total path length traveled (scalar quantity).
  • Velocity ($\vec{v}$): The rate of change of displacement (vector).
  • Acceleration ($\vec{a}$): The rate of change of velocity with time (vector). $\vec{a} = \Delta\vec{v}/\Delta t$.

Reference Frames and Circular Motion

Reference Frame: A coordinate system used to describe motion. It defines the origin and axes from which position, velocity, and acceleration are measured (Ex: a train, the Earth).

Coordinate System: Usually a Cartesian system ($x, y, z$) where we define position using coordinates.

  • Position Vector ($\vec{r}$): A vector drawn from the origin of the reference frame to the position of the particle. $\vec{r} = (x, y, z)$.
  • Trajectory / Path: The curve traced by the endpoint of the position vector as the object moves. The path length ($s$) is the distance traveled along this curve.
  • Displacement ($\Delta\vec{r}$): A vector that points from the initial position to the final position of the object. $\Delta\vec{r} = \vec{r}_2 – \vec{r}_1$ (It depends only on the start and end points, not on the path.)

Uniform Circular Motion: The object moves around a circle with constant speed. Although the speed is constant, the direction changes, so the velocity and acceleration vectors change.

  • Angular Velocity ($\omega$): Rate of change of the angle (in radians per second): $\omega = \Delta\phi / \Delta t$.
  • Linear Velocity: $v = \omega r$ (tangent to the circle).
  • Centripetal Acceleration: Directed toward the center of the circle: $a_c = v^2 / r = \omega^2 r$.

Derivatives in Motion:

  • Velocity: Time derivative of position $\vec{v} = d\vec{r}/dt$.
  • Acceleration: Time derivative of velocity $\vec{a} = d\vec{v}/dt$.

Dynamics: Causes of Motion and Forces

Newton’s Laws of Motion

Dynamics: The branch of mechanics that studies the causes of motion—how and why bodies move when forces act on them.

  • Newton’s First Law (Law of Inertia): A body remains at rest or in uniform motion in a straight line unless acted upon by an external force. It defines inertia and the concept of an inertial reference frame. $\sum \vec{F} = 0 \Rightarrow \vec{v} = \text{constant}$.
  • Newton’s Second Law: The acceleration of a body is directly proportional to the net force and inversely proportional to its mass. The direction of acceleration is the same as that of the force. $\vec{F} = m\vec{a}$. Unit of force: newton (N) $\rightarrow 1 \text{ N} = 1 \text{ kg}\cdot\text{m}/\text{s}^2$.
  • Newton’s Third Law (Action–Reaction): When one body exerts a force on another, the second body exerts an equal and opposite force on the first. $\vec{F}_{AB} = – \vec{F}_{BA}$. The two forces act on different bodies, never cancel each other.

Principle of Independence (Superposition) of Forces: If several forces act on a body at the same point, their total (resultant) effect equals the vector sum of all forces. Each force acts independently of the others. $\vec{F}_{resultant} = \sum_i \vec{F}_i$.

Gravitation and Contact Forces

Newton’s Law of Gravitational Forces: Every particle in the universe attracts each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them: $F = G \frac{m_1 m_2}{r^2}$.

Kepler’s Laws (Planetary Motion):

  • K. Law of Ellipses: Planets move in elliptical orbits with the Sun at one focus.
  • K. Law of Equal Areas: The radius vector from the Sun to the planet sweeps equal areas in equal times.
  • K. Law of Harmonies: The squares of orbital periods are proportional to the cubes of their semi-major axes.

Forces Related to Earth/Contact:

  • Gravitational Force: The attractive force due to Earth’s mass.
  • Weight (W): The force with which a body presses on its support. $W = mg$.
  • Elastic Force: The force that allows some material to return to its original shape after being stretched or compressed. $\vec{F} = -k\vec{x}$.
  • Constraint Forces: Forces that ensure the constraint conditions for the movement of the body.
  • Free Forces: Defined by universal laws.
  • Sliding Friction: $\vec{F}_f = \mu \vec{N}$. Opposes motion, acts parallel to the surface.
  • Adhesion Force (Static Friction): When bodies are in contact but not moving; $F_t \le \mu_0 F_N$.

