AP Physics 1 Comprehensive Notes

AP Physics 1 Notes: 2024-2025 Course

Unit 1: Kinematics

Motion Basics

• Vector: A measurement with both magnitude and direction (e.g., velocity, acceleration). Vectors can be added/subtracted along their axes if they represent the same variable. Angular vectors can be separated into components using trigonometric functions (sine, cosine, tangent).
• Scalar: A measurement with magnitude only (e.g., mass, temperature).
• Frame of Reference: A coordinate system where the observer is stationary, extending infinitely in all directions and equipped to measure positions/velocities with necessary accuracy.
  1. Extends infinitely far in all directions
  2. The observer is at rest within the reference frame
  3. The observer is sufficiently equipped and able to measure positions and velocities to any level of accuracy needed.
• Distance: Total path traveled (scalar).
• Displacement: Change in position with direction (vector).
• Speed: Rate of change of total position over time (scalar).
• Velocity: Speed and direction; calculated using displacement (vector).
• Acceleration: Rate of change of velocity over time.

Kinematics Equations

• Kinematics equations are in one dimension unless stated otherwise.
• Equations can be interchanged between x- and y-directions if all variables are respective to the axis. Time (t) is the only shared variable in 2-dimensional equations.

Constant Velocity

• Average Velocity: v = \frac{xf - xi}{\Delta t}

Accelerating Motion

• Final displacement/ position : xf = xi + v_i \Delta t + \frac{1}{2} a (\Delta t)^2

• Acceleration: a = \frac{vf - vi}{\Delta t}

• Final Velocity: vf = vi + a \Delta t

• Time: \Delta t = \frac{vf - vi}{a}

• Final velocity w/o Time: vf^2 = vi^2 + 2 a \Delta x

• Variables:

  • x: displacement
  • v: velocity
  • a: acceleration
  • t: time
  • x_f: final displacement/position
  • x_i: initial displacement/position
  • g: Gravitational acceleration (9.8 m/s²)

Graphing Motion

• Slope: \frac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1}
  • Position Graphs: Slope = velocity.
  • Velocity Graphs: Slope = acceleration, Area under curve = distance.
  • Acceleration Graphs: Slope is found from the slopes of velocity graphs, Area under curve= velocity.

• Areas under the curve can be positive or negative relative to the origin of the y-axis variable.

Unit 2: Forces and Translational Dynamics

Types of Forces

• Weight (w, F_g, or mg): Force of gravitational acceleration on a mass, pointing towards the center of a planetary object.
• Spring (F_s): Force exerted by an elastic object returning to its original position.
• Tension (F_T or T): Force exerted by a non-elastic object (cable, rope, string, chain).
• Normal (F_N or N): Supportive force acting perpendicular (90°) to the surface of contact.
• Friction (F_f or f): Contact force resisting motion.
• Drag (D or F_{drag}): Specific friction due to fluids (air, water).
• Thrust (F_{thrust}): Force of expelled fluid pushing an object forward.
• Buoyant (F_B): Upward force of a fluid supporting objects.

Free Body Diagrams

• Model showing all forces acting on an object:
  • Arrow tails touch the dot representing the object.
  • Arrowheads point away from the dot.
  • Known values/forces labeled; arrow lengths proportionate to known values.
  • Forces in the same direction originate from the dot.
  • Forces pointing up/right are typically positive; down/left are negative (but choose a convenient frame of reference).

Net Force

• The net force (F_{net}) is the sum of all forces in all directions, often separated into x- and y-directions.
• Angled force vectors have x- and y-components determined by geometric functions of right triangles.

Newton's Laws

• 1st Law (Inertia): Object in equilibrium remains at constant motion or at rest, object resists changes in its state of motion.
  • Balanced Forces:
    • F_{net} = 0
    • a = 0
    • v = constant
• 2nd Law: Unbalanced forces cause motion changes according to respective masses.
  • Unbalanced Forces:
    • F_{net} \neq 0
    • F_{net} = ma
    • a = \frac{F_{net}}{m}
    • a = \frac{F{net}}{m{sys}}
• 3rd Law: Paired Forces are equal and opposite; F{1 on 2} = F{2 on 1}.

