Physics-2

Section 1: General Physics

Chapter 1: Making Measurements

The System International (SI) unit of length is the metre, represented by the symbol $\text{m}$. Measurements of length require specific instruments determined by the physical magnitude and shape of the object. A standard metre rule measures straight lengths between $1\,\text{cm}$ and $1\,\text{m}$, but it cannot accommodate curved surfaces. For curved lengths or dimensions exceeding $1\,\text{m}$, a measuring tape is employed. For high-precision measurements of extremely small dimensions, a micrometre screw gauge is used, which can record measurements down to $0.01\,\text{mm}$. The foundational SI base units include mass measured in kilograms ($\text{kg}$), length measured in metres ($\text{m}$), time measured in seconds ($\text{s}$), electric current measured in amperes ($\text{A}$), and thermodynamic temperature measured in kelvin ($\text{K}$).

Zero error occurs when a measuring instrument displays a non-zero reading when the actual true value is zero. On a metre rule or measuring tape, zero error arises if the starting edge of the object is not aligned with the baseline zero mark. In a micrometre screw gauge, zero error exists if the zero marks on the main scale and the thimble scale do not coincide precisely when the jaws are fully closed. Time is measured in SI units of seconds ($\text{s}$) using instruments such as pendulums, mechanical clocks, and digital stopwatches. A simple pendulum operates through repeated to-and-fro movements where a complete single path back and forth is designated as one oscillation. The period of a pendulum is defined as the time required to complete one full oscillation. To eliminate timing inaccuracies, the period is determined by measuring the total duration for $20$ complete oscillations using a stopwatch and dividing that total time by $20$ to compute the average period. Time measurements taken with manually operated stopwatches introduce human reaction error due to the reaction time delay when pressing the start and stop buttons.

Volume is measured using a upright measuring cylinder, where the liquid level is observed directly at eye level at the bottom of the curved liquid surface known as the meniscus. To determine the volume of an irregularly shaped solid, the liquid displacement method is applied. First, a measuring cylinder is filled with a sufficient volume of water to submerge the solid completely, and this initial volume is recorded. Next, the object is gently submerged into the liquid, causing the water level to rise, and the new final volume is recorded. The exact volume of the irregular solid is computed by subtracting the initial volume from the final volume. For example, if a measuring cylinder filled with water displays an initial volume reading of $18\,\text{cm}^3$ and the reading rises to $26\,\text{cm}^3$ after a stone is immersed, the volume of the stone is calculated as $26\,\text{cm}^3 - 18\,\text{cm}^3 = 8\,\text{cm}^3$

Physical quantities consist of a numerical magnitude and an accompanying unit, such as $56\,\text{km}$, where $56$ is the magnitude and $\text{km}$ is the unit. Physical quantities are broadly categorized into scalar and vector quantities. Scalar quantities possess magnitude only and do not have a spatial direction. Examples of scalar quantities include distance, speed, time, mass, energy, and temperature. Vector quantities possess both a numerical magnitude and a specific directional vector. Examples of vector quantities include displacement, velocity, acceleration, force, weight, momentum, electric field strength, and gravitational field strength. Displacement describes a distance in a specified direction, such as $36\,\text{km}$ North.

Scalar quantities are combined using ordinary algebraic addition. Vector quantities must account for direction and are resolved either graphically or mathematically. In vector diagrams, arrows represent vectors: the arrowhead points in the direction of the vector, and the length of the arrow represents its magnitude. To calculate the resultant of two perpendicular vectors graphically, a suitable scale must be selected, such as $1\,\text{cm} = 1\,\text{N}$. The first vector is drawn from a origin point in its given direction. The second vector is drawn starting from the origin point perpendicular to the first vector. A complete rectangle is constructed around these two component vectors. The resultant vector is represented by a diagonal line drawn from the point of intersection of the vector origins to the opposite corner of the rectangle. The length of this diagonal is measured with a ruler and converted back to units using the scale factor, while its directional angle relative to the baseline is measured using a protractor.