Basic Equation of Dynamics: $\sum \vec{F} = m\vec{a}$. On an inclined plane with friction, $a = g(\sin \alpha – \mu \cos \alpha)$.

Work, Energy, and Power

  • Work (W): Work is done when a force causes displacement. $W = \vec{F} \cdot \vec{s}$. Unit: Joule (J) = N$\cdot$m.
  • Power (P): The rate of doing work or the rate of energy transfer. $P = W/t = Fv$. Unit: Watt (W) = J/s.
  • Work of Gravity: For a height difference $h$: $W_g = mgh$. Independent of the path $\rightarrow$ gravity is a conservative force.
  • Work of a Spring Force: For an extension $x$: $W = \frac{1}{2} kx^2$.
  • Work of a Coercive Force: If the surface giving the coercion does not accelerate, then the coercive force is perpendicular to the surface, so $\cos 90^{\circ} = 0$; $W = 0$.
  • Work of Gravitational Force (Field): When moving a mass $m$ in the field of another mass $M$: $W = G \frac{Mm}{r}$, or considering potential energy, $U = – G\frac{Mm}{r}$.

Energy Concepts

  • Kinetic Energy: The energy a body has due to its motion. It depends on the body’s mass ($m$) and velocity ($v$). Unit: Joule (J). $E_k = \frac{1}{2} mv^2$.
  • Work-Energy Theorem: The net work done by all forces on an object equals the change in its kinetic energy. $W_{\text{net}} = \Delta E_k = E_{k2} – E_{k1}$.
  • Potential Energy: The energy stored in a system due to its position or configuration.
  • Conservation of Mechanical Energy: In the absence of non-conservative forces (like friction), the total mechanical energy of a system remains constant.
  • Conservative Forces: Type of force where the work done by the force is independent of the path taken and depends only on the initial and final positions of an object (Gravity, Elastic force are examples).

Momentum and Collisions

  • Momentum ($\vec{p}$): A vector quantity that measures the quantity of motion of a body.
  • Calculation for a Point of Mass: $\vec{p} = m\vec{v}$. Unit $\rightarrow \text{kg}\cdot\text{m}/\text{s}$.
  • Momentum of the Centre of Mass: For a system of particles $\vec{p}_{cm} = M\vec{v}_{cm}$, where $M$ is the total mass, and $\vec{v}_{cm}$ is the velocity of the centre of mass.
  • Conservation of Momentum: If no external forces act on a system, the total momentum remains constant. Initial Total Momentum = Final Total Momentum.

Collisions:

  • Elastic Collision: Momentum and kinetic energy are conserved. $\vec{p}: m_1 \vec{v}_{1i} + m_2 \vec{v}_{2i} = m_1 \vec{v}_{1f} + m_2 \vec{v}_{2f}$; $E_k$: $\frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$. Example: a billiard ball bouncing off the cushion.
  • Inelastic Collision: Momentum is conserved, but kinetic energy is not. $m_1 \vec{v}_{1i} + m_2 \vec{v}_{2i} = (m_1+m_2)\vec{v}_f$.
  • Collision with a Large Wall: A large wall has very large mass, so its velocity doesn’t change. The object’s momentum is not conserved (external force). The object rebounds with the same speed, but reversed direction.

Rotational Dynamics and Equilibrium

Rigid Body Motion

Rigid Body: Does not suffer deformation under the effect of force. It is in equilibrium if the sum of all the forces acting on it and the sum of the moments of the forces, with respect to an arbitrary point, is zero.