Special Forces

• Friction: The magnitude of friction is determined by the material and the normal force.
  • Ff = \mu FN
  • F_f: friction force
  • \mu: coefficient of friction (static or kinetic)
    • Static Friction (f_s): Prevents an object from moving.
    • Kinetic Friction (f_k): Resists an object already moving.
• Spring Force:
  • F_s = -k \Delta x
  • F_s: spring force
  • k: spring constant (material characteristic)
  • \Delta x: stretch/compression distance (the negative constant indicates that the spring force acts in the opposite direction of the displacement of the spring).
• Gravitational Force (Newton's Law of Gravity):
  • Fg = \frac{G m1 m_2}{r^2}
  • a_g = \frac{GM}{r^2}
  • G: gravitational constant
  • m1, m2: masses of two objects
  • r: distance between centers of mass
• Objects in a system affect each other but are not affected by objects outside of the system.

Center of Mass

• Determined by the sum of products of mass and position, divided by total mass.
  • x{cm} = \frac{\Sigma mi xi}{\Sigma mi}
• In uniform density, symmetrical objects, the center of mass is along lines of symmetry.

Circular Motion/Orbits

• Motion in a uniform circular path (often constant velocity but not always).
• Change in direction counts as acceleration (even if speed is constant).
• Centripetal Acceleration: Acceleration towards the center of the orbital path.
  • a_c = \frac{v^2}{r}
• Centripetal Force: Net force providing centripetal acceleration.
  • F_{net} = \frac{mv^2}{r}
• v = \frac{2 \pi r}{T}
  Where T is the period.
• Frequency: f= \frac{1}{T}
• v = 2 \pi r f

Unit 3: Work, Energy, and Power

Energy

• Capacity to do work, measured in Joules (J).
• Two main categories:
  • Potential Energy (U).
  • Kinetic Energy (K).
• Mechanical Energy: Total potential and kinetic energy of a system.

Potential Energy

• "Energy stored".
• Gravitational Potential Energy: Energy due to position above a reference point; U_g = mg \Delta y
• Elastic/Spring Potential Energy: Energy stored in deformation of elastic object; \Delta U_s = \frac{1}{2} k (\Delta x)^2

Kinetic Energy

• "Energy of motion".
• Two specific motions:
  • Translational (linear) Kinetic Energy.
  • Rotational (angular) Kinetic Energy.
• Translational Kinetic Energy: Energy needed to move mass (m) at velocity (v); K = \frac{1}{2} m v^2
• Rotational Kinetic Energy: Energy to move an object in circular motion at angular velocity some distance from its center; K = \frac{1}{2} I \omega^2
  Translational Kinetics can use linear kinematic equations
  vx = v{xo} + ax t x = xo + v{xo}t + \frac{1}{2} ax t^2
  vx^2 = v{xo}^2 + 2 ax(x - xo)
  Rotational Kinetics can use the same kinematic equations with their angular variables
  \omegax = \omega{xo} + \alphax t \theta = \thetao + \omega{xo}t + \frac{1}{2} \alphax t^2
  \omegax^2 = \omega{xo}^2 + 2 \alphax(\theta - \thetao)
• Conversions:
  • v = r \omega
  • a = r \alpha

Conservation of Energy

• In a closed system, total energy is conserved but transforms between objects/forms. ($\Sigma Ei = \Sigma Ef$)
• In an open system, energy transfers to the outside environment via:
  • Work: mechanical energy transfer by pushing/pulling.
  • Heat: nonmechanical energy transfer due to temperature difference.
• Efficiency: Energy maintained by the system after energy is transferred out:
  • \text{Efficiency} = \frac{\text{Useful energy}}{\text{Total input energy}} \times 100\%

Work-Energy Theorem

• Total work done on/by an object equals the total change in energy: \Sigma W = \Sigma \Delta K + \Sigma \Delta U
• Along a linear path: W = Fd = Fd \cos{\theta}
• Along a circular path: W = \tau \theta The mass of a rotating object has less to do with its absolute mass and more to do with its rotational inertia, which is the rotational mass in motion, similar to finding the center of mass.
  • Where Torque: \tau = r_{\perp} F = r F \sin{\theta}
  • Rotational Inertia: I=\Sigma mi ri^2

Power

• Rate at which work is done (or rate of change in energy):
  • P_{avg} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}
  • P_{inst} = F v = F v \cos{\theta}

Unit 4: Linear Momentum

Momentum vs Inertia

• All objects with mass have inertia, which is the tendency of an object to continue doing what it is already doing.
• Only moving objects have momentum, which is defined as mass in motion.