Vectors can also be resolved analytically using right-angled triangle geometry. A vector diagram is sketched with the components and resultant clearly labeled. The magnitude of the resultant vector $c$ formed by two perpendicular vector components $a$ and $b$ is calculated using Pythagoras' Theorem, given by the formula $c^2 = a^2 + b^2$, which yields $c = \sqrt{a^2 + b^2}$. The directional angle $\theta$ formed between the base component $b$ and the resultant hypotenuse $c$ is calculated using trigonometric ratios. The sine ratio is given by $\sin(\theta) = \frac{a}{c}$, the cosine ratio is given by $\cos(\theta) = \frac{b}{c}$, and the tangent ratio is given by $\tan(\theta) = \frac{a}{b}$.

Chapter 2: Kinematics

Distance is defined as the total length of the path covered by a moving object regardless of the direction of travel, making it a scalar quantity. Displacement is defined as the straight-line distance measured from a specific starting point to a final position in a specified direction, making it a vector quantity. If an object travels along a path and returns directly to its starting location, its total displacement is equal to $0\,\text{m}$.

Speed is defined as the distance traveled per unit time and is expressed mathematically as $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$. Speed is a scalar quantity, measured in the SI unit of metres per second ($\text{m/s}$), and is always non-negative. Average speed accounts for speed variations over an entire trip and is calculated as $\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$. Velocity is defined as the speed of an object in a specified direction, calculated as $\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$. Velocity is a vector quantity, measured in metres per second ($\text{m/s}$), and can be positive or negative depending on the chosen coordinate orientation. Constant speed or constant velocity occurs when an object covers equal distances in equal time intervals without speeding up, slowing down, or altering its direction.

Acceleration is defined as the rate of change of velocity per unit time, indicating how rapidly an object accelerates or decelerates. Acceleration is a vector quantity with the SI unit of metres per second squared ($\text{m/s}^2$). When an object's velocity decreases over time, its acceleration is negative, which is referred to as deceleration or retardation. Acceleration is calculated using the equation $a = \frac{\Delta v}{t}$, where $\Delta v$ represents the change in velocity. Expanding this expression yields $a = \frac{v - u}{t}$, where $v$ represents the final velocity, $u$ represents the initial velocity, and $t$ represents the elapsed time. Uniform or constant acceleration occurs when an object's velocity changes by equal magnitudes in equal time intervals. Non-uniform acceleration occurs when the rate of change of velocity changes over time. For example, if an object accelerates from an initial velocity $u = 10\,\text{m/s}$ to a final velocity $v = 25\,\text{m/s}$ over a duration of $t = 10\,\text{s}$, its average acceleration is calculated as $a = \frac{25\,\text{m/s} - 10\,\text{m/s}}{10\,\text{s}} = 1.5\,\text{m/s}^2$

In a distance-time graph, the gradient or slope of the line represents the speed of the object. A horizontal straight line on a distance-time graph indicates that the object is stationary ($\text{speed} = 0\,\text{m/s}$). A straight diagonal line with a constant, unchanged gradient represents motion at a constant speed. A curve with an increasing gradient curving upward represents increasing acceleration, whereas a curve with a decreasing gradient flattening out represents decreasing acceleration or deceleration. The instantaneous speed at any specific point on a curved distance-time graph is determined by calculating the gradient of a tangent line drawn to the curve at that specific point in time.

In a speed-time graph, the gradient of the line represents the acceleration of the object. The gradient is determined by evaluating the ratio of the change in velocity to the change in time, $\frac{\Delta v}{t}$. A horizontal line indicates motion at a constant speed with zero acceleration. A flat line along the horizontal time axis at zero speed represents an object at rest. A straight diagonal line represents uniform acceleration, a curve curving upward represents increasing acceleration, and a curve bending downward represents decreasing acceleration. The absolute total distance traveled by an object during a motion sequence is calculated by finding the total geometric area bounded beneath the speed-time curve. This total area is found by partitioning the region under the graph into standard geometric shapes, calculating their individual areas using spatial formulas, and summing those values.