  • $\sum \vec{F}_i = 0$
  • $\sum \vec{M}_i = \sum (\vec{r}_i \times \vec{F}_i) = 0$

Torque (Moment):

  • Torque about a Fixed Axis: If a body rotates around a fixed axis, only the perpendicular component of the force to the radius produces torque: $M = F_{\perp} r$.
  • Torque about a Point: When several forces act on a body, each produces a torque around the same point; their vector sum gives the resultant moment: $\vec{M}_{resultant} = \sum_i (\vec{r}_i \times \vec{F}_i)$.
  • Torque as a Vector: It is the vector product of the position vector and the force, giving the axis and direction of rotation. $\vec{M} = \vec{r} \times \vec{F}$.

Simple Machines: Inclined plane, Screw, Wedge, Lever, Pulley, and Wheel and Axle.

Pair of Forces: Two equal and opposite forces whose lines of action do not coincide form a pair of forces.

Composition of Forces Acting on a Rigid Body: The forces acting on the rigid body can be replaced by a resultant force $\vec{F}$ acting at an arbitrary point O and a torque depending on the choice of point O.

Centre of Mass and Inertia

Centre of Mass: The center of gravity of a freely moving body always moves in a straight line; if it is supported at this point, it will be in equilibrium.

Equilibrium Position of Rigid Bodies: A rigid body is in equilibrium when the net force and the net torque acting on it are both zero: $\sum \vec{F}=0$; $\sum \vec{M}=0$. When the body is slightly displaced, the new forces/torques determine the type of equilibrium: Stable, Unstable, Neutral.

Centre of Mass Motion Theorem: The motion of the center of mass of a system behaves as if all mass were concentrated at that point and all external forces acted on it. Ex: the center of mass of a high jumper moves on a parabolic path perhaps lower than the bar.

Methods to Find Centre of Mass:

  • Suspension Method: The Centre of Mass is the intersection of the vertical lines drawn from different suspension points of the body.
  • Horizontal Axis Method: If a body placed on a horizontal axis remains in equilibrium in any position, the axis passes through the Centre of Mass.
  • Symmetry Rule: For plane figures, the Centre of Mass is the point where the axes of symmetry intersect.

Moments of Inertia: The moment of inertia ($I$) measures an object’s resistance to rotational acceleration about a particular axis. It is the rotational analogue to mass.

  • Solid Cylinder (about symmetry axis): $I = \frac{1}{2} mr^2$.
  • Hoop (about symmetry Axis): $I = mr^2$.
  • Steiner’s Theorem (Parallel Axis Theorem): Determines the moment of inertia of a rigid body about any axis, provided you know the moment of inertia about a parallel axis that passes through the body’s center of gravity.

Motion of Tops and Spinners: A spinner is a rigid body rotating about a free axis of symmetry. Stability is greatest when rotating about the axis with the greatest moment of inertia. When an external torque is applied, the axis moves perpendicular to the force, undergoing precession.

Conservation Theorems in Mechanics

  • Mechanical Energy Conservation Theorem: If only conservative forces act on a mechanical system, the total mechanical energy is constant. $E_{\text{mech}} = U+K = \text{constant}$. Both external and internal forces can change the kinetic energy.
  • Conservation of Momentum Theorem: If the sum of the torques of the external forces is zero, then the momentum of the mechanical system is constant ($\sum \vec{N} = \text{constant}$). A closed system is said to exist if the mechanical system is subjected only to internal forces, in which case the momentum of the system is also constant.

Non-Inertial Frames and Relativity

Galilei’s Principle of Relativity: This principle states that inertial frames exist, and the same laws of physics apply in all inertial frames of reference, regardless of one frame’s straight-line, constant-speed motion relative to another.

Transformations between inertial frames (where $\vec{v}_0$ is constant):

  • Position: $\vec{r} = \vec{r}’ + \vec{v}_0 t$.
  • Velocity: $\vec{v} = \vec{v}’ + \vec{v}_0$.
  • Acceleration: $\vec{a} = \vec{a}’$ (because $\vec{v}_0$ is constant).

Accelerating and Rotating Coordinate Systems

When a frame of reference is accelerating (non-inertial), fictitious forces appear.

