Physics 3C Fundamental Principles

[S]PEC

(1) sketch a relevant, labeled visual(s) for the problem
(2) state important info and what you are solving for

S[P]EC

(1) the name(s) of the principle(s)
(2) the equation from the principle in its mathematical form
(3) the required assumption to use the principle

SP[E]C

(1) show your work and obtain a final answer
(2) if stuck, explain how you are stuck and what other information you need to be able to obtain a final answer
(3) if still stuck, write a second plan

SPE[C]

(1) for vector-involved problems, estimate the direction of the answer with a drawing or free-body diagram to scale
(2) estimate order of magnitude (power of 10) of an answer
(3) use dimensional analysis to solve equations mathematically by using the units only to see if they result int he expected unit of the answer
(4) reason through the problem using different principles/procedure
(5) check the symmetry of the solution. especially useful for answers in equation form
(6) evaluate what happens a variable approaches a limit or in other special cases. especially useful for answers in equation form.

∆

change: the change in a variable between two times is its value at the final time minus its value at the initial time

coordinate system

an axis or set of axes (e.g. x, y, z) and a choice of which instant of time is called t=0

reference frame

a coordinate system that also specifies the velocity and acceleration of the coordinate system (and for relativistic problems, time)

inertial reference frame

a reference frame for which the observer is not accelerating (approach problems differently when the observer does accelerate)

⃗r

position: a vector that points from the origin of the coordinate system to the object’s location in space [meters (m)]

⃗v ≡ ∆⃗r/∆t

velocity: rate of change of the position vector over the time elapsed ((∆t = t_f − t_i) so velocity is a vector that points in the direction the object is moving on average during the time interval ∆t. if initial and final times are almost the same instant, it is instantaneous velocity because velocity is essentially being measured at one point in time. in that case, the magnitude of velocity is the speed of the object [meters per second (m/s)]

⃗v (absolute value)

instantaneous speed: an object’s speed at a given time is defined as the magnitude of its instantaneous velocity vector at that time

⃗a ≡ ∆⃗v/∆t

acceleration: the rate of change of the velocity vector in time [meters per second squared (m/s²)]

a_c = v²/r

centripetal acceleration: when an object moves in a circle of radius r, one component of linear acceleration points towards the center. that acceleration keeps the object moving in a circle. in other words, centripetal acceleration keeps redirecting the velocity vector, or changing its direction so the object keeps moving in a circle. when centripetal acceleration is zero, an object’s path does not curve, but moves in a straight line.

F

force: a push of pull on an object. typically falls in two categories: field forces and contact forces. contact forces can only be exerted by something touching an object and cannot be exerted after the objects lose contact. for field forces for objects, there are only two types of field forces for objects larger than subatomic particles: force of gravity and electromagnetic forces. [Newton (N)]

V_net ≡ ΣV

principle of superposition: the net vector of a physical quantity at a given location (for a single object) is the vector sum of each contribution to the physical quantity at that location. in electrostatics, the net electrostatic force on an object in the presence of many other charged objects is the vector sum of the electrostatic forces between the object and each other charged object. true for velocity, acceleration, forces, and fields

F_net ≡ ΣF

net force: the sum of all forces acting on an object (using the principle of superposition). a free-body diagram is typically used to help visualize and do a correct calculation of the net force.

∆F_net = 0

equilibrium state: a state in which an object experiences zero net force (and zero net torque about all possible pivot points)

x_eq

equilibrium position: the location where an object or system is in a sate of equilibrium, meaning that if the system is placed in this position at rest, it will remain at rest

F_g = mg directed towards earth

force of gravity (due to earth): gravitational force is the force between two objects with mass. for objects on earth, we often neglect gravitational forces due to any mass except the earth since the mass the earth, and therefore the gravitational force due to the earth, is much larger than all other masses and the gravitational forces caused by other masses. the magnitude of the gravitational force on an object due to the earth is the mass (m) of the object multiplied by the acceleration due to gravity on earth (g). the term “weight” is often used to mean the total gravitational force on an object. assume that the object located near the surface of the earth

F_N

normal force: a contact force between two surfaces that are touching, it is directed perpendicular to the surface. assume two objects’ surfaces are in contact

