AP Physics Conceptual Stuff

Adding Vectors

you need to define directions on a coordinate plane

Then, use the directions to add together to find the sum

Displacement

Vector quantity, This represents the change in an object’s position.

can be found by subtracting final position from initial position

Distance

Scalar quantity - total length traveled (does not account for direction like displacement does)

position vs time graph

Represents an object's position over a period of time

slope = velocity of object

Linear graph = constant velocity

Velocity vs Time Graphs

Measures the velocity of an object over a given period of time

Slope = average acceleration of object

Linear graph = constant acceleration

The  area underneath a velocity vs time graph is the displacement of the object

Objects in Free Fall

Objects that are only being impacted by gravity is considered to be in “free fall” 

Air resistance in these types of problems are typically negligible because of how minimal of impact it has

All objects, no matter their mass, will have an acceleration with a magnitude of g if they are near Earth’s surface

NOTE: Accelration will remain the same when an object is tossed up and then falls down, but velocity will change (at highest point, velocity will equal 0)

Objects in 2D

When a vector points in multiple directions (ex: up and rightward), you can form a right triangle to find the vx and vy components.(sine for y and cosine for x)

To find  find a total vector when only given the vx and vy  components, you can use the pythagorean theorem

Projectile Motion

Projectile motion is where an object is moving in both the horizontal and vertical directions, with the only force acting upon it being gravity

You will be given information in both dimensions, but only one factor can link the two dimensions together (time), since time is the same for both x and y dimensions. 

Projectile Motion II What is Always True

• Objects in projectile motion will have a horizontal acceleration of 0 (no acceleration/velocity is constant) 

• Objects in the vertical direction will have an acceleration due to gravity 

• At the highest point, the vertical velocity will equal 0

center of mass

where the mass is most heavily concentrated based on the masses and positions of all counterparts/objects involved

Normal force

force caused by a surface, perpendicular to surface (forms 90 degree angle with surface)

Tension force

 force occurring with strings, cables, ropes, etc, 

• Ideal ropes/strings → mass is negligible and tension is the same for all points in the rope

• Ropes/strings with mass will not always have the same tension force

• Forces of tension in the vertical direction will be weaker at the bottom than the top, since the top has to support the weight of the rope as well as additional weight attached

Newton’s First Law

“An object at rest will remain at rest and an object in motion will remain in motion (constant velocity) unless acted upon by a net external force”

N1L Balanced Forces

When an object is not accelerating, the forces acting upon the object are considered to be balanced, meaning the sum of all forces will equal 0.

Newton’s Second Law

Acceleration is equal to the net force acting upon object over the mass of the object (or system)

Newton’s Third Law

If  an object (Object A) were to exert a force (Force of A on B) on another object (Object B), the other object will exert a force equal and opposite from the other object (Force of B on A)

No matter the different masses of objects or velocities of each object, the forces exerted on each other will always be equal

Forces at an Inclined Angle

When an object is at an inclined angle, the force of gravity will continue to remain pointing downward, but the normal force will be pointing downward, as the normal force is perpendicular to surface

This means that there will be a disbalance, allowing for acceleration to occur.

To determine the coordinates, we need to make a new coordinate system (x’ and y’) at an angle. The y direction will be the axis perpendicular to the horizontal and the x axis will be the axis parallel

In an inclined angle, the normal force will equal

mgcostheta

In an inclined angle, force applied will equal

mgsintheta

If there is friction on an inclined angle problem…

you must make it opposite the direction of the objects motion (tries to resist objects motion)

gravitational force

When an object with mass interact with another object with mass, it can be described by gravitational force/weight (the planet with gravitational field and object of interest)

Fg will always point downward, meaning that it will always point towards the center of mass of the planet

Friction

Friction is the force that always opposes a sliding motion and will always remain parallel to the surface of the object of interest that is sliding

This force attempts to resist the motion of the force applied

Kinetic friction

The only factors that impact the magnitude of kinetic friction are the magnitudes of the normal force (surface on object) and the material of the surface the object is moving on

The coefficient of kinetic friction (μ) varies based on surfaces, but is typically greater than 0 and less than 1.