Momentum

• Mass in motion, quantified as mass × velocity: p = mv (kg m/s).

Conservation of Momentum

• Momentum is conserved in closed systems (no external force).

• Initial momentum equals final momentum: pi = pf

• (p1 + p2 + … pn)i = (p1 + p2 + … pn)f

• (m1 v1 + m2 v2)i = (m1 v1 + m2 v2)f

Impulse (J=Ft)

• Changes in momentum caused by a force applied over time.
  • J = F \Delta t
    *Specifically
    J = F_{avg} \Delta t

Impulse-Momentum Theorem (J=p)

• Equivalence between impulse and change in momentum.
  • J = \Delta p
• Which also means
  • \Delta p = Ft
  • Ft = mv

Elastic vs Inelastic Collisions

• Elastic Collisions: Objects separate before and after impact, retaining their original shape and Kinetic Energy is conserved.
  KEi=KEf
  (\frac{1}{2} m1v1^2 + \frac{1}{2} m2v2^2)i = (\frac{1}{2} m1v1^2 + \frac{1}{2} m2v2^2)f
• Inelastic Collisions: Objects become entangled, stuck, or trapped together upon contact, and Kinetic Energy is not conserved.
• Pi = Pf
• (m1v1 + m2v2)i = (m1 + m2)vf

Unit 5: Torque and Rotational Dynamics

Linear to Rotational Conversions

Rotational Kinematics

Treat our kinematics equations are treated exactly the same regardless of linear or rotational motion, as long as the variables are converted appropriately. Variables need to be either all rotational or all linear.
Linear Equations:
vf = vo + at
x = xo + vot + \frac{1}{2}at^2
vf^2 = vo^2 + 2 a (x-x0) Rotational Equations: \omegaf = \omegao + \alpha t \theta = \thetao + \omegaot + \frac{1}{2} \alpha t^2 \omegaf^2 = \omegao^2 + 2 \alpha (\theta - \theta0)

Centripetal Force and Acceleration

Sum-force (net force) that pulls or pushes the object towards the center of its rotation. It does NOT need to be physically attached to the center
For an object to have a circular motion, the TOTAL NET FORCE must be centripetal ("center-seeking")
If the net force on an object is centripetal, it will provide a centripetal acceleration according to Newton's Second Law
F=ma becomes Fc = mac
Centripetal acceleration can also be directly solved if we know the linear velocity and the
radius of the turn using the equation a=\frac{v^2}{r}

Vertical and Horizontal Rotation

Apparent weight is the force pushing up on an object by a surface (typically the Normal Force), but for an object in a vertical rotation, it can be experienced by the centripetal force as well
Force at the Top: FN = mg - \frac{mv^2}{r} = 0 Force at the Bottom: FN = mg + \frac{mv^2}{r}
The velocity at which both of these apparent weights are achieved can be derived from the equations and is equal to
v= \sqrt{rg}

Without sliding: v= \sqrt{\mu gr}
Without breaking free: v=\sqrt{gr \tan{\theta}}

Torque

• Net force causing rotation about an axis.
  Finding perpendicular radii gives the equation T = r{\perp} F Finding perpendicular force components gives the equation T=rF{\perp}
  *For an object to continue horizontal rotation without slipping or breaking free of its circular path, the maximum velocity is dependent on the horizontal component of its motion
  *Counterbalances can result in rotational equilibrium ($\tau = 0$).

Rotational Inertia

Rotational inertia, I, is the rotational equivalent of mass and can indicate how difficult or easy it is to cause a change in rotational motion.

• Rotational inertia can be used with torque to find the rotational acceleration according to Newton's Second Law F = ma, which converts to T = I a.
  The Parallel Axis Theorem relates axes of a uniform density body and allows us to calculate new rotational inertias about parallel axes through the equation I' = I{cm} + Md^2 where I{cm} is the inertia about the center of mass, M is the total mass of the rotating body, and d is the distance between the parallel axes.

Unit 6: Energy and Momentum of Rotating Systems

Rotational Energy

• Rotational kinetic energy: K = \frac{1}{2} I \omega^2
• Total energy is conserved but may be divided between translational and rotational forms.

Work-Energy Theorem

• Change in rotational kinetic energy equals torque applied over a time: W{rot} = \Delta KE{rot}
  • W_{rot} = \frac{1}{2} I \omega^2
  • \tau \Delta \theta = KE_{rot}
  • \tau \Delta \theta = \frac{1}{2} I \omega^2
• Work can increase translational kinetic energy without affecting rotational energy.