F_net = ma

newton’s 2nd law: the net force on an object is equal to the object’s mass multiplied by the object’s acceleration because force and acceleration are vectors, the direction of the net force is the same direction as the object’s acceleration. assume analysis from inertial reference frame (reference frame that is not accelerating)

F_AonB = F_BonA

newton’s 3rd law: when two objects interact, they exert forces on one another that are equal in magnitude but opposite in direction. these two forces are often called “third-law pair” or “interaction pair” forces. third-law pair forces must be between the same two objects, and be the same type of force (e.g. gravitational, normal, etc.) a gravitational force and a normal force can never be third-law pair forces because they are not the same type of force.

system

a collection of objects that must be chosen before analyzing problems using energy or momentum

E_tot = KE + PE +…

total energy: the sum of all types of energy present in the system. often we ignore types of energy that do not change over the course of the problem (such as thermal energy). [Joule (J)]

∆E_tot = ∆KE + ∆PE +...

change in total energy: change comes from a change in one of the forms of energy in the system. in other words, if the total energy of a system changes, we can detect this energy by measuring all types of energy before and after the event. system’s energy (kinetic, potential, thermal, etc, or a combination) will be changed by the exact amount that the system’s energy changed

∆E_sys = W_ext + Q

law of conservation of energy: the total amount of energy in the universe never changes. for a system, the total amount of energy in the system can increase or decrease only if work is done on the system by external forces or if heat is added to or removed from the system

KE_t = ½ mv²

kinetic energy (translational): energy due to (the center of mass of an object) moving in a straight line. assume that the objects that have mass and have v < < c (speed of light)

PE

potential energy: energy that is stored in a system due to the relative position of objects in the system. it’s only possible to have potential energy if the energy it takes/work done to move the object does not depend on the path taken, but only depends on the initial and final location. assume that the force associated with the energy must be conservative (the proper name for a force for which the work done by the force depends only on initial and final locations), and the objects exerting and feeling the force must be part of the system

PE_g = mg(∆h)

gravitational potential energy: when both the earth and an object are part of a system, the system has energy due to its configuration (location of the two objects). to use gravitational potential energy, a coordinate system including the location where PE_g = 0 must be defined. this is because the value of gravitational potential energy for a particular configuration of a system doesn’t have physical meaning, only changes in potential energy between two configurations of the system have physical meaning. assume it must be near the surface of the earth so that F_g = mg

P ≡ ∆E/∆t

power: the energy used (or work done) per unit time [Watts (W)]

p ≡ mv

momentum: a quantity defined as an object’s mass multiplied by its velocity. since velocity is a vector, momentum is also a vector. therefore, momentum has both a magnitude and direction. assume non-relativistic speeds (v < < c) [(kgm/s)]

q, Q

electric charge: a fundamental physical quantity of which there are two types: positive and negative. objects with the same type of charge repel one another; objects with the opposite type of charge attract one another. hard to define, but can detect its presence and predict motion of charged objects [Coulomb (C)]

static electricity

a phenomena in which charge is transferred between objects through the process of rubbing against each other.

neutral

an object is neutral if it has no net electric charge. note that the object can still contain charge, but the number of positive and negative charges must be equal in the object overall

∆Q_universe = 0

law of conservation of charge: the net amount of electric charge produced in any process is zero. no net electric charge can be created or destroyed

proton

a particle with a positive charge (+e) found in the nucleus of atoms

neutron

a neutral particle found in the nucleus of atoms

electron

an elementary particle with a negative charge (-e) often found surrounding the nucleus in an atom

e

elementary charge: the amount of charge on one electron or proton is the elementary charge, e = 1.602 × 10^-19C. the property charge is quantized. this means e is also the smallest amount of charge known, and all charge measured is a multiple of e. assume value and quantization determined experimentally

ion

an atom which has lost or gained electron(s), giving it a net charge

point charge

an object whose spatial size is much smaller than other distances in a problem or question

polar

an object is polar if it is neutral overall, but does not have the charge uniformly distributed. this means they have one location that is more positively-charged and another location that is more negatively charged.