Freefall refers to the motion of an object falling under the sole influence of gravity, devoid of air resistance. Under freefall, all objects experience identical constant gravitational acceleration regardless of their mass or size. Near the surface of the Earth, the acceleration due to gravity is denoted by the symbol $g$ and maintains a constant value of $9.8\,\text{m/s}^2$. This value indicates that during freefall, an object's downward velocity increases by $9.8\,\text{m/s}$ during every second of fall.

Chapter 3: Mass and Weight

Mass is defined as the measure of the quantity of matter in an object. Mass is a scalar quantity measured in the SI base unit of kilograms ($\text{kg}$). Masses recorded in grams ($\text{g}$) are converted to kilograms using the relation $1\,\text{kg} = 1000\,\text{g}$. The mass of an object provides an intrinsic property known as inertia, which resists any change in the object's state of rest or motion. Consequently, an object with a larger mass requires a greater net force to alter its speed or direction of motion. Masses are directly compared and measured experimentally using a beam balance or equal-arm balance.

A gravitational field is a region of space in which a mass experiences an attractive force due to gravity. The Earth produces a surrounding gravitational field that pulls any mass situated within it toward its center. Gravitational field strength, denoted by $g$, is defined as the gravitational force exerted per unit mass at a specific location, expressed mathematically as $\text{Gravitational Field Strength} = \frac{\text{Weight}}{\text{Mass}}$.

Weight is defined as the force of gravity acting on a mass. Weight is a vector quantity measured in newtons ($\text{N}$) that acts vertically downward toward the center of the gravitational field. The magnitude of an object's weight depends directly on its mass and the local gravitational field strength, as expressed by the equation $W = m \times g$, where $W$ is weight in newtons, $m$ is mass in kilograms, and $g$ is gravitational field strength in newtons per kilogram ($\text{N/kg}$). On Earth's surface, the gravitational field strength $g$ is equal to $9.8\,\text{N/kg}$, which is frequently rounded to $10\,\text{N/kg}$ for standard estimations. Weight is measured using a force meter or a spring balance, wherein the attached object causes a calibrated spring to extend proportionally to the gravitational force.

Chapter 4: Density

Density is defined as mass per unit volume of a substance, expressed mathematically by the equation $\rho = \frac{m}{V}$, where $\rho$ represents density, $m$ represents mass, and $V$ represents volume. Density is measured in units of grams per cubic centimetre ($\text{g/cm}^3$) or kilograms per cubic metre ($\text{kg/m}^3$). To convert a density value expressed in $\text{g/cm}^3$ into $\text{kg/m}^3$, the value is multiplied by $10^3$ (or $1000$). The mass of a liquid or solid sample is determined using an electronic balance.

For regularly shaped solids, volume is calculated using standard geometric formulas based on precise length measurements taken with rulers or calipers. For irregularly shaped solids, volume is determined via liquid displacement in a measuring cylinder. The density is subsequently computed by dividing the measured mass by the displacement volume.

Pure water has a standard density of $1000\,\text{kg/m}^3$, which is equivalent to $1\,\text{g/cm}^3$. An object or substance will sink in a liquid if its average density exceeds the density of the liquid. Conversely, an object will float on a liquid if its average density is less than the density of the liquid. When two immiscible liquids that do not mix are combined in a container, the liquid with the lower density floats on top of the denser liquid.

Chapter 5: Forces

A force is defined as a push or pull exerted by one body on another. Forces can cause an object at rest to move, accelerate, decelerate, come to a complete stop, or change its direction of motion. The SI unit of force is the newton ($\text{N}$). Forces are vector quantities and are represented in diagrams by straight arrows. The length of the vector arrow is drawn proportional to the force's magnitude using a scale, while the arrowhead indicates the precise line of action and direction.

When multiple parallel forces act along a single straight line, they are combined algebraically. Parallel forces acting in the same direction are added together to find the resultant force. Parallel forces acting in opposite directions are subtracted, and the net resultant force acts in the direction of the larger force with a magnitude equal to their numerical difference. Equal and opposite parallel forces cancel out completely, producing a net resultant force of $0\,\text{N}$.