  • Fictitious Force (General Acceleration $\vec{a}_0$): Arises from the frame’s acceleration. $m\vec{a}’ = \sum \vec{F}_i – m\vec{a}_0$.
  • Centripetal Force (Real): The real force required to keep an object moving in a curved path (circular motion). It is always directed inward toward the center of rotation. $F_{cp} = m\omega^2 r$.
  • Centrifugal Force (Fictitious): An inertial force that appears in a rotating reference frame, acting outward away from the center of rotation. $F_{cf} = m r’ \omega^2$.
  • Coriolis Force (Fictitious): An inertial force that acts only on objects that are moving within a rotating frame. It causes a deflection of the trajectory. $\vec{F}_{co} = 2m\vec{v}’ \times \vec{\omega}$.

Oscillations and Waves

Simple Harmonic Motion (SHM)

Simple harmonic motion occurs when the acceleration of an object is proportional to its displacement from the equilibrium position and is directed opposite to it ($\vec{a} = – \omega^2\vec{x}$).

  • Time-Displacement Function: The position varies sinusoidally with time: $x = A\sin(\omega t+\phi)$.
  • Amplitude (A): The maximum displacement from the equilibrium position.
  • Periodic Time (T): The time needed for one complete oscillation.
  • Frequency ($\nu$): The number of oscillations per unit time ($\nu = 1/T$).
  • Angular Frequency ($\omega$): Related to the period by $\omega = 2\pi/T$.
  • Phase ($\phi$): The phase constant depends on the initial position and velocity at $t = 0$.

Pendulums:

  • Simple Pendulum: A point mass suspended by a massless string of length $l$. Its period depends only on length and gravity, not mass. $T = 2\pi\sqrt{l/g}$.
  • Physical Pendulum: A rigid body that pivots about a point that does not go through its center of mass. $T = 2\pi\sqrt{I/mgd}$.
  • Torsional Pendulum: A body suspended by a wire that twists. The restoring torque is proportional to the angular displacement. $T = 2\pi\sqrt{I/\kappa}$ (where $\kappa$ is the torsion constant).

Composition of Vibrations

Composition of Harmonic Vibrations: This refers to the superposition (sum) of two or more waves.

  • Same Direction & Frequency (Interference): The resulting amplitude depends on the phase difference ($\Delta\phi$).
    • Constructive Interference: Waves arrive in phase.
    • Destructive Interference: Waves arrive out of phase.
  • Same Direction & Different Frequencies (Beats): Produces Beating, which is the periodic variation in intensity due to the superposition of waves with slightly different frequencies.
  • Perpendicular Vibrations: The sum of two mutually perpendicular vibrations usually forms an ellipse or a complex curve known as a Lissajous curve, depending on the frequency ratio and phase difference.
  • Fourier Theorem: We can represent any periodic function as a series of sine and cosine terms: $f(t) = \sum_{i=0}^{\infty} A_i \sin(i \omega t+ \phi_i)$.

Damped and Forced Vibrations

Damping Vibrations: It is the decrease in the amplitude of a wave over time. Damping is the result of friction reducing the energy of a wave. As damping occurs, frequency and wavelength remain the same if no other energy is added.

  • Overdamped ($c^2 – 4mk > 0$): The system returns to equilibrium without oscillating.
  • Critical Damping ($c^2 – 4mk = 0$): The system returns to equilibrium as quickly as possible without oscillating.
  • Underdamped ($c^2 – 4mk < 0$): The system oscillates with decreasing amplitude.

Forced Vibration: To produce undamped vibration (or maintain it), the energy loss must be replaced by an external source. Coercive Force: An external periodic force is applied, described as $F(t) = F_0\sin(\omega t)$.

Resonance: A phenomenon that occurs when an object or system is subjected to an external force or vibration that matches its natural frequency. Ex: Tacoma Bridge collapses due to wind.

Attached Vibrations: Occurs when two or more systems capable of vibration are connected by a coupling device (such as a weak spring or a weight on a rope).