1. BANKING OF ROADS Banking of roads is defined as the phenomenon in which the outer edges are raised for the curved roads above the inner edge to provide the necessary centripetal force to the vehicles so that they take a safe turn. Now, let us recall, what is centripetal force? It is the force that pulls or pushes an object toward the center of a circle as it travels, causing angular or circular motion. In the next few sections, let us discuss the angle of banking and the terminologies used in the banking of roads. Another terminology used is banked turn which is defined as the turn or change of direction in which the vehicle inclines towards inside. The angle at which the vehicle is inclined is defined as the bank angle. The inclination happens at the longitudinal and horizontal axis. Angle of Banking Consider a vehicle of mass ‘m’ with moving speed ‘v’ on the banked road with radius ‘r’. Let ϴ be the angle of banking, with frictional force f acting between the road and the Tyres of the vehicle. Total upwards force = Total downward force NcosΘ =mg+fsinΘ Where, NcosΘ: one of the components of normal reaction along the verticle axis mg: weight of the vehicle acting vertically downward fsinΘ: one of the components of frictional force along the verticle axis therefore, mg = NcosΘ – fsinΘ (eq.1) mv2/r = NsinΘ +fcosΘ (eq.2) Where,

2. NsinΘ: one of the components of normal reaction along the horizontal axis fcosΘ: one of the components of frictional force along the horizontal axis

3. KINETIC THEORY The kinetic theory of gases is a theoretical model that describes the molecular composition of the gas in terms of a large number of submicroscopic particles, which include atoms and molecules. Further, the theory explains that gas pressure arises due to particles colliding with each other and the walls of the container. The kinetic theory of gases also defines properties such as temperature, volume and pressure, as well as transport properties such as viscosity and thermal conductivity and mass diffusivity. It basically explains all the properties that are related to the microscopic phenomenon. The significance of the theory is that it helps in developing a correlation between the macroscopic properties and the microscopic phenomenon. In simple terms, the kinetic theory of gases also helps us study the action of the molecules. Generally, the molecules of gases are always in motion, and they tend to collide with each other and the walls of the containers. In addition, the model also helps in understanding related phenomena, such as the Brownian motion. Kinetic Theory of Gases Assumptions The kinetic theory of gases considers the atoms or molecules of a gas as constantly moving point masses with huge inter-particle distances and may undergo perfectly elastic collisions. Implications of these assumptions are as follows: i) Particles Gas is a collection of a large number of atoms or molecules. ii) Point Masses Atoms or molecules making up the gas are very small particles like a point (dot) on a paper with a small mass. iii) Negligible Volume Particles Particles are generally far apart such that their inter-particle distance is much larger than the particle size, and there is large free unoccupied space in the container. Compared to the volume of the container, the volume of the particle is negligible (zero volume). iv) Nil Force of Interaction Particles are independent. They do not have any (attractive or repulsive) interactions among themselves. v) Particles in Motion The particles are always in constant motion. Because of the lack of interactions and the free space available, the particles randomly move in all directions but in a straight line. vi) Volume of Gas

4. Because of motion, gas particles occupy the total volume of the container, whether it is small or big, and hence the volume of the container is to be treated as the volume of the gases. vi) Mean Free Path This is the average distance a particle travels to meet another particle. vii) Kinetic Energy of the Particle Since the particles are always in motion, they have average kinetic energy proportional to the temperature of the gas. viii) Constancy of Energy/Momentum Moving particles may collide with other particles or containers. But the collisions are perfectly elastic. Collisions do not change the energy or momentum of the particle. ix)Pressure of Gas The collision of the particles on the walls of the container exerts a force on the walls of the container. Force per unit area is the pressure. The pressure of the gas is thus proportional to the number of particles colliding (frequency of collisions) in unit time per unit area on the wall of the container. Kinetic Theory of Gases Postulates The kinetic theory of gas postulates is useful in understanding the macroscopic properties from the microscopic properties. Gases consist of a large number of tiny particles (atoms and molecules). These particles are extremely small compared to the distance between the particles. The size of the individual particle is considered negligible, and most of the volume occupied by the gas is empty space. These molecules are in constant random motion, which results in colliding with each other and with the walls of the container. As the gas molecules collide with the walls of a container, the molecules impart some momentum to the walls. Basically, this results in the production of a force that can be measured. So, if we divide this force by the area, it is defined to be the pressure. The collisions between the molecules and the walls are perfectly elastic, which means when the molecules collide, they do not lose kinetic energy. Molecules never slow down and will stay at the same speed. The average kinetic energy of the gas particles changes with temperature; i.e., the higher the temperature, the higher the average kinetic energy of the gas. The molecules do not exert any force of attraction or repulsion on one another except during collisions. A)Understanding Gas Laws of Ideal Gases i) Pressure α Amount or Number of Particles at Constant Volume

