Units

ä a theory is an explanation of natural phenomenon based on observations, rather than a hypothesis, which ispurely speculative

ä no theory is ever considered the final or ultimate truth, but will always at least have some degree of validity,i.e. Newton’s laws

Range of Validity:the range or set of phenomenon that a theory accurately explains

Target Variable: the quantity you’re trying to find in a problem

Model: a simplified version of a physical system that would be too complicated to analyze in full detail

Particle: an idealization of an object to have no shape or volume; an object that is contained at a single point inspace

Physical Quantity: any number that is used to describe a physical phenomenon quantitatively

Operational Definition: a definition for a physical quantity so fundamental that it can only be defined by describinghow to measure it

International System (SI): the Systeme International; the international standard set of physical units

Unit: a definite magnitude of a physical quantity

Prefix: multiplies a base unit by some power of 10 corresponding to the specific prefix

The Three Fundamental Units of Mechanics:

ä Time⇒ Second (s): the time required for 9192631770 cycles of the microwave radiation at the frequencyused to transition cesium atoms between its two lowest energy states

ä Length⇒ Meter (m): the distance that light travels in a vacuum in 1/299792458 seconds

ä Mass⇒ Kilogram (kg): the mass of the standard kilogram made of a platinum-iridium alloy at the Interna-tional Bureau of Weights and Measures

Dimensionally Consistent: terms having the same or equivalent dimensions such as liters and meters cubed

ä equations must ALWAYS be dimensionally consistent

ä terms can only be added or equated if they’re dimensionally consistent

Uncertainty (Error): the difference between the measured value and the actual true value

Fractional Error: the ratio of the error to the measurement

Percent Error: the fractional error times 100Significant Figures: how many meaningful digits are used in a measurement, calculation, or answer; you can’thave more significant figures than the accuracy and precision of your measurements allow

Accuracy: how close a measured value is likely to be to the true value

Precision: how detailed a measured value is, how many decimal places the measurement is to

Order-of-Magnitude Estimates: rough approximations of quantities based on significant idealizations and esti-mates of values to make quick, simple calculations to get a general idea of things

Vectors

1

Scalar Quantity: a physical quantity fully described by a single number

Vector Quantity: a physical quantity fully described by a magnitude and a direction in space

Magnitude: how much or how large a quantity is

Displacement: a change in position of a point

Negative of a Vector: a vector having the same magnitude as the original vector but the opposite direction

Vectors Equal: vectors with the same direction and magnitude

Parallel Vectors: vectors with the same direction, regardless of their magnitudes

Antiparallel Vectors: two vectors with opposite directions regardless of their magnitudes

Resultant (Vector Sum): the vector created by adding two vectors with their tails at the same point

Component Vectors: the vectors added together to get a resultant vector; parts of a larger vector

Components: the numbers Ax and Ay of the vector ~A; components aren’t intrinsically vectors themselves

Unit Vector: a vector with a magnitude of 1 and no units; used to formally make component vectors

ä Graphically Add Vectors By:

• place the vectors tip to tail in any order and then connect the first tail to the last head

• for two vectors, construct a parallelogram and connect the diagonal with the tails to the constructedcorner

ä add and subtract vectors by adding and subtracting corresponding individual components, usually 〈x, y, z〉components

ä multiple vectors by scalars by multiplying every component by the scalar

ä ~A =⟨

Ax i, Ayj, Azk⟩

(component form of a vector)

ä A =∣∣∣~A∣∣∣=√A2

x +A2y +A2

z (magnitude of a vector)

ä ~R = ~A+~B =⟨Ax +Bx, Ay +By

⟩(vector addition)

ä ~A−~B = ~A+(−~B)

(vector subtraction)

ä ~A+~B = ~B+~A (commutative law)

ä ~R =(~A+~B

)+~C = ~A+

(~B+~C

)(distributive law)

Converting Between Magnitude and Direction Form and Component Form:

ä same as converting between rectangular and polar coordinates

ä requires that the angle θ is measured from the positive x-axis in the standard counterclockwise direction

ä Ax = Acosθ Ay = Asinθ (convert to component form)

ä∣∣∣~A∣∣∣=√A2

x +A2y θ = tan−1

(Ay

Ax

)(convert to magnitude and direction form)

Scalar Product (Dot Product):

ä yields a scalar quantity

2

ä a measure of how close two vectors are or how small the angle between them is

ä the dot product of perpendicular vectors is zero; if θ = 90 then ~A ·~B = 0

ä useful for finding work

ä ~A ·~B =∣∣∣~A∣∣∣ ∣∣∣~B∣∣∣cosθ = AxBx +AyBy +AzBz (dot product)

ä ~A ·~B = ~B ·~A (commutative law)

Vector Product (Cross Product):

ä yields a vector quantity

ä a measure of how far apart two vectors are or how large the angle between them is

ä the cross product of parallel, antiparallel, or identical vectors is zero; if θ = 0, 180 then ~A×~B = 0

ä useful for finding torque

ä ~A×~B =∣∣∣~A∣∣∣ ∣∣∣~B∣∣∣sinθ =

∣∣∣∣∣∣Ax Ay AzBx By Bz

i j k

∣∣∣∣∣∣ (cross product)

ä ~A×~B =−~B×~A (not commutative)

Right-Hand Rule: the direction of the cross product can be determined by taking your right hand, putting yourfingers along the direction of the first vector, and then curling them in towards the direction of the second vector.Your thumb will point in the direction of the resultant

One Dimensional Motion

ä speed is the absolute value of velocity

vav−x =x2− x1

t2− t1=

∆x∆t

vx = lim∆t→0

∆x∆t

=dxdt

aav−x =v2x− v1x

t2− t1=

∆vx

∆t

ax = lim∆t→0

∆vx

∆t=

dvx

dt

ä the acceleration on earth due to gravity is g

ä the value of g varies slightly from place to place on the earth’s surface

ä g = 9.80665m/s2

Constant Linear Acceleration:

ä first two equations are intuitive

ä third equation is derived from solving the first equation for t, substituting into the second equation, andrearranging

ä forth equation is derived by setting two equations for vav−x equal to each other and rearranging

3

ä vx = v0x +at (standard for velocity)

ä x = x0 + v0xt +12

axt2 (standard for postition)

ä v2x = v2

0x +2ax (x− x0) (time is unknown)

ä x− x0 =

(v0x + vx

2

)t (acceleration is unknown)

ä vav−x =x− x0

t=

v0x− vx

2= v0x +

12

axt (constant acceleration; lesser importance)

Varying Acceleration:

ä vx = v0x +

ˆ t

0ax dt

ä x = x0 +

ˆ t

0vx dt

Two Dimensional Motion

ä acceleration parallel to a particle’s path is parallel to the velocity and tells the change in speed

ä acceleration perpendicular to to a particle’s path tells the change in direction

ä the key to analyzing motion in multiple dimensions is to analyze the components separately

ä ~r = xi+ yj+ zk

~vav =r2−~r1t2− t1

=∆~r∆t

~v = lim∆t→0

∆~r∆t

=d~rdt

vx =dxdt

vy =dydt

vz =dzdt

~aav =v2−~v1t2− t1

=∆~v∆t

~a = lim∆t→0

∆~v∆t

=d~vdt

ax =dvx

dtay =

dvy

dtaz =

dvz

dt

Projectile Motion:

