United Grade 11 AP Physics 1: Algebra-Based
Chapters: 8
1. Kinematics
Scalars and Vectors in One Dimension · Displacement, Velocity, and Acceleration · Representing Motion · Reference Frames and Relative Motion · Vectors and Motion in Two Dimensions
- 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.
- Scalars and Vectors for Motion in a Plane (Class 11) – A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.
2. Force and Translational Dynamics
Systems and Center of Mass · Forces and Free-Body Diagrams · Newton's Third Law · Newton's First Law · Newton's Second Law · Gravitational Force · Kinetic and Static Friction · Spring Forces · Circular Motion
- Centre of Mass: Two Particles, Rigid Body and Uniform Rod – The centre of mass is the one point that moves as if all the mass of a system were packed there. For two particles on a line, x_cm = (m₁x₁ + m₂x₂)/(m₁ + m₂). It lies on the line joining them, closer to the heavier one. For many particles, x_cm = Σmx/Σm (same for y and z). For a uniform rod of length L, the centre of mass is at L/2, its middle. Internal forces cannot move the centre of mass; only an outside force can: M·a_cm = F_ext.
- 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.
- 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.
- 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).
- 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.
- Oscillations of a Spring and a Simple Pendulum – A block on a spring feels a pull back F = −kx, where k is the force constant (stiffness). It does SHM with T = 2π√(m/k). A simple pendulum, for small swings, feels a pull back mg sinθ ≈ mgθ and does SHM with T = 2π√(L/g). The pendulum’s period does not depend on the bob’s mass or (for small swings) on the amplitude.
- 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)].
3. Work, Energy, and Power
Translational Kinetic Energy · Work · Potential Energy · Conservation of Energy · Power
- 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.
4. Linear Momentum
Linear Momentum · Change in Momentum and Impulse · Conservation of Linear Momentum · Elastic and Inelastic Collisions
- 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.
- 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.
- Elastic and Inelastic Collisions in 1D and 2D – In every collision, total momentum is conserved (no outside force during the short hit). In an elastic collision kinetic energy is also conserved. In an inelastic collision some kinetic energy becomes heat, sound or dent energy; if the bodies stick together it is perfectly inelastic. In 1D, elastic collision gives v₁ = (m₁ − m₂)u₁/(m₁ + m₂) and v₂ = 2m₁u₁/(m₁ + m₂) when body 2 starts at rest. In 2D, momentum is conserved separately along x and y.
5. Torque and Rotational Dynamics
Rotational Kinematics · Connecting Linear and Rotational Motion · Torque · Rotational Inertia · Rotational Equilibrium and Newton's First Law in Rotational Form · Newton's Second Law in Rotational Form
- Moment of Inertia, Radius of Gyration and Rotational Motion – Every idea of straight-line motion has a turning twin: θ for x, ω for v, α for a, I for m, τ for F, L = Iω for p. Moment of inertia I = Σmr² tells how hard it is to change a body's spin; it grows fast when mass sits far from the axis. Radius of gyration k is the distance at which all mass could sit to give the same I: I = Mk². Standard values: ring MR², disc ½MR², solid sphere ⅖MR², rod about centre ML²/12. With constant α: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ, and τ = Iα.
- 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. Energy and Momentum of Rotating Systems
Rotational Kinetic Energy · Torque and Work · Angular Momentum and Angular Impulse · Conservation of Angular Momentum · Rolling · Motion of Orbiting Satellites
- Moment of Inertia, Radius of Gyration and Rotational Motion – Every idea of straight-line motion has a turning twin: θ for x, ω for v, α for a, I for m, τ for F, L = Iω for p. Moment of inertia I = Σmr² tells how hard it is to change a body's spin; it grows fast when mass sits far from the axis. Radius of gyration k is the distance at which all mass could sit to give the same I: I = Mk². Standard values: ring MR², disc ½MR², solid sphere ⅖MR², rod about centre ML²/12. With constant α: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ, and τ = Iα.
- 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).
- Escape Speed, Orbital Velocity and Energy of an Orbiting Satellite – Throw something fast enough sideways and it keeps falling around the Earth without landing: that speed is the orbital velocity, v₀ = √(GM/r), about 7.9 km/s just above the surface. Throw it faster, at the escape speed vₑ = √(2GM/R) = √(2gR) ≈ 11.2 km/s, and it leaves Earth for ever. vₑ = √2 × v₀. A satellite in a circular orbit has KE = GMm/2r, PE = −GMm/r and total energy E = −GMm/2r. The total is negative, so the satellite is bound. Higher orbits are slower and take longer: T = 2π√(r³/GM).
7. Oscillations
Defining Simple Harmonic Motion (SHM) · Frequency and Period of SHM · Representing and Analyzing SHM · Energy of Simple Harmonic Oscillators
- Simple Harmonic Motion (SHM) – A motion that repeats after a fixed time is periodic. If the object goes to and fro about a middle point and the force pulling it back is proportional to its distance from the middle (F = −kx), the motion is simple harmonic. Its position is x = A sin(ωt + φ), with ω = 2π/T = 2πf. Speed is largest in the middle, acceleration is largest at the ends, and total energy ½kA² stays constant.
8. Fluids
Internal Structure and Density · Pressure · Fluids and Newton's Laws · Fluids and Conservation Laws
- Fluids: Internal Structure and Density – A fluid is any substance that can flow: liquids and gases. Its particles can slide past one another, so a fluid takes the shape of its container. Density ρ = m/V tells how much mass is packed into each cubic metre (water: 1000 kg/m³). Relative density = ρ / ρwater. Physicists often model a liquid as an ideal fluid: incompressible (constant density) with no internal friction.
- 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.