5. The collision of the particles on the walls of the container creates pressure. Larger the number of the particle (amount) of the gas, the more the number of particles colliding with the walls of the container. At constant temperature and volume, the larger the amount (or the number of particles) of the gas, the higher will be the pressure. ii) Avogadro’s Law – N α V at Constant Pressure When there is a greater number of particles, it increases the collisions and the pressure. If the pressure is to remain constant, the number of collisions can be reduced only by increasing the volume. At constant pressure, the volume is proportional to the amount of gas. ii) Boyle’s Law – Pressure � 1� at Constant Temperature At a constant temperature, the kinetic energy of particles remains the same. If the volume is reduced at a constant temperature, then the number of particles in unit volume or area increases. If there is an increased number of particles in the unit area, then it increases the frequency of collisions per unit area. At constant temperature, the smaller the volume of the container, the larger the pressure. ii) Amonton’s Law: P α T at Constant Volume The kinetic energy of the particle increases with temperature. When the volume is constant, with increased energy, particles move fast and increase the frequency of collisions per unit time on the walls of the container and hence the pressure. At constant volume, the higher the temperature, the higher will be the pressure of the gas. iv) Charles’s Law – V α T at Constant Pressure The change of temperature changes proportionately to the pressure. If the pressure also has to remain constant, then the number of collisions has to be changed proportionately. At constant pressure and a constant amount of substance, collisions can be changed only by changing the area or volume. At constant pressure, volume changes proportionally to temperature. v) Graham Law of Diffusion –

6. SURFACE TENSION Definition and Causes of Surface Tension • Surface tension is the tension of the surface film of a liquid caused by the attraction of particles in the surface layer by the bulk of the liquid. • It is influenced by the forces of attraction between particles within the liquid and the forces of attraction of solid, liquid, or gas in contact with it. • Surface tension is typically measured in dynes/cm, with the force required to break a film of length 1 cm. Causes of Surface Tension • Intermolecular forces such as Van der Waals force draw liquid particles together. • Surface tension is defined as the ratio of the surface force F to the length L along which the force acts.

7. • The SI unit of Surface Tension is Newton per Meter or N/m. Unit of Surface Tension • The dimensional formula of surface tension is MT-2. • The SI unit is Newton per Meter or N/m. Examples of Surface Tension • Water striders, water striders, and other insects can walk on water due to their weight being considerably less to penetrate the water surface. • Surface tension disinfectants, washing clothes by soaps and detergents, and washing with cold water can also contribute to surface tension. Calculation of Surface Tension • The formula for calculating surface tension is T = F/L. Methods of Measurement • Methods include the spinning drop method, pendant drop method, Du Noüy–Padday method, Du Noüy ring method, and Wilhelmy plate method etc. FREQUENCY • Defined as the number of complete oscillations made by any wave element per unit of time. • Defined as a parameter describing the rate of oscillation and vibration. Relation between Frequency and Period • Equation: f=1/T. • Represented by equation: y (0,t) = -a sin (ωt). Angular Frequency in Sinusoidal Wave • Refers to the angular displacement of any wave element per unit of time. • Represented by ω. • Formula: SI unit: rads^-1. • Frequency: Defined as the number of complete cycles of waves passing a point in unit time. • Time Period: Defined as the time taken by a complete cycle of the wave to pass a point. • Angular Frequency: Defined as the angular displacement of any element of the wave per unit of time. • Sinusoidal or Harmonic Motion: Periodic in nature, the graph of an element of the wave repeats itself at a fixed duration. • Time Period: Defined as the time taken by any string element to complete one such oscillation. • Simple Harmonic Motion: Represented by the equation y (0, t) = -a sin(ωt). Understanding Simple Harmonic Motion • Simple harmonic motion is a body's motion that experiences a restoring force proportional to its displacement from equilibrium position and directed towards it.