Friction (Static)

Static friction will increase as applied force increases, but can reach a maximum/extent (hence the less then or equal symbol in the equation)

• The magnitude of the normal force and material of the surfaces determine the maximum of the static frictional force

The coefficient of kinetic friction (μ) varies based on surfaces, but is typically greater than 0 and less than 1.

Spring Force/Hooke’s Law

When a spring has no forces acting upon it, the spring’s length is considered to be at equilibrium.

When forces are applied to a spring, the spring will stretch or compress by a certain displacement from its equilibrium length.

The magnitude of force a spring will exert is related to the change in length from equilibrium

• When at equilibrium, it does not exert any force

• When compressed or stretched, the spring will exert force

  • The further the spring is stretched for compressed, the stronger the force it will exert

circular motion

When an object moves around in a circle, the velocity will always be tangent to the circular course the objects takes.

The net force will ALWAYS point towards the center of the circular course

The second equation determines that if an object moves faster the acceleration will be greater If the acceleration is greater, the radius must be smaller. Both equations will be set equal to one another for circular motion

Circular Orbits / Newtons Law of Gravitation

When it comes to orbits around a central point (like the world revolving around the sun), the centripetal acceleration occurs from the gravitational force of the central point, which is represented by the equation.

G is represented as universal gravitational constant

There are two masses (m1 and m2, which are the planet’s mass and the mass of the object rotating it)

this equation can also be used to measure distance between two objects in space

Work

The ability of an outside system force to change the system

Ex: lifting a box, exercising, moving an item

Work is determined by the force applied multiplied by the distance the traveled

Scalar value

Force vs Displacment Graph

you can find the area underneath the curve to determine the work done by a force at a certain distance. 

so area under curve = work done

Bar Charts

Energy cannot be created nor destroyed. Due to this concept, energy must be conserved and transferred from one form of energy to another.

Since energy is conserved, one (or more) forms of initial energy must be equivalent to the heights of the other bars in the chart when each side is added To put it simply = initial energy amount MUST equal the final energy amount

Using these bar charts can aid in making conservation of energy equations, which can be used to plug in equations and determine a desired value

Kinetic Energy

The big YES question: Is the object moving?

is determined by the motion of the object, as well as its mass

Gravitational Potential Energy

The big YES questions: Is there height? Are you NOT at h=0?

Gravitational potential energy is determined by the force of gravity (mass and acceleration due to gravity) as well as height. This is typically used for interactions between Earth and smaller objects. Due to the interaction between Earth and the object(s), there will be potential energy, which is found with the first equation.

There is also another equation for UG, which typically involves a system of multiple celestial objects interacting with one another (such as two planets)r As a result of interaction, the system will have potential energy, which can be determined with the second equation

Elastic Potential Energy

The big YES question: Is something being stretched for compressed?

Elastic potential energy is defined by the spring constant as well as the change in distance (compression or stretch) of the object (typically would be a spring).

When the spring is at its relaxed length/rest,  it has no elastic potential energy. 

When stretched or compressed further from the relaxed length, the Us is increasingly positive.

Internal Energy

The big YES question: If there friction?

Internal energy allows for an object to change temperature and be impacted by a frictional force. Friction serves as a dissipative force in mechanical energy, so work done by friction will always reducing the overall mechanical energy. 

Mechanical Energy

Mechanical energy accounts for all of the total kinetic and potential energies involved in a system. 

Friction and air resistance, however, do not have potential energy and is not contained as potential energy. They often become thermal energy

Work is also dependent of path taken from initial to final configurations, indicating that the it only depends on the initial and finals configurations. 

Power

Power is the rate at which work id sone or at which a force does work and is measured in Watts.