free electron

an electron in a material which is found very loosely and moves freely within the material (but cannot easily leave the material)

conductor

a material which has many available charge carries (generally free electrons or ions). when charges are at rest, there is no net force on charges inside in the conductor. if not the case, the free electrons would reposition themselves (electric field inside a conductor is zero) and any net charge on a conductor distributes itself on the surface. once in equilibrium and charges are at rest, there cannot be forces directed along the conductor’s surface. if there were, free electrons would move in response to these forces until there were no forces parallel to the surface (electric field is always perpendicular to the conductor’s surface)

insulator

a material which has very few free charge carriers (generally free electrons or ions)

semiconductor

a material which has some free electrons (fewer than a conductor but more than an insulator)

charging by conduction

the process by which an object acquires a charge by coming into contact with a charged object. this is accomplished by transferring electrons between objects

induced charge

charge separated onto one side of an object. this occurs when a second contact is placed near, but not touching, the original object, causing electrons in the original object to move.

ground

the earth, or another object which has an essentially unlimited ability to absorb or supply electrons. also, a common conductor in a circuit that provides continuity.

F_E,qQ = k(|q||Q|/r²)rˆ

coloumb’s law: the force, called “electrostatic force,” between two charged point particles at rest. it is a field force, meaning the charged objects can exert force on each other without touching. the strength of the force depends on the magnitude of the two particles q and Q, and the distance between the particles, r. note that it does not depend on the size (radius) of the charges, as Coulomb’s Law assumes the charges are point charges and therefore do not take up space. all forces have direction. the direction of the electrostatic force is always along the line joining the two charges (ˆr). since this can point in two possible directions, we choose which with the knowledge that opposite charges attract and like charges repel. coulomb’s law is an “empirical law” (determined experimentally). assume that point charges are at rest

ε₀ ≡ 8.85 × 10⁻¹² C²/Nm²

permittivity of free space: a universal constant used in many equations of electromagnetism, including coulomb’s law (when written with 1/4πε₀ instead of k)

k = 1/4πε₀ = 8.99 × 10⁹ Nm²/C²

coulomb constant: the constant of proportionality in coulomb’s law, which can also be written in terms of the permittivity of free space, ε₀

(vector) field

a field is a mathematical quantity which as a value at locations in space. this can be a scalar value like temperature (scalar field) or a vector value like electric field (vector field). because vectors have both magnitude and direction, a vector field is a collection of vectors—one for each location in space—which each have a direction and magnitude

test charge

a small charge, usually positive. this charge is so small that the force it exerts does not significantly affect the charges that create the field

E_P = F_E/q

electric field: is a vector at each point in space with a magnitude and direction. is “the force per unite charge” so if a charge were located at P, total electric force is its charge (including sign) x electric field vector. electric field at P tells magnitude of the force on a charge at P per coulomb of charge. P’s electric field is created by the charged particles surrounding P. any charge located at P do not contribute to P’s electric field, but is instead created by the environment (surrounding charge). P’s electric field points in the direction of the force of a positively-charged object located at P (always true regardless of what charge is actually located at P since electric field is solely dependent on surrounding charge) [Newtons per Coulomb (N/C) OR Volts per meter (V/M)]

gravitational field

the field created by an object with mass (i.e. the earth) which is equal in magnitude to the force on an object per unit mass. at the surface of the earth, the earth’s gravitational field has a magnitude of g and is directed towards the center of the earth.

electric field lines

a series of lines which indicate the direction and magnitude of the electric field at different points in space. the electric field at a location points in the direction tangent to the field line at that location (field lines never cross). number of lines starting on a positive test charge or ending on a negative charge is proportional to the magnitude of the charge. the closer together the lines are, the stronger the electric field in that region.

E = k(|Q|/r²)rˆ

electric field due to a point charge: the electric field at location P due to charge Q that is distance r from a point charge Q has the magnitude shown. the electric field direction points along the line between the charge Q and the location so if Q is positive, E points away from Q. if Q is negative, E points towards Q. because electric field at P points in the direction that a positive charge placed at P would feel a force. for a point charge at rest at any point in space, electric field is directed radially into or outwards from the charge and permeates all space. assume that point charge is at rest.