ä assumes no forces other than gravity (such as air resistance)

ä always a parabolic trajectory

ä maximum height occurs when vy = 0

ä total distance needs time at y = 0 to plug t into x position equation

ä if starting and ending heights are different, change y accordingly and beware multiple solutions

4

x = (v0 cosα0) tvx = v0 cosα0

ax = 0

y = (v0 sinα0) t− 12

gt2

vy = v0 cosα0−gtay =−g

ä v =√

v2x + v2

y (magnitude of velocity)

ä tanα =vy

vx(trajectory angle measured counterclockwise from standard position)

ä R =v2

0g

sin2θ (range of projectile; flat ground)

Newtonian Relativity:

Relative Velocity: the velocity seen by a particular observer

Frame of Reference: a coordinate system and a time scale

ä vP/A−x = vP/B−x + vB/A−x (relative velocity along a line)

ä ~rP/A =~rP/B +~rB/A

ä ~vP/A =~vP/B +~vB/A (Galilean velocity transformation; relative velocity in space)

ä ~vA/B =~vB/A

Forces

Dynamics: the relationship of motion to the forces that cause it

Kinematics: the language for describing motion

Force: an interaction between two objects or between an object and its environment; a push or a pull; force is avector quantity with magnitude and direction

Contact Force: a force involving direct contact between two bodies

Normal Force: a force perpendicular to a surface

Friction Force: a force exerted on an object moving parallel to a surface it is in contact with

Tension Force: a force pulling on an object

Long-Range Force: a force that acts between bodies separated by empty space

Weight: the gravitational force that the earth exerts on objects; ~w = m~g (weight of a body of mass m)

Apparent Weight: the weight an object feels from accelerations in the y direction

Apparent Weightlessness: the feeling of weightlessness an object experiences when its acceleration in the ydirection equals the acceleration from the earth’s gravity g

ä n = m(g+ay) (apparent weight of an object)

ä the SI unit of mass is the kilogram

ä the SI unit of the magnitude of force is the Newton; 1N = 1kg ·m/s2

Superposition of Forces: any number of forces applied at a point on a body have the same effect as a singleforce equal to the vector sum of the forces

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ä any force can be replaced by its component vectors, acting at the same point

Net Force: the vector sum (resultant) of all the forces acting on a body

ä R =√

R2x +R2

y +R2z (net force in three dimensions is Pythagorian of component nets)

Newton’s First Law of Motion

Newton’s First Law of Motion: a body acted on by no net force moves with constant velocity (which may also bezero) and zero acceleration

ä zero net force is equivalent to no force at all

Inertia: the tendency of a body to keep moving once it is set into motion

Equilibrium: when a body is at rest or moving with constant velocity in an inertial frame of reference

ä ∑~F = 0 (body in equilibrium)

ä ∑~Fx = 0 and ∑~Fy = 0 (body in equilibrium; implied by ∑~F = 0)

Inertial Frame of Reference: a frame of reference in which Newton’s first law is valid; the earth’s surface is anapproximation of an inertial frame

ä if one frame of reference is inertial, then every other frame moving relative to it with constant velocity is alsoinertial

ä for non-zero velocity equilibriums, direction doesn’t matter, just that the velocity is constant

Newton’s Second Law of Motion

Newton’s Second Law of Motion: if a net external force acts on a body, the body accelerates. The direction ofthe acceleration is the same as the direction of the net force. The mass of the body times the acceleration of thebody equals the net force vector.

ä acceleration m~a is the result of a nonzero net force, but is not a force itself

ä accelerations, including m~a, do not belong in free body diagrams

Mass (Inertial Mass): a quantitative measure of inertia; m =

∣∣∣∑~F∣∣∣

~a

ä ∑~F = m~a (Newton’s second law of motion)

ä ∑~Fx = m~ax and ∑~Fy = m~ay and ∑~Fz = m~az (implied by ∑~F=m~a)

ä only external forces are included in ∑ F

ä valid only when mass is constant

ä valid only in inertial frames of reference

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Newton’s Third Law of Motion

Newton’s Third Law of Motion: If a body A exerts a force on body B (an “action”), then body B exerts a force onbody A (a “reaction”). These two forces have the same magnitude but are opposite in direction. These two forcesact on different bodies.

ä ~FA on B =−~FB on A (Newton’s third law of motion)

ä two forces in a an action-reaction pair never act on the same body

Tension: the magnitude of the force acting at a point; not the same as the tension force

ä if the rope is in equilibrium and if no forces act except at its ends, the tension is the same at both ends andthroughout the rope

ä if there’s X amount of force at each end of a rope, the tension on the rope is X , not 2X

ä if a rope has mass, include a downward force from its weight and if the rope isn’t horizontal, the tension won’tbe the same at both ends

ä in a system with a vertical or horizontal and oblique ropes, the oblique rope takes the entire load in onedirection and the other rope counterbalances the component of the oblique rope that is in the same directionas it is

Free-Body Diagrams

Free-Body Diagrams: a diagram showing the chosen body by itself, “free” of its surroundings, with vectors drawnto show the magnitudes and directions of all the forces applied to the body by the various other bodies that interactwith itself

ä must include all the forces acting on the body, but equally careful not to include any forces that the bodyexerts on any other body

ä forces that a body exerts on itself are never included since these can’t affect the body’s motion

ä multiple FBDs may be needed to solve a single problem

ä do not include accelerations, just forces

Friction

Friction Force(~f)

: the component of a vector parallel to the surface of contact and antiparallel to the direction ofmotion

ä the friction and normal forces are really components of a single contact force

ä friction and normal forces are always perpendicular

7

Kinetic Friction Force(~f)

: the kind of friction force that acts when a body slides over a surface

Static Friction Force(~f)

: the kind of friction force that acts when there is no relative motion

Coefficient of Kinetic Friction (µk): a unitless scalar constant that relates the normal force to the kinetic frictionforce; higher equals more friction

Coefficient of Static Friction (µs): a unitless scalar constant that relates the normal force to the maximum staticfriction force; higher equals more friction

Coefficient of Rolling Friction (Tractive Resistance) (µr): a unitless scalar constant that relates the normalforce to the rolling friction force; higher equals more friction

ä fk = µkn (magnitude of kinetic friction force)

ä fs ≤ µsn (magnitude of static friction force)

Fluid Resistance: the force that a fluid (a gas or liquid) exerts on a body moving through it

ä the direction of the fluid resistance force acting on a body is always opposite the direction of the body’svelocity relative to the fluid

ä an object falling in a fluid does not have a constant acceleration

Air Drag (Drag): the fluid resistance in air at moderate to high speeds

ä air drag is for speeds equal to or greater than that of a tossed tennis ball

ä y = vt

[t− m

k

(1− e−(

km)t)]

( f = kv low speed free fall in a fluid)

ä vy = vt

[1− e−(

km)t]

( f = kv low speed free fall in a fluid)

ä ay = ge−(km)t ( f = kv low speed free fall in a fluid)

Terminal Speed (vt): the maximum speed an object falling in a fluid reaches due to the fluid resistance forceequaling the force of gravity

ä vt =mgk

(terminal speed, fluid resistance f = kv)