The average power is found by taking the change in energy over the change in time (or work given over change of time)

Instantaneous power, on the other hand, is a specific amount of work done in a certain measure of time. This is measured by finding the force and multiplying it by the velocity (as well as any angles between velocity and force

Impulse - Momentum Theorem

Momentum is a VECTOR quantity

J = impulse (Force x change in time) or Delta P = change in momentum (mass x change in velocity) → linear momentum

Force vs Change in Time

The area under a force vs change in time is the impulse value

Open System

 at least one interacting object is NOT in system so impulse is applied.

closed system

every interacting objects is in the system. No impulse is applied.

Sigma Pi = Sigma Pf → conservation of linear momentum in closed system

systemw with no external forces acting upon them or energy transfers  will be referred to as an isolated system 

Center of Mass and Momentum

The center of mass of the system is concentrated in a specific point in a system. 

The center of mass of the system moves as if the system’s mass were located within that point

You can measure velocity of the center of mass through the different objects involved in the system (or total momentum of system)

Elastic Collisions

Objects that collide where motion remains unchanged 

• allows for conservation of both kinetic energy and momentum totals

• Think PV = nRT, where particles are assumed to have perfectly elastic collisions with no losses in energy

Inelastic collisions

Objects that collide where motion CHANGES

• Allows for conservation of ONLY momentum, while KE is lost 

• Think of ACTUAL particle collisions

• Typically in physic problems,, the two objects will stick together and function as a system aka one entity

Conservation of Momentum

• Total linear momentum remains constant from start to finish

• This only occurs if NO external forces are acting on a system that could change the total momentum

• If an object breaks down into multiple different components, momentum is conserved by equaling the components of both parts to 0

• Conservation of momentum will differ for x and y directions, if handling a 2-dimensional problem

Torque

measure of the rotational force acting on an object)

• This is measured through the amount of force applied to the object and the radius of rotation aka distance from pivot where force is appleid)

  • Remember that it is not squared since it corresponds to the arc length

    • If it was squared, it would correspond  to double the needed value!

• The distance FURTHEST away from the pivot )or where something rotates about its axis) will result in the highest angular acceleration

Newton’s Second Law → Rotations

For determining newton's second law of rotations, the net torque is divided by the rotational inertia

• Inertia (moment of inertia) is equal to the sum of masses and radii (or distance from axis) (mr^2)

  •  Greater inertia = less angular acceleration

  • Lesser inertia value = greater angular acceleration

If mass is concentrated around a rim, the inertia value will INCREASE and be HARDER to rotate

If mass is concentrated in the center of an object, it will be EASIER to rotate and have a SMALLER interia value

mass concentration vs rotational inertia

If mass is concentrated around a rim, the inertia value will INCREASE and be HARDER to rotate

If mass is concentrated in the center of an object, it will be EASIER to rotate and have a SMALLER interia value

Rotational Kinetic Energy

When an object is rolling DOWN, force of friction causes angular acceleration in clockwise direction, so angular and linear velocities will most speed up/increase (Ug → K + Krotational) ← assumed to be released from rest

When object is rolling upward, angular acceleration will be occurring in the COUNTERCLOCKWISE direction, so angular and linear velocity will decrease

(K + Krotational → Ugf)

Angular momentum and impulse

When there is an external torque acting upon a rotating object, the angular momentum will change

When there is no net torque acting upon a rotating object, angular momentum will be conserved and the change in angular momentum will be 0

Gravitation/Motion of Orbiting Satellites

gravitational force of one planet exerts on another are going to be equal and opposite to one another and will orbit a center mass between them.

Acceleration points towards the object that another object is orbiting while velocity is tangential

Oscillations

• A mass attached to a spring (where it will oscillate up and down or left to right depending on the setup)

• A pendulum/mass swinging back and forth (like a clock pendulum’s motion)

All of these setups have an equilibrium position, which is the point where there is no net force acting on the oscillating object

When an object is oscillating, there is a restoring force, which is always attempting to restore the system back to its equilibrium position (more on that in the next slide…)

Simple Harmonic Motion

Fsp and displacement will ALWAYS be in opposite directions (restoring force)

when a mass is attached to a spring, the motion occurring as a result will go up and downward. , it’s important to remember that…

• There is NO subtraction or addition of forces (so no additional forces applied or friction)

• Theoretically will go on forever (end at infinity)

What impacts SHM Period?