Non-parallel vectors are resolved graphically using either the tip-to-tail method or the parallelogram method. In the tip-to-tail method, the tail of the second vector arrow is positioned at the tip of the first vector arrow. The resultant vector is drawn from the tail of the first vector to the tip of the second vector. In the parallelogram method, both vector tails are placed at a common origin point. Parallel lines are drawn from the tip of each vector to complete a four-sided parallelogram. The resultant vector is represented by the diagonal drawn from the common origin point to the opposite corner where the extended parallel lines intersect.

Friction is an opposing force that resists motion between two surfaces in contact, converting kinetic energy into thermal energy and generating heat. Air resistance, or drag, is a form of fluid friction experienced by a body moving through air. Drag opposes motion, reduces acceleration, and causes heating. When a skydiver leaps from an aircraft, weight acts downward while air resistance acts upward. Initially, air resistance is minimal, so the downward force of weight causes the skydiver to accelerate downward. As velocity increases, air resistance increases proportionally. Eventually, the upward air resistance force grows until it equals the downward force of weight. At this point, the forces are balanced, the resultant force becomes zero, acceleration ceases, and the skydiver falls at a steady terminal velocity. When the parachute opens, its surface area increases upward drag, creating an unbalanced upward resultant force that decelerates the jumper to a lower terminal velocity.

Vehicle dynamics depend on stopping distances, which comprise thinking distance and braking distance. Thinking distance is defined as the distance traveled by a vehicle during the driver's reaction time, from the moment an obstacle is spotted until the brakes are applied. Thinking distance depends on driver alertness and reaction time, remaining independent of road surface conditions. Braking distance is the distance traveled by the vehicle after the brakes are applied until it comes to a complete stop, which depends on road conditions, tire tread, brake condition, and vehicle mass.

Newton's Laws of Motion govern mechanical dynamics. Newton's First Law of Motion states that an object continues in its initial state of rest or uniform motion along a straight line unless acted upon by a net external resultant force. Newton's Second Law of Motion states that when a net resultant force acts on an object of mass $m$, it causes the object to accelerate in the direction of the resultant force. The resultant force is proportional to the product of mass and acceleration, expressed as $F = m \times a$. Newton's Third Law of Motion states that for every action force, there is an equal and opposite reaction force. When object $1$ exerts a force on object $2$, object $2$ exerts an equal magnitude force in the opposite direction on object $1$, satisfying $F_1 = F_2$ or $m_1 \times a_1 = m_2 \times a_2$.

When an object moves at a constant speed in a circular path, its direction changes continuously. Because velocity is a vector quantity that depends on direction, a changing direction implies that velocity changes continuously, meaning the object accelerates. The net resultant force required to sustain uniform circular motion acts inward toward the center of the circular path.

Mechanical deformation occurs when loads are applied to elastic bodies such as springs. Connecting two identical springs end-to-end in series doubles the overall extension for a given load, whereas mounting two identical springs side-by-side in parallel halves the extension because the applied load is divided equally between them. Hooke's Law states that the extension $x$ of a spring is directly proportional to the applied stretching force $F$, provided the force does not exceed the limit of proportionality. Hooke's Law is expressed mathematically as $F = k \times x$, where $k$ represents the spring constant measured in newtons per metre ($\text{N/m}$) or newtons per millimetre ($\text{N/mm}$). On a load-extension graph, motion obeying Hooke's Law produces a straight line passing through the origin. The point where the graph curves away from a straight line marks the limit of proportionality, beyond which the material deforms non-linearly.