ä vt =

√mgD

(terminal speed, fluid resistance f = Dv2)

Rolling Friction: the kind of friction that acts when a body rolls over a surface

Occurs If:

ä the rolling body is deformable

ä the rolling surface is deformable

ä the contact forces on the object act over an area instead of at a single point

ä the normal force exerts a torque that opposes the rotation

ä sliding occurs

8

Circular Motion

Period (T ): the time it takes for one complete revolution

Centripetal Acceleration: acceleration always directed towards the center of the circle

Centripetal Force: the force normal to the velocity, directed towards the center of the circle that pulls the objecttowards the center and makes circular motion possible

ä when an object suddenly stops circular motion, it travels in a straight, linear path

ä only the net force matters when determining the motion

ä a full circular arc is not needed

ä centripetal force acts on the object in circular motion; the object is not applying centripetal force on anything(or else it wouldn’t be in the free-body diagram)

Centrifugal Force: the fictitious force that pushes an object to the outside of its arc in circular motion

Centrifugal Force Does Not Exist in an Inertial Frame of Reference:

ä velocity is constantly changing in direction, so the body is accelerating and not in equilibrium

ä the net force would be zero and the body would move in a straight line instead of a circle

ä the value m v2

R is not a force and corresponds to m~a

Uniform Circular Motion: speed is constant, no net force parallel to velocity

ä in uniform circular motion, the acceleration and net force are always directed towards the center of the circle

ä arad =v2

R=

4π2RT 2 (uniform circular motion)

ä v =2πRT

(uniform circular motion)

ä Fnet = marad = mv2

R(uniform circular motion)

ä tanβ =arad

g=

v2

gR(horizontal turn banked at angle β; no friction)

ä through tension, centripetal force can create an upward force like a helicopter because the tension force isacting on the body pulling it in, the body isn’t applying the centripetal force

Nonuniform Circular Motion: speed changes; non-zero tangential component of acceleration

ä except at the top and the bottom of the circle, the net acceleration and net force are not directed towardsthe center of the circle

ä arad =v2

R(nonuniform circular motion)

ä atan =d |~v|dt

(nonuniform circular motion)

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Work

Work (W ): the amount of energy transferred by a force acting through a distance

ä the SI unit of work is the Joule

ä 1 J = 1 N ·m (definition of Joule)

ä W =~F ·~s = Fscosφ (constant force, straight-line displacement)

ä what matters is not the total net force, but the net force in the direction of the displacement

ä if there’s force applied to an object but it doesn’t move, no work is done on it

ä net force on an object in the direction opposite its motion is negative work

ä when a particle undergoes a displacement, it speeds up if Wtot > 0, slows down if Wtot < 0, and maintainsthe same speed if Wtot = 0

ä W =

ˆ x2

x1Fx dx (varying x-component of force; straight-line displacement)

ä W =

ˆ P2

P1

F cosφ dl =ˆ P2

P1

F‖ dl =ˆ P2

P1

~F ·d~l (work done on a curved path)

ä Wtot = K2−K1 = ∆K (work-energy theorem)

ä the work energy theorem is only valid in an inertial frame of reference

Force Constant (Spring Constant): a constant relating the force required to stretch a spring a given distance; SIunits are N/m

ä Fx = kx (Hooke’s law; force required to stretch a spring)

Power: the rate at which work is done

ä the SI unit of power is the watt

ä Pav =∆W∆t

(average power)

ä P = lim∆t→0

∆W∆t

=dWdt

=~F ·~v (instantaneous power)

Energy

ä the joule is the SI unit used for all quantities of work and energy

Kinetic Energy:

ä K =12

mv2 (linear kinetic energy)

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ä the kinetic energy of a particle is equal to the total work that was done to accelerate it from rest to its presentspeed

ä the kinetic energy of a particle is equal to the total work that particle can do in the process of being broughtto rest

ä the total kinetic energy of a system can change even if no work is done on any part of the system by anoutside force

Potential Energy (U): the energy stored in a body or a system due to its position in a force field or due to itsconfiguration

ä the work done by all forces other than gravitational force or elastic force equals the change in the totalmechanical energy E = K +U of the system

ä U =Ugrav +Uel (total potential energy)

ä K1 +U1 +Wother = K2 +U2 (valid in general)

ä Fx =−∂U∂x

Fy =−∂U∂y

Fz =−∂U∂z

(force from potential energy)

ä ~F =−~∇U =−(

∂U∂x

i+∂U∂y

j+∂U∂z

k)

(force from potential energy)

Gravitational Potential Energy (Ugrav): potential energy from position in a gravitational field, usually with theearth

ä Ugrav = mgy (gravitational potential energy)

ä we can pick anywhere we want to be y = 0 for gravitational potential energy, but the value of interest is thedifference in gravitational potential energy between two points; Ugrav, 2−Ugrav, 1 = mg(y2− y1)

ä gravitational potential energy can be less than zero at a point if it is below the origin because only thedifference between two points matters

ä the gravitational potential energy hold regardless of if the path taken is straight or curved

ä Wgrav =Ugrav, 1−Ugrav, 2 =−(Ugrav, 2−Ugrav, 1) =−∆Ugrav (work from gravitational potential energy)

ä ∆K =−∆Ugrav (kinetic energy versus gravitational potential energy)

ä E = K1 +Ugrav, 1 = K2 +Ugrav, 2 (if only gravity does work)

Elastic Body: a body that returns to its original shape and size after being deformed

Elastic Potential Energy (Uel): potential energy from the elastic deformation of an elastic body

ä unlike gravitational potential energy, the origin for elastic potential energy must be where the spring is neitherstretched nor compressed

ä an ideal spring is massless

ä Uel =12

kx2 (elastic potential energy)

ä Wel =12

kx21 −

12

kx22 =Uel, 1−Uel, 2 =−∆Uel (work from elastic potential energy)

11

ä K1 +Uel, 1 = K2 +Uel, 2 (if only elastic energy does work)

Mechanical Energy (E): the sum of kinetic and potential energy

ä the work done by all forces other than the gravitational force equals the change in the total mechanical energyof the system

ä friction may reduce kinetic energy and thus mechanical energy

Internal Energy: the energy associated with this change in the state of the materials

ä the increase in the internal energy is exactly equal to the absolute value of the work done by friction

ä ∆Uint =−Wother (internal energy versus work from nonconservative forces)

Conservation of Energy

Conserved Quantity: a quantity that always has the same value

Law of Conservation of Energy: energy is never created or destroyed, it only changes form

ä ∆K +∆U +∆Uint = 0 (law of conservation of energy)

Conservation of Mechanical Energy: when gravity is the only force that does work, mechanical energy is con-served

Conservative Force: a force that allows the two-way lossless conversion between kinetic and potential energies

ä always acts to push the system towards lower potential energy

ä includes gravitational, spring, and electric forces

Work Done By a Conservative Force:

ä it can be expressed as the difference between the initial and final values of a potential-energy function

ä it is reversible

ä it is independent of the path of the body and only depends on the starting and ending points

ä when the starting and ending points are the same, the total work is the same

ä when the only forces that do work are conservative forces, the total mechanical energy E =K+U is constant

Nonconservative Force: a force that is not conservative; a force that is not reversible