Only the mass and the spring constant will impact an oscillating motion. This is best represented in the equation in the diagram. 

Larger mass increases inertia, which makes the period longer, thus slowing oscillation

A larger spring constant (aka a stiffer spring) will result in a greater restoring force, which makes the period shorter and thus speeding up the oscillation

SHM Acronym

DA-V-DA-V

So.. at most compressed/streteched point = displacement ad acceleration are at their maxes (acceleration will point to where the mass will move afterward, displacement direction is based on where the mass is

At eq, velocity is at its maximum

Periods and Amplitude

Period = time needed for a cycle of something t to occur

Frequency (the inverse of period) determines how many cycles occur in a given amount of time in Hz

Amplitude = max magnitude of displacement

Pendulum motion

force of gravity served as the restoring force

These periods only depend on the acceleration due to gravity and length of the string

• Mass changes have NO effect on oscillation for

• Increasing string length will increase inertia, thus increasing period

• Displacement is measured via angles

Energy in SHM

There will be a constant shift between kinetic energy and potential energy.

• When released from rest, the potential energy is at its greatest, KE at its lowest

• Then, when the object reaches its equilibrium state, the kinetic energy will be at its highest point, potential energy at its lowest

• Then, when the object reavhes the other side of its trajectory, the potential energy will be at its maximum yet again and KE will be at its lowest

Fluids

Fluids are substances that have the ability to flow and do not have a fixed shape

• Liquids are incompressible because the molecules of liquid cannot be compressed/pushed closer together than they already are

• Gasses are compressible since they are free to move in their container

density

Determines an objects ability to float

• Densities of an object greater than that of a liquid will sink

• Densities of an object less than that of the liquid will float

Density of air will decrease going up into atmosphere

Pressure

Measures the perpendicular forces applied per surface area

• No matter the size of the area, the pressure would be the same

  • Decreasing area will reduce number of particles exerting force on the area, still upholding the exact same ratio as prior

Pressure and Depth

Pressure, as you move up into the atmosphere, will decrease

Pressure will increase when you move further down in a liquid

Guage Pressure

Pressure above the atmospheric pressure, does NOT factor atmospheres into equation

Bouyancy Force

Forces acting upward upon an object will eb greater than the forces acting downward (based on the weight of fluid displaced), resulting in a net upward force that allows an object to float

Weight will determine if an object sinks for floats

• If an obejct’s weight is greater than the buoyant force, it will sink

  • The weight of liquid displaced will remain the same, so the bouyant force will be the same, but the downward force will change

  • If a floating object increases weight, more of the liquid’s weight will be displaced, hence increasing bouyancy force upward to still allow for the object to be increasingly submerged, but still float

Archimedes Principle

Shape of objects determine bouyant force (because of how much vlume they displace)

• The weight of he fluid displaced from the object = bouyant force

• When weight if displaced fluid is greater than the weight of the object, it will gloat

• When weight if displaced fluid is less than the weight of the object, it will sink

Fluid Flow

Fluids flow when there are unbalance pressures/differences in pressure

• Similar to that of N2L

Equation of Conntinuity

The volume of fluid flowing inward (in a pipe with larger radius) should equal the exact same volume of of fluid flowing out of the pipe (into another pipe with a smaller radius)

• Since the surface areas will differ, the speed of the fluid’s flow will ALSO be different

Flow Rate

Flow rate equals the area times the acceleration, which should remain constant within pipes

Bernoulli’s Principle

Essentially conservation of energy in fluid form

• Pressure is equal to pressure energy, while the ρgy = potential energy (very similar to mgh). Then, the ½ρv² is kinetic energy.

• Both parts equal one another since energy will be conserved.