Romania Clasa a IX-a Physics (mathematics-informatics)
Chapters: 8
1. Introductory notions and working methods in physics
Measurement in physics · Errors and uncertainty
- Units and Measurement (Class 11) – To measure means to compare a quantity with a fixed, agreed amount called a unit. Result = number × unit. The world now uses the SI system with 7 base units: metre, kilogram, second, ampere, kelvin, mole and candela. Every other unit (like newton or joule) is a derived unit made by multiplying or dividing base units. Prefixes like kilo (10³) and milli (10⁻³) make very big or very small numbers easy.
- Significant Figures and Errors in Measurement – No measurement is perfect. Significant figures are the digits we trust plus the first doubtful digit. Errors tell how far a reading may be from the true value: absolute error Δa, relative error Δa/a and percentage error (Δa/a) × 100. When we add or subtract, absolute errors add. When we multiply or divide, percentage errors add. For a power aⁿ, the percentage error becomes n times.
2. Elements of kinematics
Describing motion · Velocity and acceleration · Types of motion
- Motion in a Straight Line (Class 11) – To describe motion we first choose a frame of reference: an origin, a direction and a clock. Position x changes with time t. Velocity v = dx/dt is the slope of the x–t graph; acceleration a = dv/dt is the slope of the v–t graph, and the area under the v–t graph is the displacement. For constant a: v = u + at, x = ut + ½at², v² = u² + 2as.
3. Newton's principles of mechanics and their applications
Newton's laws · Contact interactions · Dynamics of circular motion · Field interactions: gravitation
- Inertia, First Law, Momentum, Second Law and Impulse – A body keeps its state of rest or uniform motion unless a net external force acts on it (first law). Inertia is this laziness to change, and mass measures it. Momentum p = mv. The rate of change of momentum equals the net force: F = dp/dt, which gives F = ma when mass is constant (second law). Impulse J = F × Δt = Δp; a longer stopping time means a smaller force.
- Friction: Static, Kinetic and Rolling Friction, Laws and Lubrication – Friction is the force that opposes relative motion (or its start) between surfaces in contact. Static friction adjusts itself up to a maximum, the limiting friction fs,max = μs N. Once sliding starts, kinetic friction fk = μk N acts, and μk < μs. Friction depends on the normal force and the nature of the surfaces, not on the area of contact. Rolling friction is much smaller than sliding friction; lubricants and ball bearings reduce friction.
- Centripetal Force, Car on a Level Road and on a Banked Road – A body moving in a circle is always changing direction, so it needs a net force towards the centre: the centripetal force F = mv²/r. It is not a new kind of force; tension, gravity, friction or a part of the normal force supplies it. On a level road only friction supplies it, so vmax = √(μs r g). On a road banked at θ, a part of the normal force helps: with no friction the ideal speed is v₀ = √(r g tanθ), and with friction vmax = √[r g (μs + tanθ)/(1 − μs tanθ)].
- Newton's Universal Law of Gravitation – Every mass in the universe pulls every other mass. The pull between two point masses m₁ and m₂ a distance r apart is F = G m₁ m₂ / r². It acts along the line joining them. The two bodies pull each other with equal and opposite forces. G = 6.67 × 10⁻¹¹ N m² kg⁻² is the same everywhere, so it is called the universal gravitational constant. When many masses pull one body, the forces add as vectors (superposition).
4. Variation theorems and conservation laws in mechanics
Mechanical work and power · Mechanical energy · Momentum
- Work, Kinetic Energy, Work–Energy Theorem and Power – Work is done when a force moves something along its direction: W = F·s = F s cos θ. For a changing force, work is the area under the F–x graph. A moving body has kinetic energy K = ½mv². The work–energy theorem says: net work done on a body = change in its kinetic energy. Power is how fast work is done: P = W/t = F·v.
- Potential Energy, Spring Energy and Conservative Forces – Potential energy U is energy stored because of position or shape. Near the Earth U = mgh; in a stretched or squeezed spring U = ½kx². A force is conservative if its work depends only on the start and end points, not on the path (gravity, spring force). Then F = −dU/dx and mechanical energy K + U stays constant. Friction and air drag are non-conservative: they turn mechanical energy into heat.
- Third Law, Conservation of Momentum and Equilibrium of Forces – Forces always come in pairs: if A pushes B, B pushes A with an equal and opposite force at the same instant (third law). The pair acts on different bodies. For a system with no net external force, the total linear momentum stays constant, which explains recoil, rockets and collisions. A particle is in equilibrium when all forces on it add to zero; three concurrent forces in equilibrium form a closed triangle.
5. Mechanical equilibrium
Equilibrium of bodies
- Torque, Angular Momentum and Equilibrium of Rigid Bodies – Torque is the turning effect of a force: τ = r × F, size rF sinθ, unit N m. Angular momentum is the turning version of momentum: L = r × p; for a body spinning about a fixed axis L = Iω. Torque changes angular momentum: τ = dL/dt. If the outside torque is zero, L stays constant, so pulling mass in makes a body spin faster. A rigid body is in equilibrium when the total force is zero (no sliding) and the total torque about any point is zero (no turning).
6. Integrating theme: road safety
Physics of road safety
- Road Safety: The Physics of Stopping and Staying Safe – A vehicle cannot stop at once. First the driver needs time to notice and react (reaction time, about 1 s). During this time the vehicle keeps moving: this is the thinking distance = speed × reaction time. Then the brakes slow it down: this is the braking distance = speed² ÷ (2 × deceleration). Stopping distance = thinking distance + braking distance. Double the speed and the braking distance becomes four times longer. Wet, icy or worn roads and tyres give less grip (friction), so braking is longer and bends are harder to take. In a crash, seat belts, helmets, airbags and crumple zones make the stop take longer, which makes the force on the body smaller. Pedestrians, cyclists, scooter riders and passengers all stay safer by following traffic rules, being seen, and never using a phone while moving.
7. Elements of celestial mechanics (optional)
Kepler's laws and orbits
- Kepler's Laws of Planetary Motion – Kepler gave three rules for how planets move. 1) Each planet moves on an ellipse with the Sun at one focus. 2) The line from the Sun to the planet sweeps equal areas in equal times, so the planet moves faster when it is near the Sun. 3) The square of the time for one round (T²) is proportional to the cube of the semi-major axis (a³). The second law is really conservation of angular momentum. The third law follows from Newton's law of gravitation.
8. Notions of fluid mechanics
Fluid statics · Fluid dynamics
- Pressure in Fluids and Pascal's Law – A fluid (liquid or gas) pushes on every surface it touches. Pressure is this normal force per area, P = F/A. Because of gravity, the fluid above a point has weight, so pressure grows with depth: P = P₀ + ρgh. Points at the same depth in a still liquid have the same pressure, whatever the vessel's shape. Pascal's law says an extra pressure applied to an enclosed fluid reaches every point equally. The hydraulic lift and brakes use this: a small force on a small piston becomes a big force on a big piston, F = f × A/a.
- Viscosity, Stokes' Law and Bernoulli's Theorem – A moving fluid is like a stack of layers sliding over each other. The friction between layers is viscosity: F = ηA(dv/dx). A small ball falling through a fluid feels a drag F = 6πηrv (Stokes' law); when drag plus buoyancy balance the weight, it falls at a steady terminal velocity. Slow flow is streamline; above a critical velocity (Reynolds number about 2000) it becomes turbulent. For steady streamline flow, Av is constant (continuity) and P + ½ρv² + ρgh is constant (Bernoulli). So fast fluid has low pressure. This explains Torricelli's efflux speed √(2gh), the venturimeter, and the lift on wings and spinning balls.