ä nonconservative forces can increase mechanical energy at times such as a chemical explosion

ä includes friction and fluid resistance

Dissipative Force: a force that causes mechanical energy to be lost or dissipated

Energy Diagram: a graph of potential energy versus position

Equilibrium Position: a point on an energy diagram where the slope and force are zero

Restoring Force: a positive force towards the origin when position and slope are negative

Stable Equilibrium: any minimum in a potential-energy curve

Unstable Equilibrium: any maximum in a potential-energy curve

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Momentum

Momentum (~p): the product of mass times velocity

ä the SI unit of momentum is kg ·m/s

ä the momentum of a particle is equal to its impulse that accelerated it from rest to its current speed

ä the total net momentum and the net momentum in a given direction can be different

ä momentum is about the time a force acts, while kinetic energy is about the distance a force acts

ä ~p = m~v (definition of momentum)

ä px = mvx py = mvy pz = mvz (components of momentum)

ä ∑~F =d~pdt

(Newton’s Second Law in terms of Momentum)

ä only valid in inertial frames of reference

Impulse: the net force times the time interval; the change in momentum per unit time

ä the SI unit of impulse is N · s = kg ·m/s

ä the total net impulse and the net impulse in a given direction can be different

ä impulse from rest equals final momentum, so it offers another way to find momentum

ä ~J = ~p2−~p1 (impulse-momentum thereom)

ä ~J = ∑~F(t2− t1) = ∑~F∆t (constant net force)

ä ~J =~Fnet (t2− t1) (even with non-constant force)

ä ~J =

ˆ t2

t1∑~F dt (general definition of impulse)

ä Jx =

ˆ t2

t1∑Fx dt = (Fav)x (t2− t1) = p2x− p1x = mv2x−mv1x (impulse components)

ä Jy =

ˆ t2

t1∑Fy dt = (Fav)y (t2− t1) = p2y− p1y = mv2y−mv1y (impulse components)

ä Jz =

ˆ t2

t1∑Fz dt = (Fav)z (t2− t1) = p2z− p1z = mv2z−mv1z (impulse components)

Internal Force: the forces that the particles of the system exert on each other

External Force: forces exerted on any part of the system by some object outside it

Isolated System: when there are no external forces

Law of Conservation of Momentum: if the vector sum of the external forces on a system is zero, the totalmomentum of the system is conserved

ä conservation of momentum means conservation of its components

13

ä conservation of momentum is valid even when the internal forces are not conservative, an advantage overthe conservation of mechanical energy

ä even though kinetic energy may not be conserved, momentum still can be in inelastic collisions

ä ~P =~pA +~pB + · · ·= mA~vA +mB~vB + · · · (total momentum of a system of particles)

Collisions

Elastic Collision: a collision in which BOTH momentum AND kinetic energy are conserved

ä if the forces between the bodies in an isolated system are conservative, the total kinetic energy of the systemis the same after the collision as before

ä in an elastic collision, the relative velocity of the two bodies has the same magnitude before and after thecollision

ä in two dimensional collisions, you cannot use these equations with x subscripts and must instead use energyand standard and component based conservation of momentum

ä12

mAv2A1x +

12

mBv2B1x =

12

mAv2A2x +

12

mBv2B2x (kinetic energy; elastic collision)

ä mAvA1x +mBvB1x = mAvA2x +mBvB2x (elastic collision)

ä vA2x = 0 and vB2x = vA1x (elastic collision; mA=mB; vB1x = 0)

ä in an elastic collision, if mA = mB and one body is at rest; the moving body will come to a complete stop andthe body at rest will acquire the kinetic energy and momentum of the moving body

ä vB2x− vA2x =−(vB1x− vA1x) (relative velocties; straight-line elastic collision)

Inelastic Collision: a collision in which momentum is conserved, but kinetic energy is NOT

ä mAvA1x +mBvB1x = mAvA2x +mBvB2x (inelastic collision)

Completely Inelastic Collision: an inelastic collision in which the colliding bodies stick together and move asone body after the collision

ä in any collision in which external forces can be neglected, momentum is conserved and the total momentumbefore equals the total momentum after; in elastic collisions only, the total kinetic energy before equals thetotal kinetic energy after

ä energy equations can be useful in finding values for momentum equations when solving problems

ä mA~vA1 +mB~vB1 = (mA +mB)~v2 (completely inelastic collision)

Rocket Propulsion: system mass is constant, but rocket mass decreases over time from ejection of the propellant

ä assuming no gravitational force and no air resistance

ä isolated system; momentum is conserved

ä m dv =−dm vex−dm dv (momentum of rocket plus fuel; rocket propulsion)

14

ä F = mdvdt

=−vexdmdt

(force from propellant; rocket propulsion)

ä v f − vi = vex ln(

mi

m f

)(change in speed; rocket propulsion)

Effective Rockets Have:

ä high propellant velocity relative to the rocket

ä ratio of fuel mass to rocket mass is as large as possiblemi

m f

ä minimal external resistance from air, etc

Rotational Motion

Fixed Axis: an axis that is at rest in some inertial frame of reference and does not change direction relative to thatframe

Rigid Body: an idealized body that has a perfectly definite and unchanging shape and size

Radian (rad): the angle subtended at the center of a circle by an arc with a length equal to the radius of the circle

ä use radians in all rotation equations

ä no subscripts means magnitude like speed, not a vectorized quantity

ä θ =sr

or s = rθ (angle θ in radians on circle with radius r and arc length s)

ä 1 rad =360

2π≈ 57.2958

ä 1 rev = 2π rad

ä 1 rev/s = 2π rad/s≈ 10 rpm

ä 1 rev/min = 1 rpm =2π

60rad/s

ä ωav−z =θ2−θ1

t2− t1=

∆θ

∆t(average angular velocity)

ä ωz = lim∆t→0

∆θ

∆t=

dθ

dt(definition of instantaneous angular velocity)

ä αav−z =ω2z−ω1z

t2− t1=

∆ωz

∆t(average angular acceleration)

ä αz = lim∆t→0

∆ωz

∆t=

dωz

dt(definition of instantaneous angular acceleration)

Constant Angular Acceleration:

ä first two equations are intuitive

ä third equation is derived from solving the first equation for t, substituting into the second equation, andrearranging

15

ä forth equation is derived by setting two equations for ωav−θ equal to each other and rearranging

ä ωz = ω0z +αzt (standard for angular velocity)

ä θ = θ0 +ω0zt +12

αzt2 (standard for angular postition)

ä ω2z = ω

20z +2αz (θ−θ0) (time is unknown)

ä θ−θ0 =12(ω0z−ωz) t (angular acceleration is unknown)

ä ωav−z =θ−θ0

t=

ω0z−ωz

2= ω0z +

12

αzt (constant angular acceleration; lesser importance)

Angular-Linear Conversions:

ä v = rω (relationship between linear and angular speeds)

ä atan =dvdt

= rdω

dt= rα (tangental acceleration of a point on a rotating body)

ä arad =v2

r= ω

2r (centripetal acceleration of a point on a rotating body; true even for nonconstant ω and v)

Torque

Line of Action: the line along which the force vector lies

Lever Arm (Moment Arm) (l): the distance from the point of rotation to the force vector, measured perpendicularto the force vector extended as an infinite line