λ ≡ Q/l

σ ≡ Q/A

ρ ≡ Q/V

charge density: the charge per unit space. can be charge per length, charge per area, or charge per unit volume respectively. assume uniformly charged object.

|E| = σ/2ε₀

electric field due to one large, uniform plate of charge: electric field due to a single, large, uniform plate of chart. electric field is perpendicular to the plate, and points away from a positive plate or towards a negative plate. assume infinite plate, uniform distribution of charge, plate is at rest

|E| = σ/ε₀

electric field between parallel plates: electric field between two large, oppositely-charged parallel plates which is perpendicular to the plates and points from the positive plate toward the negative plate. assume location between two very large, charged parallel plates with constant charge density and that plates are of equal and opposite charge.

PE_E

electric potential energy: the potential energy of a charge at a given location due to a configuration of other charges. electrostatic force is a conservative force, meaning the work done by electrostatic force depends only on the initial and final configurations of the system and not the path taken to get from one configuration to another. therefore, we can define electrostatic potential energy for a system that includes all charges and their fields. values of potential energy only have physical meaning relative to another (true for all other forms of potential energy), meaning the only differences in potential energy are physically meaningful so location where PE_E = 0 can be chosen arbitrarily

V = PE_E/q

electric potential: the electric potential energy per unit charge, a property of the configuration of a system. sometimes expressed by saying the potential is a location’s property (in a system of charges). same with potential energy, only differences in electric potential are physically meaningful and measurable so location where V = 0 can be chosen arbitrarily. often location of V = 0 is chosen as either: (1) as infinitely far away from a charged object (r -> ∞) OR (2) the earth (ground). electric potential is a scalar, not a vector, so often easier to work with than electric field. if an object with a charge of 1 coulomb moves between two locations with a potential difference of 1 Volt, its potential energy changes by 1 Joule (unit Volt will always have the same symbol V as electric potential, but a volt is the unit of electric potential and electric potential is the physical quantity) [Volt (Joule/Coulomb)]

∆V or V

electric potential difference: the difference between electric potential V at two locations (∆V = V₂ - V₁). only differences in electric potential and electric potential energy are physically meaningful. note that ppl can use V to mean both electric potential AND electric potential difference

equipotential line or surface

a line or surface on which all points have the same electric potential

∆V = −∫ E · dl

potential difference & field relationship: related to the def’n of electric potential V, def’n of potential energy PE, and the relationship between electric force and electric field. physically, it means that the electric field points from a location of higher electric potential to a location of lower electric potential

V_P = k(Q/r)

electric potential of a point charge: electric potential due to a point charge Q at a location P that is a distance r from Q. can be derived given the electric field due to a point charge and the relationship between field and potential difference. assume (1) single point charge, (2) V = 0 at r = ∞

1 eV = 1.602 × 10^-19 J

electron volt: a very small unit of energy equal to the amount of energy gained (or lost) by an electron that moves across a 1V potential difference. electron volts are useful because a Joule is a unit of energy that is very large for many electrostatics problems

resistor

a circuit element that dissipates energy, usually in the form of light or heat. light bulbs can act as resistors in circuits

capacitor

a device that can keep charge separated. it normally consists of two conductors placed near each other but not touching. these two conductors carry equal and opposite charges because it requires energy to keep charge separated, capacitors store energy

∆V_b = V₊ − V_-

ideal battery: a device that acts as a constant source of electric potential difference

I ≡ ∆Q/∆t

current (conventional current): the net charge that passes a location in a given amount of time. the direction of “conventional current” is the direction positive charges in a circuit would flow. this is what most problem-solving strategies use, but physically most currents are actually due to the motion of electrons (electron current) in the opposite direction of the conventional current we imagine [ampere (amp) = 1 coulomb of charge passing a location each second (1 A ≡ 1(C/s)]

∆V₁ = ∆V₂

parallel (circuit connection): configuration of devices in circuits in which two branches (1 & 2) both have the same start and end points. the same wire connects the ‘top’ sides of two branches and a different wire connects the ‘bottom’ sides of the two branches. since the two branches directly connect the same two locations, there is the same potential difference across the two branches