8. • Periodic motion is a motion that repeats identically after a fixed interval of time, such as the orbital motion of the earth around the sun, the motion of arms of a clock, or the motion of a simple pendulum. • Oscillatory motion is a periodic motion that takes place to and fro or back and forth about a fixed point. Harmonic Oscillation • Harmonic oscillation is an oscillation expressed in terms of a single harmonic function, such as sine or cosine function. • Simple harmonic oscillation is a harmonic oscillation of constant amplitude and single frequency under a restoring force whose magnitude is proportional to the displacement and always acts towards the mean position. Terms Related to Simple Harmonic Motion • Time Period: The time taken by the body to complete one oscillation. • Frequency: The number of oscillations completed by the body in one second. • Angular Frequency: The product of frequency with factor 2π. • Displacement: A physical quantity that changes uniformly with time in a periodic motion. • Amplitude: The maximum displacement in any direction from mean position. • Phase: The physical quantity expressing the position and direction of motion of an oscillating particle. Important Formulae of Simple Harmonic Motion • Displacement in SHM at any instant is given by y = a sin ωt or y = a cos ωt. • Velocity of a particle executing SHM at any instant is given by v = ω √(a2 – y2). • Acceleration of a particle executing SHM at any instant is given by A or α = – ω2 y. • Time period in SHM is given by T = 2π √Displacement / Acceleration. Force and Energy in SHM • Force in SHM is proportional and opposite to displacement. • Kinetic and potential energies of a particle are calculated. • The frequency of kinetic or potential energy of a particle executing SHM is double that of the frequency in SHM. • The frequency of total energy of particles executing SHM is zero as total energy in SHM remains constant at all positions. Simple Pendulum • A simple pendulum consists of a heavy point mass suspended from a rigid support by an elastic inextensible string. • The time period of a simple pendulum is given by T = 2π √l / g, where l is the effective length of the pendulum and g is acceleration due to gravity. • The time period increases with the effective length of the pendulum, when the bob of the simple pendulum is suspended by a metallic wire. Second’s Pendulum • A simple pendulum with a time period of 2 seconds is called a second’s pendulum. • The effective length of a second’s pendulum is approximately 1 metre on earth.

9. Conical Pendulum • A simple pendulum fixed at one end with a rotating bob. • Time period T = 2π √mr / T sin θ. Compound Pendulum • Any rigid body capable of swinging in a vertical plane about an axis passing through it. • Time period T = 2π √l / mg l. Torsional Pendulum • Time period T = 2π √I / C. • Time period I = moment of inertia of the body about an axis passing through the centre of suspension. Physical Pendulum • A rigid body capable of oscillating about an axis. • Time period T = 2π √I / mgd. Equivalent Simple Pendulum • A simple pendulum whose time period is the same as that of a physical pendulum. • Time period l = I / md. Spring Pendulum • A point mass suspended from a massless or light spring. • Time period and frequency of oscillations T = 2π √m / k or v = 1 / 2π √k / m. Oscillations of Liquid in a U-tube • Time period of oscillation T = 2π √h / g. • Oscillations of a floating cylinder in liquid T = 2π √l / g. Vibrations of a Loaded Spring • Restoring force acts on a spring when compressed or stretched. • Time period of a loaded spring is given by T = 2π √m / k. Free Oscillations • Oscillations with a decreasing amplitude with time. • Un-damped Oscillations with a constant amplitude with time. Forced Oscillations • Oscillations of any object with a frequency different from its natural frequency under a periodic external force. Resonant Oscillations • Oscillations of an external force applied on a body whose frequency is an integer multiple of the natural frequency.

10. Lissajous’ Figures • If two SHMs are acting in mutually perpendicular directions, the resultant motion is a curve loop. WORK ENERGY THEOREM • The Work-Energy Theorem states that net work done on a body equals the change in kinetic energy. • The equation is represented as: Kf – Ki = W, where Kf is final kinetic energy, Ki is initial kinetic energy, and W is net work done. • This equation follows the law of conservation of energy, stating that energy can only be transferred from one form to another. • Work done by all forces, including gravity, friction, and external forces, equals the change in kinetic energy. • Work done by a constant force produces constant acceleration, resulting in the equation W = F.ds.

11. • Work done by a non-uniform force is valid only when force remains constant throughout displacement. • A graphical approach to this is to find the area between F(x) and x from xi to xf.