Angular Momentum

• Mass in angular motion: L = I \omega

Angular Momentum of A Point Particle

An object moving in translational motion past a fixed point will have an angular momentum relative to that point, based on the radius, mass, and translational velocity of the object, given by the equation:

• L = r m v \sin{\theta}

Conservation of Momentum

L₁ = L₁
Iω₁ = Iω
romvormvf

Unit 7: Oscillations

Period and Frequency

• Frequency: f = cycles/second
• Period: T = seconds/oscillation
  T = \frac{1}{f}
• Oscillation of a spring:
  • T = 2 \pi \sqrt{\frac{m}{k}}
  • f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}
• Pendulums (gravity does restoring work):
  • T = 2 \pi \sqrt{\frac{L}{g}}
  • f = \frac{1}{2 \pi} \sqrt{\frac{g}{L}}

Forces Involved

• Simple harmonic motion experiences a restoring force proportional to displacement from equilibrium.
• Pendulum: Restoring force provided by gravity.
• Spring: Restoring force is the spring force (F = -kx).

Graphing

• Full oscillation includes every position of motion.
  Amplitude (A) is the distance from equilibrium to maximum displacement

Finding Position

Sine graphs represent oscillation cycles that begin by passing through the equilibrium position (x = 0)
Cosine graphs represent oscillation cycles that begin from the maximum displacement position, X_{max}
Formulas:
y = A\sin(2 \pi f t)
x = A\cos(2 \pi f t)
Position equations for either function utilize amplitude, frequency, and time, along with their respective functions.

SHM and Energy

• Objects in SHM convert between potential and kinetic energy.

*Pendulum: ME=U+K
At maximum displacement (amplitude), the pendulum will have only gravitational potential U{g max}\ At its equilibrium, or lowest position in its oscillation, it will have only kinetic energy. K{max}
There may still be some gravitational potential energy if it is above ground level, but this does not contribute to the oscillation. Set your reference point before calculating the GPE!

Spring: Spring potential energy and kinetic energy.
Regardless of the oscillation type, the total mechanical energy is
always conserved in the absence of friction.
ME{initial} = ME{final}

Unit 8: Fluids

Density

• Density: \rho = \frac{m}{V}
  • \rho = density (kg/m³), m = mass (kg), V = volume (m³).
  • Can be rearranged into m = \rho V
    Volume of a rectangular solid = lwh
    Volume of a cylinder = πr²l
    Volume of a sphere = (4/3πr^2)

Pressure

• Pressure: P = \frac{F}{A}
  • P = pressure (N/m² or Pa), F = force (N), A = area (m²).
  • Scalar, acting perpendicular to the surface.
    *Atmospheric pressure (Po) is the pressure of the Earth's atmosphere pushing down on objects at its surface.
    All questions are assumed to be at sea level, where atmospheric pressure = 100,000 Pa
    Absolute pressure is the total pressure exerted at a given point, which can include multiple fluid pressures and atmospheric pressure.
• P=Po+pgy
• Fluid Pressure: P = \rho g y
  • \rho = density (kg/m³), g = gravitational acceleration (m/s²), y = height of fluid column.

Buoyancy

• Upward pressure on an object submerged in a fluid.
• Buoyant Force: FB = mf g = \rho V g
  • F_B = buoyant force (N),\rho= fluid density (N/m³), V = volume of object/displaced fluid (m³), g = gravitational acceleration (m/s²).

Archimedes' Principle

Upward buoyant force equals the weight of the displaced fluid: FB = m \rho g = \rhoF g V

Bernoulli

• Flow rate (Q) is constant: Q = Av = \frac{m}{\rho A v}

  • A = area, v = velocity.
  • A1 v1 = A2 v2

• Bernoulli's Principle:

  • P + \rho g y + \frac{1}{2} \rho v^2 = constant
  • P1 + \rho g y1 + \frac{1}{2} \rho v1^2 = P2 + \rho g y2 + \frac{1}{2} \rho v2^2

Torricelli's Theorem

Conditions where P₁ = P₂ = Po and vi can be set to zero.

• Then, v_2 = \sqrt{2g \Delta y}
  • v_2 exactly the same velocity that an object would have if dropped from a height of Ay = y₁-Y2, allowing us to apply kinematic equations to fluid motion.