Torque (τ): the tendency of a force to cause or change a body’s rotational motion

ä the SI unit of torque is the newton-meter N ·m

ä counterclockwise rotation is usually considered positive

ä clockwise rotation is usually considered negative

ä torque is always defined with reference to a specific point. If we shift the position of this point, the torque ofeach force may also change

ä τ = Fl = rF sinφ = Ftanr (magnitude of torque)

ä ~τ =~r×~F (definition of torque)

ä ∑τz = Iαz (rotational analog of Newton’s second law for a rigid body)

ä The rotational Newton’s second law is valid even when the axis of rotation moves provided:

• the axis of rotation through the center of mass must be an axis of symmetry

• the axis must not change direction

ä vcm = Rω (condition for rolling without slipping)

ä assumes a symmetrical wheel with the center of mass at the geometric center

16

ä assumes an inertial frame of reference in which the surface on which the wheel rolls is at rest

Angular Work and Power

ä W = τz (θ2−θ1) = τz∆θ (work done by a constant torque)

ä W =

ˆθ2

θ1

τz dθ (work done by a torque)

ä correct for any force, regardless of its components

ä an axial component (parallel to the rotation axis) or a radial component (directed toward or away from theaxis) would do no work because the displacement of the point of application has only a tangential component

ä Wtot =

ˆω2

ω1

Iωz dωz =12

ιω22 −

12

Iω21 (angular work-kinetic energy theorem)

ä P = τzωz (power of a torque)

Angular Momentum

ä the SI units of angular momentum are kg ·m2/s

ä depends on choice of origin

ä ~L =~r×~p =~r×m~v (angular momentum of a particle)

ä in an inertial frame of reference

äd~Ldt

=~r×~F =~r (for a particle acted on by net force~F)

ä ~L = I~ω (for a rigid body rotating around a symmetry axis)

ä when the axis of rotation is not an axis of symmetry, ~L changes and thus there is a net external torque onthe body, even if ω is constant

ä if the body is not rigid, I may change and thus L will change, even if ω is constant

ä ∑~τ =d~Ldt

(for any system of particles)

Law of Conservation of Angular Momentum: when the net external torque acting on a system is zero, the totalangular momentum of the system is conserved

ä I1ω1z = I2ω2z (zero net external torque)

äd~Ldt

= 0 (zero net external torque)

Precession

17

Precession: a change in the orientation of the rotation axis of a rotating body; the rotational analog of uniformcircular motion

Spinning: rotation of the object around itself

Pitching: rotation around the gimbal axis

Rotating: rotation around the vertical pivot axis

Nutation: the up and down wobble of an object as it precesses around a pivot

Precession Angular Speed (Ω): the speed an object rotates around a pivot point due to rotating in the planeperpendicular to the pivot point

ä Ω =dφ

dt=

∣∣∣d~L∣∣∣/ ∣∣∣~L∣∣∣dt

=τz

Lz=

wrIω

(precession angular speed)

ä the upward normal force~n exerted by the pivot must be just equal in magnitude to the object weight

ä F = MΩ2r (magnitude of the centripetal force from the pivot tension in precessional motion)

Center of Mass

Center of Mass (CM): the mass-weighted average position of a system of particles

ä ~rcm =

∑imi~ri

∑imi

(center of mass)

ä xcm =

∑imixi

∑imi

(xcoordinate of the center of mass)

ä ycm =

∑imiyi

∑imi

(ycoordinate of the center of mass)

ä zcm =

∑imizi

∑imi

(zcoordinate of the center of mass)

ä whenever a homogeneous body has a geometric center, the center of mass is at the geometric center

ä whenever a body has an axis of symmetry, the center of mass always lies on that axis

ä the center of mass does NOT have to be within the body

ä ~P = M~vcm = m1~v1 +m2~v2 +m3~v3 + · · · (system of particles)

ä the total momentum is equal to the total mass times the velocity of the center of mass

ä ∑~Fext = M~acm (body or collection of particles)

18

ä ∑~Fext =d~Pdt

(extended body or system of particles)

ä when a body or a collection of particles is acted on by external forces, the center of mass moves just asthough all the mass were concentrated at that point and it were acted on by a net force equal to the sum ofthe external forces on the system

ä every possible motion of a rigid body can be represented as a combination of translational motion of thecenter of mass and rotation about an axis through the center of mass. This is true even when the center ofmass accelerates, so that it is not at rest in any inertial frame

ä U = Mgycm (gravitational potential energy of an extended body)

ä this applies to any extended body, even if it is not rigid

Moment of Inertia

Axis of Rotation (ri): an axis perpendicular to the plane of motion and and always a constant distance from everyparticle in a system during a rotation if there is no linear, translational motion

Moment of Inertia (I): the tendency of a body to maintain its angular velocity; even if it is zero

ä the SI unit of moment of inertia is the kg ·m2

ä the moment of inertia depends on the choice of axis

ä you can NOT assume that all of the mass is concentrated at the center of mass and multiply by the squareof the distance from the center of mass to the axis; this is WRONG

ä the greater the moment of inertia, the greater the kinetic energy of a rigid body rotating with a given angularspeed ω

ä I = ∑i

mir2i (definition of moment of inertia)

ä Ir = Icm +Md2 (parallel axis theorem)

ä Iz = Ix + Iy (perpendicular axis theorem)

ä K =12

Iω2 (rotational kinetic energy of a rigid body)

ä K =12

mcmv2 +12

Icmω2 (rigid body with both translation and rotation)

Equilibrium

Statistically Indeterminate: a problem that is impossible to solve with equilibrium conditions alone

ä a particle is in equilibrium whenever the vector sum of the forces acting on it is zero

Static Equilibrium: when an object has no translational or rotational motion

First Condition of Equilibrium: in an inertial frame of reference, the vector sum of all external forces acting onthe body is zero

19

ä ∑~F = 0 (first condition for equilibrium)

ä ∑Fx = 0 ∑Fy = 0 ∑Fz (first condition for equilibrium)

Second Condition of Equilibrium: in an inertial frame of reference, the sum of the torques due to all externalforces acting on the body, with respect to any specified point, must be zero

ä the choice of reference point for calculating torques in ∑τz is completely arbitrary, but the same point mustbe used to calculate all of the torques on the body

ä if a force has a line of action that goes through a particular point, the torque of the force with respect to thatpoint is zero, allowing unknown forces or components to be eliminated from the torque equation

ä the body doesn’t actually have to be pivoted about an axis through the chosen point

ä ∑~τ = 0 (second condition for equilibrium; about any point)

ä the conditions for equilibrium also apply for non-static equilibrium where there is translational motion, but notrotational motion

Center of Gravity

Center of Gravity (CG): the gravitational force weighted average position of a system of particles; the point throughwhich the gravity on an object acts

ä in a uniform gravitational field, the center of gravity is the same location as the center of mass

ä ~τ =~rcm×M~g =~rcm×~w (the total gravtitational torque)

ä if~g has the same value at all points on a body, its center of gravity is identical to its center of mass

ä when a body acted on by gravity is supported or suspended at a single point, the center of gravity is alwaysat, directly above, or below the point of suspension

ä a body supported at several points must have its center of gravity somewhere within the area bounded bythe supports or else it will tip over