A moment is defined as the turning effect of a force about a pivot point. Moments occur when an applied force causes an object to rotate around a fixed axis. The magnitude of a moment depends on the applied force and the perpendicular distance from the pivot to the line of action of the force, calculated as $\text{Moment} = \text{Force} \times \text{Perpendicular Distance}$. The SI unit of a moment is the newton-metre ($\text{N}\cdot\text{m}$). Common examples of turning moments include opening doors or turning spanners. The Principle of Moments states that for a body in rotational equilibrium, the sum of all clockwise moments about any pivot point must equal the sum of all anticlockwise moments about that same pivot point, expressed as $F_1 \times d_1 = F_2 \times d_2$

The centre of mass of an object is defined as the theoretical point at which its entire weight appears to act. For symmetrical objects of uniform density, the centre of mass lies at the geometric intersection of their axes of symmetry. To locate the centre of mass of an irregular flat sheet called a lamina, the lamina is freely suspended from a pin near its edge alongside a hanging plumb line. A line is traced along the plumb line using a pencil. The lamina is then suspended from a different point, and a second plumb line is traced. The point where these drawn lines intersect marks the centre of mass. An object remains stable as long as the vertical line of action passing downward through its centre of mass falls within its base area. If the vertical line of action falls outside the base area, an unbalanced moment is created, causing the object to topple over. Stability is enhanced by lowering the centre of mass or widening the base area.

Chapter 6: Momentum

Momentum is defined as the product of an object's mass and its velocity, expressed mathematically by the equation $p = m \times v$, where $p$ represents momentum, $m$ represents mass in kilograms ($\text{kg}$), and $v$ represents velocity in metres per second ($\text{m/s}$). Momentum is a vector quantity measured in kilogram metres per second ($\text{kg}\cdot\text{m/s}$) and takes positive or negative signs according to direction. For example, a body with a mass of $2\,\text{kg}$ traveling rightward at a velocity of $10\,\text{m/s}$ possesses a momentum of $20\,\text{kg}\cdot\text{m/s}$.

The Principle of Conservation of Momentum states that in the absence of external forces, such as friction, the total momentum of a closed system remains constant. In a collision or explosion involving interacting bodies, the total vector momentum before the event equals the total vector momentum after the event. Objects moving to the right are assigned positive momentum, while objects moving to the left are assigned negative momentum.

Impulse is defined as the product of the net force acting on an object and the time duration over which it acts, given by $\text{Impulse} = F \times t$. Impulse causes a change in momentum in an object, yielding the relationship $\text{Impulse} = \Delta p = m \times v - m \times u$, where $u$ is initial velocity and $v$ is final velocity. Consequently, $\text{Force} \times \text{time} = m \times v - m \times u$, showing that net force equals the rate of change of momentum.

Chapter 7: Work, Power, and Energy

Energy is defined as the capacity to do work. Energy is a scalar quantity measured in the SI unit of joules ($\text{J}$). The Law of Conservation of Energy states that energy cannot be created or destroyed, only transformed from one form into another, keeping the total amount of energy in an isolated system constant. Energy exists in various forms, including kinetic, gravitational potential, chemical, elastic strain, nuclear, electrostatic, and internal or thermal energy.

Kinetic energy is the energy possessed by an object due to its motion. Gravitational potential energy is the energy stored in an object due to its vertical position within a gravitational field. Chemical energy is energy stored within chemical bonds, such as in fuels, food, and electrical batteries. Elastic strain energy is energy stored in objects subjected to elastic deformation, such as stretched or compressed springs. Nuclear energy is energy stored within the atomic nucleus. Thermal energy represents the total kinetic energy of microscopic particles within a body due to its temperature. Electrostatic energy is stored in separated electrical charges and transferred via electrical currents. Internal energy represents the total sum of microscopic kinetic and potential energies within a substance.

Energy transformations occur through four mechanisms: mechanical working (a force acting through a distance), thermal working (heating via conduction, convection, or radiation), electrical working (charges moving through a potential difference), and wave working (transfers via electromagnetic or sound waves). In a battery connected to a motor, stored chemical energy transfers via electrical working into mechanical kinetic energy. In a boiler, chemical energy stored in fuel converts via thermal working into internal energy in water. A cyclist transforms chemical energy stored in muscle tissue via mechanical working into kinetic energy.

Gravitational Potential Energy ($GPE$) depends on mass, gravitational field strength, and height above a reference level, calculated as $GPE = m \times g \times h$. Changes in gravitational potential energy are calculated using $\Delta E_p = m \times g \times \Delta h$, where $\Delta h$ represents the net height change ($\text{final height} - \text{initial height}$). For example, if