ä symmetries are very useful in finding the center of gravity

Locate the Center of Gravity of Any Object:

ä suspend the object from a single point and make a vertical line from this point

ä suspend the object again from a different point and make another vertical line from this point

ä the center of gravity is at the intersection of the two points

Stress and Strain

Stress: the quantity of the strength of the forces causing a deformation on a force per unit area basis

ä the SI unit of stress is the pascal

20

ä 1Pa = 1N/m2 = 1.450×10−4 psi = 9.8692×10−6 atm

Strain: the deformation resulting from stress; the ratio of the change in size to the original size

Elastic Modulus: the proportionality constant between stress and strain when they’re directly proportional

äStressStrain

= Elastic Modulus (Hooke’s Law)

Hooke’s Law: the proportionality of stress and strain under certain circumstances

Tension: when an object has pulling forces on its ends perpendicular to the surface

Tensile Stress =F⊥A

Tensile Strain =l− l0

l0=

∆ll0

Y =Tensile StressTensile Strain

=F⊥/A∆l/l0

=F⊥A

l0∆l

(Young’s modulus)

Compression: when an object has pushing forces on its ends perpendicular to the surface

Compressive Stress =F⊥A

Compressive Strain =∆ll0

Pressure (p): the force F⊥ per unit area that the fluid exerts on the surface of an immersed object; the force isuniform on all sides ignoring gravity

Bulk Stress = ∆p Bulk (Volume) Strain =∆VV0

B =Bulk StressBulk Strain

=− ∆p∆V/V0

(bulk modulus)

Compressibility (k): the reciprocal of the bulk modulus; the fractional decrease in volume −∆V/V0 per unitpressure

ä the SI units of compressibility are Pa−1

ä k =1B=−∆V/V0

∆P=− 1

V0

∆V∆p

(compressibility)

Shear: when an object has forces in opposite directions tangent to opposite surfaces of it

Shear Stress =F‖A

Shear Strain =xh

S =Shear StressShear Strain

=F‖/Ax/h

=F‖A

hx

(shear modulus)

ä for a given material, S is usually 13 to 1

2 as large as Young’s modulus Y for tensile stress

ä shear only applies to solid materials

Material Deformation

21

Proportional Limit: the point at which a material no longer follows Hooke’s Law

Elastic Behavior: when a material will return to it’s original shape and size after a load is removed and the stress-strain curve is retraced as energy is conserved

Yield Point: the point on a stress-strain curve where the deformation changes from elastic to plastic

Elastic Limit: the stress at the yield point

Permanent Set: when a material has undergone permanent deformation

Fracture: the local separation of an object or material into two, or more, pieces due to stress

Plastic Flow (Plastic Deformation): when a material is stressed beyond the yield point and irreversible deforma-tion occurs; energy is not conserved

Ductile: a material where a large amount of plastic deformation takes place between the elastic limit and thefracture point

Brittle: a material where fracture occurs soon after the elastic limit is passed

Elastic Hysteresis: when a material follows different stress-strain curves for increasing and decreasing stress

Breaking Strength (ultimate strength or Tensile Strength): the stress required to cause actual fracture of amaterial

Gravitation

Celestial Mechanics: the study of the dynamics of objects in space

Newtonian Synthesis: the concept that Newtonian physics is valid for the entire universe, not just on Earth

ä Fg =Gm1m2

r2 (law of gravtitation)

ä gravitational forces always act along the line joining the two particles in an action-reaction pair

ä even if the masses are different, both particles experience the same force of Fg

ä spherically symmetric mass distributions are equivalent to a particle with all of the mass at the center ofmass in gravitation

ä at any point in a spherically symmetric mass distribution, the gravitational force on a mass m is the sameas though we removed all the mass at points farther than the distance r of the object from the center andconcentrated all the remaining mass at the center

ä G = 6.6274210×10−11N ·m2/kg2 = m3/(kg · s2) (universal gravitational constant)

Law of Superposition of Forces: the total gravitational force on a body is the vector sum of the forces

ä w = Fg =GmEm

R2E

(weight of a body of mass m at the earth’s surface)

ä g =GmE

R2E

(acceleration due to gravity at the earth’s surface)

True Weight (~w0): the total gravitational force exerted on the body by all other bodies in the universe

Apparent Weight (~w): due to centripetal force from the Earth rotating, the perceived weight is less than the truevalue; this effect is greatest at the equator

22

ä w = w0−mv2

RE(at the equator)

ä g = g0−v2

RE(at the equator)

Apparent Weightlessness: weightlessness that occurs when an object is orbiting the earth with~arad =~g0

True Weightlessness: weightlessness that occurs when an object is infinitely far from all other masses so thattheir is no gravitational force acting on it

ä ~w = ~w0−m~arad = m~g0−m~arad (apparent weight or weightlessness of a body)

ä U =−GmEmr

(gravitational potential energy)

ä gravitational potential energy is always conserved

Escape Speed: the speed required for a body to escape completely from a planet

ä v =

√2GM

R(escape speed)

Orbits

Closed Orbit: an orbit that forms a closed loop and will orbit the body indefinitely; an elliptical or circular orbit

Open Orbit: an orbit where the satellite eventually keeps moving further away from the body; a parabolic orhyperbolic orbit

ä v =

√GmE

r(circular orbit)

ä T =2πr

v= 2πr

√r

GmE=

2πr32

√GmE

(circular orbit)

ä E =−GmEm2r

(circular orbit)

Kepler’s Laws:

1. each planet moves in a elliptical orbit, with the sun at one focus of the ellipse

2. a line from the sun to a given planet sweeps out equal areas in equal times

3. the periods of the planets are proportional to the 32 powers of the major axis lengths of their orbits

ä angular momentum is conserved in orbits

Perihelion: the point in a planet’s orbit closest to the sun

Aphelion: the point in a planet’s orbit farthest from the sun

ädAdt

=12

rvsinφ (sector velocity; elliptical orbit)

23

ä T =2πa

32

√Gms

(elliptiral orbit around the sun)

Black Holes

Black Hole: an object that exerts a gravitational force on other bodies but cannot emit any light of its own

ä we know that black holes exist because of their gravitational effects on planets and other objects in space,they emit x-rays, and produce accretion disks

Schwarzschild Radius: the critical radius which if a body of mass M is contained within, then nothing, includinglight, can escape it’s gravitational field from within the event horizon

ä Rs =2GM

c2 (Schwarzschild radius)

Event Horizon: the surface of a sphere with radius Rs surrounding a black hole

Gravitational Red Shift: the relativistic effect where the wavelength of electromagnetic radiation leaving a gravi-tational field is decreased

Tidal Forces: the forces that result from the varying strength of a black hole’s gravitational field on different partsof an object

Accretion Disk: the disk of gas and dust in the proximity of a black hole that gets pulled in by the gravitational fieldand swirls around like a whirl pool

Periodic Motion

Displacement (x): the distance of the body from the equilibrium position

Restoring Force: the force that tends to bring a body displaced from equilibrium back to the equilibrium position

ä Fx =−kx (restoring force exerted by an ideal spring)

Amplitude (A): the maximum magnitude of displacement from equilibrium

Cycle: a complete vibration; a round trip from any starting place, example: A to 0 to −A to 0 to APeriod (T ): the time needed to complete one cycle

Frequency ( f ): the number of cycles in a unit of time

ä the SI unit of frequency is the hertz

ä 1Hertz = 1Hz = 1cycle/s = 1s−1

ä f =1T

and T =1f

(relationships between frequency and period)

Angular Frequency (ω): ; 2π times the frequency

ä ω = 2π f =2π

T(angular frequency)

24

Phase Angle (φ): the point in radians where t = 0Harmonic Oscillator: a body that undergoes simple harmonic motion

Reference Circle: the circle in which an object would move so that its projection matches the motion of theoscillating body

Phasor: a rotating vector

Simple Harmonic Motion

Simple Harmonic Motion (SHM): sinusoidal motion where the restoring force is directly proportional to the dis-placement form equilibrium

ä simple harmonic motion is the projection of uniform circular motion onto a diameter

ä in simple harmonic motion the period and the frequency do not depend on the amplitude A and any oscillatingbody where period does depend on amplitude is not simple harmonic motion

ä E =12

mv2x +

12

kx2 =12

kA2 = constant (total mechanical energy in SHM)

ä Fx =−kx (restoring force exerted by an ideal spring)

ä x = Acos(ωt +φ) (displacement in SHM)

ä vx =dxdt

=−ωAsin(ωt +φ) (velocity in SHM)

ä ax =dvx

dt=

d2xdt2 =− k

mx =−ωAcos(ωt +φ) (acceleration in SHM)

ä vmax =

√km

A = ωA (maximum velocity in SHM)

ä displacement versus time is shifted .25 periods from velocity versus time and .5 periods from accelerationversus time

ä maximum velocity in SHM occurs at x = 0

ä A =

√x2

0 +v2

0xω2 (amplitude in SHM)

ä φ = arctan(− v0x

ωx0

)(phase angle in SHM)

ä T =1f=

2π

ω= 2π

√mk

(period in SHM)

ä f =ω

2π=

12π

√km

(frequency in SHM)

ä in vertical SHM, everything is the same except that the equilibrium position is k∆l = mg instead of where thespring is unstretched

ä θ = Θcos(ωt +φ) (angular displacement in angular SHM)

25

ä ω =

√km

(angular frequency in SHM)

ä ω =

√κ

iand f =

12π

√κ

I(angular SHM)

Pendulums

Simple Pendulum: an idealized model consisting of a point mass suspended by a massless, unstretchable string

ä can be approximated by simple harmonic motion

ä Fθ =−mgsinθ (true exact restoring force, simple pendulum)

ä T =2π

ω=

1f= 2π

√Lg

(simple pendulum, small amplitude)

ä f =ω

2π=

12π

√gL

(simple pendulum, small amplitude)

ä ω =

√km

=

√mg/L

m=

√gL

(simple pendulum, small amplitude)

ä T = 2π

√Lg

(1+

12

22 sin2 Θ

2+

12 ·32

22 ·42 sin4 Θ

2+ · · ·

)(true exact period, simple pendulum)

Physical Pendulum: any real pendulum that uses an extended body

ä can be approximated by simple harmonic motion

ä τz =−(mg)(d sinθ) (true exact restoring torque, physical pendulum)

ä ω =

√mgd

I(physical pendulum, small amplitude)

ä T = 2π

√I

mgd(physical pendulum, small amplitude)

Damped and Driven Motion

Damping: the decrease in amplitude caused by dissipative forces

ä x = Ae−(b

2m)t cos(

ω′t +φ

)(oscillator with little damping)

ä the variable b stands for the damping constant

ä ω′=

√km− b2

4m2 (oscillator with little damping)

26

Critical Damping: damping where b is equal to the critical value of b = 2√

kmOverdamping: damping where b is greater than the critical value of b = 2

√km

Underdamping: damping where b is less than the critical value of b = 2√

km

ädEdt

= vx (−bvx) =−bv2x (damped oscillations)

Driving Force: an additional external force added to an oscillating system

Forced Oscillation (Driven Oscillation): the motion that occurs from a periodic driving force

Natural Angular Frequency (ω′): the frequency the system naturally assumes when displaced from equilibriumand then left alone; dependent on m, k, and b

ä A =Fmax√(

k−mω2d

)2+b2ω2

d

(amplitude of a driven oscillator)

Resonance: an amplitude peak at driving frequencies close to the natural frequency of the system

Fluid Statics

Fluid Statics: the study of fluids at rest in equilibrium situations

Density (ρ): mass per unit volume

ä the SI unit of density is the kilogram per cubic meter 1kg/m3

ä ρ =mv

(definition of density)

Specific Gravity: the (unitless) ratio of the density of a material to the density of water at 4.0C, which equals1000kg/m3

Average Density: the density over a volume of a material whose density varies from point to point

Pressure (p): the normal force per unit area

ä pressure is a scalar and has no direction

ä other than height and gravity, pressure is independent of the shape of the container

ä the SI unit of pressure is the pascal

ä 1Pascal = 1Pa = N/m3 = 9.8692×10−6 atm = 1×10−5 bar

ä p =dF⊥dA

(definition of pressure)

ä p =F⊥A

(if pressure is the same at all points of a finite plane surface of area A)

Atmospheric Pressure (pa): the pressure exerted by the earth’s atmosphere

ä p2− p1 =−ρg(y2− y1) (pressure in a fluid of uniform density)

ä p = p0 +ρgh (pressure in a fluid of uniform density)

27

Pascal’s Law: pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid andthe walls of the containing vessel

ä in a hydraulic lift, the output force can be greater than the input force, but the smaller force must go a longerdistance due to the conservation of energy

ä p =F1

A1=

F2

A2and F2 =

A2

A1F1 (Pascal’s Law)

Gauge Pressure: the excess pressure above atmospheric pressure

Absolute Pressure: the total pressure

Manometer: an instrument to measure pressure by comparing the pressure a vessel of fluid exerts on a fluid ofknown density such as water against atmospheric pressure by looking at the height difference of the known fluidaround a vertical U-shaped bend

ä p+ρgy1 = patm +ρgy2 (manometer equation)

Mercury Barometer: an instrument to measure air pressure by seeing how high a column of mercury rises in atube with a vacuum at the top and open to the air at the surface of a dish holding the mercury column

Sphygmomanometer: an instrument to measure blood pressure by using a sort of mercury manometer

Archimedes’s Principle: when a body is completely or partially immersed in a fluid, the fluid exerts an upwardforce on the body equal to the weight of the fluid displaced by the body

ä Fy = ρV g (upward force from Archimede’s Principle)

Buoyant Force: the upward force exerted by a fluid on a body completely or partially immersed in the fluid

ä the line of action of the buoyant force passes through the center of gravity of the displaced fluid, not the body

Surface Tension: the surface of a liquid behaves like a membrane under tension because interior molecules haveequal attractive forces in all directions but the surface molecules don’t have attraction from the air above, so thesides are pulled in and down

Fluid Dynamics

Fluid Dynamics: the study of fluids in motion

Ideal Fluid: a fluid that is incompressible and has no internal friction

Incompressible: the fluid’s density cannot change

Viscosity: the internal friction in a fluid

Boundary Layer: a thin layer of fluid near the surface of an object which is nearly at rest with respect to thesurface; develops from the viscosity of the fluid

Flow Line: the path of an individual particle in a moving fluid

Steady Flow: flow where the overall flow pattern does not change with time

Streamline: a curve whose tangent at any point is in the direction of the fluid velocity at that point

Flow Tube: a tube formed by the flow lines passing through the edge of an imaginary element of area

Laminar Flow: flow in which adjacent layers of fluid slide smoothly past each other and the flow is steady

Turbulent Flow: flow that is irregular and chaotic with no steady state pattern, but rather changes with time

Continuity Equation: the mass of a moving fluid doesn’t change as it flows

28

ä A1v1 = A2v2 = constant (continuity equation, incompressible fluid)

ä ρ1A1v1 = ρ2A2v2 = constant (continuity equation; compressible flow)

ädVdt

= Av (volume flow rate)

Bernoulli’s Equation: the work done on a unit volume of fluid by the surrounding fluid is equal to the sum of thechanges in kinetic energy and potential energies per unit volume that occur during the flow

ä only valid for ideal, incompressible, steady state flowing fluids with no viscosity

ä p1 +ρgy1 +12

ρv21 = p2 +ρgy2 +

12

ρv22 = constant (Bernoulli’s equation)

Torricelli’s Theorem: the speed of efflux from an opening at a distance h below the top surface of the liquid is thesame as the speed a body would acquire in falling freely through a height h

ä can be derived from Bernoulli’s Equation

ä v =√

2gh (Torricelli’s Theorem)

ä ∆p =LR4 (pressure difference to sustain flow rate; viscous fluid)

ä a ball with spin will push against the air flow in one direction and with the air flow in the other direction, sothe direction being resisted will have a lower velocity and higher pressure

ä increasing turbulence can decrease resistance and thus increase velocity of an object

Special Relativity

Michelson-Morley Experiment: an effort to detect the motion of the Earth relative to the ether, but no ether motionwas detected and the concept of the ether was discarded

ä Einstein saw that if Maxwell’s equations are valid in all inertial frames of reference, then the speed of light ina vacuum must be constant in all inertial frames of reference

Einstein’s First Postulate (Principle of Relativity): the laws of physics are the same in every inertial frame ofreference

Einstein’s Second Postulate (Principle of Invariant Light Speed): the speed of light in vacuum is the same inall inertial frames of reference and is independent of the motion of the source

ä it is impossible for an inertial observer to travel at c, the speed of light in vacuum

ä massless particles (zero rest mass) always travel at the speed of light

ä x = x′+ut y = y′ z = z′ t = t ′ (Galilean coordinate transformation; no y or z motion)

ä vx = v′x +u (Galilean velocity transformation; no y or z motion)

Event: an occurrence that has a definite position and time

Relativity of Simultaneity: simultaneity is not an absolute concept but rather depends upon the frame of reference

29

ä in general, two events that are simultaneous in one frame of reference are not simultaneous in a secondframe of reference moving relative to the first, even if both are inertial frames

• each observer is correct in his or her own frame of reference

• whether or not two events at different x axis locations are simultaneous depends on the state of motionof the observer

• the time interval between two events may be different in different frames of reference

ä γ =1√

1− v2

c2

(Lorentz factor)

• β =vc

β2 =

v2

c2 (abbriviation occasionally used)

Time Dilation: observers measure any clock to run slow if it moves relative to them

Proper Time: the time interval ∆t0 between two events that occur at the same time

ä ∆t = γ∆t0 (time dilation)

Twin Paradox: even though all inertial frames are equivalent, accelerations of a body can allow time to pass moreslowly for it

ä for a pair of identical twins, if one person stays in an inertial frame of reference and the other accelerates ina non-inertial frame of reference, then the accelerating twin will age more slowly

Length Contraction: the length measured in any frame moving relative to the resting frame of reference S′ is lessthan the length in that frame l0

ä this is not an optical illusion, the lengths really are shorter in a reference frame moving relative to the restframe of the body

ä there is no length contraction perpendicular to the direction of motion relative to the coordinate systems

Proper Length: a length measured in the frame which the body measured is at rest

ä l =l0γ

(length contraction)

Pole and Barn Paradox (Ladder Paradox):

ä x′ = γ(x−ut) y′ = y z′ = z t ′ = γ

(t− ux

c2

)(Lorentz coordinate transformation; 1D motion)

ä v′x =vx−u1− uvx

c2

(Lorentz velocity transformation)

ä vx =v′x−u

1+ uv′xc2

(Lorentz velocity transformation)

ä frame S′ is moving in the positive x-direction with velocity u relative to frame S, while the body is moving withvelocity v in frame S and velocity v′ in frame S′

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ä objects moving by a reference frame appear distorted because light can reach parts that would be blockedand light is blocked to parts it could normally reach if it was at rest with respect to the reference frame

Relativistic Kinetics

Rest Mass (m): the mass of a particle m when it is at rest relative to the reference frame of the measurement

Relativistic Mass (mrel): the mass of a particle mrel when it is moving at a speed v relative to the reference frameof the measurement

Material Particle: a particle that has a nonzero rest mass

ä mrel = γm (relativistic mass)

ä ~p = γm~v (relativistic momentum)

ä ~F =d~pdt

(Newton’s second law; form valid in special relativity)

ä F = γ3ma (~F and~v along the same line)

ä F = γma (~F and~v perpendicular)

ä unless the net force on a relativistic particle is either along the same line as the particle’s velocity or perpen-dicular to it, the net force and acceleration vectors are not parallel

ä constant force does not cause constant acceleration

Total Energy: the sum of the rest energy and the kinetic energy of a particle

Rest Energy: the energy that a particle contains when it is at rest and therefore has zero kinetic energy

ä K = (γ−1)mc2 (relativistic kinetic energy)

ä E = K +mc2 = γmc2 (total energy of a particle)

ä E2 =(mc2)2

+(pc)2 (total energy, rest energy, and momentum)

• E = pc (zero rest mass)

ä f =

√c+uc−u

f0 (Doppler effect; electromagnetic waves; source approaching observer)

ä f =

√c−uc+u

f0 (Doppler effect; electromagnetic waves; source moving away from observer)

• ∆ ff≈ u

c(approximation for

uc 1; positive u is towards observer)

Conservation of Mass and Energy: neither total mass alone nor total energy alone must be conserved, but theircombined total as determined by special relativity must be conserved

General Relativity

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Correspondence Principle: whenever a new theory is in partial conflict with an older, established theory, the newmust yield the same predictions as the old in areas in which the old theory is supported by experimental evidence

General Theory of Relativity: extends special relativity to account for gravity

ä a person in free fall in a box (elevator, Niagara Falls, etc) cannot tell if they really are in free fall or if thegravitational interaction has vanished

ä if we cannot distinguish experimentally between a gravitational field at a particular location and an uniformlyaccelerating reference frame, then there cannot be any real distinction between the two

• this leads us to represent any gravitational field in terms of special characteristics of the coordinatesystem

General Relativity is Important For:

ä precession of the perihelion of Mercury

ä the apparent bending of light rays from distant stars when they pass near the sun

• gravitational red shift - the increase in wavelength of electromagnetic radiation leaving a massive source

ä global positioning system (GPS)

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