AP C Concepts

8/11/2019 AP C Concepts

http://slidepdf.com/reader/full/ap-c-concepts 1/25

Kinematics, Forces and Differential Eqns

Definitions of velocity and acceleration:

v =dt

dx x = vdt

a = 2

2

dt

xd

dt

dv v = adt

Differential Equations and Kinematics/Forces:

F = mdt

dv Insert forces in left hand side, then separate variables, then integrate

F = m 2

2

dt

xd

Insert forces into left hand side and compare to 022

2

xdt xd

Note: =2 /T

Ex: The force on a car of mass m is given by F(t)=4t+2, find an expression for the velocity as a function of time

v(t) if the car starts from rest.

dt

dvmt t F 24)( (set the forces in a given direction equal to

dt

dvm )

dvdt m

t )24( (separate variables to get all t on one side and all v on the other)

dvdt mt vt

00 )24( (integrate both sides and choose appropriate limits of integration)

m

t

m

t t v

2

2

4)(

2

(do integral and use correct limits)

Ex: Air resistance. Assume the force of air resistance on a falling object is given by F = -kv. Determine the

velocity as a function of time for a dropped object.

F dt

dvmmg kv (treat downward as positive since object is falling)

mdvdt mg kv )( (make the whole force side one term to separate!)

)( mg kvdv

mdt so vt

mg kvdv

mdt

00 )( (separate variables, and integrate)

mg

mg kv

k mg

k mg kv

k m

t )(ln

1)ln(

1)ln(

1 (now just use algebra to solve for v)

mg kvemg m

t k

)( and solving for v you get… ]1[)( m

t k

ek

mg t v

This expression for v(t) mankes sense since at t=0, v=0. And for t= inifinity. v=mg/k which is terminal velocity.



1 D Kinematics

2 D Kinematics

Graphing motion (Note: the graphs below do not represent the same moving object)



Forces

Newton’s Laws

1. Objects maintain constant velocity unless there is force.- No forces does not mean no motion. No forces means no acceleration (constant motion).

2. F = ma = m (dv/dt)-You need to choose only one direction at a time, e.g. Fx = max Fy = may

3. Every force on an object, has an equal and opposite force

on a different object.

- Since the forces are on different objects they do not cancel.

Some common kinds of forces



Centripetal Motion

F = m v2/r

- Forces directed into the circle are positive and forces directed out of the circle are negative

- You can always use this if something is going in a circle (but you don’t always have to use this)

- If an object is moving with constant speed in a circle then v = 2 r/Tperiod



Energy and Work

Work

xd F W

xd F U

note: U is just another symbol for potential energy PE, and W = - PE = - U

If F is constant then…

W=Fdcos Fd

(Note: Area under F vs. x graph equals work .)

- Work tells you how much energy is tranfered to an object

- If a force pushes in the direction of d (tries to speed up object), force does + work

- If the object doesn’t move d=0, or the force is perpendicular to motion, W=0 (this is why magnetic forces

and centripetal forces never do any work)

- Work done by Conservative Forces (gravity, electric, nuclear) only depends on initial and final position of

object. It does not depend on the path taken to go from the initial point to the final point.

-Work done by Non-conservative Forces (firction, airresistance) depends on the path taken. A longer path

between two points (as opposed to staight path) will cause more work to be done.

- Note: Conservative forces have potential energies associated with them. NC forces do not. Also, the work

done by any conservative force is WF = - U .

- Work-Kinetic energy principle Wtotal = KE = KEf - KEi

dx

dU F Force is the spatial derivative (not time derivative!) of the PE



Types of energy (KE, E, PE, U all represent energies)

KE = ½mv2

Ugravity = -Gm1m2/r (use for satellites/planets/etc.)

Ugravity = mgh (use if near the surface of the Earth)

Uspring = ½kx2 (x is the compression or extension of spring)

Uelectric = kq1q2/r

Uelectric = qV (V is the electric potential in J/C or Volts)

Ecapacitor = ½ CV2

= ½ QV

Einductor = ½ L I2



Common Energy/Work problems



Momentum and Collisions

Momentum

p = mv [kg m/s] -Momentum is a vector and direction matters (+ if going right, - if left)

-Momentum is always conserved, unless an outside force is exerted

dt

pd F

-This is a way to relate changes in momentum to forces

Impulse = p = F t [Ns] or p =∫Fdt -Area under F vs. time graph is the Impulse (change in momentum)

Collisions-momentum is conserved in every type of collision (elastic and inelastic)

a. Elastic collisions

- KE must be conserved to be an elastic collision,

- just because objects bounce off each other, doesn’t mean it has to be elastic

-Note: For elastic collisions, (v1 – v2)initial = -(v1 – v2)final

b. Inelastic collisions

- KE is not conserved, KE gets lost in the collision (turns into thermal energy, etc)

-objects can stick together ( perfectly inelastic ) or bounce off each other

Note: To determine whether a collision is elastic or inelastic just compare the

KEtot before and after the collision.

- If KEtot initial = KE tot final, then it is elastic. If not, it is inelastic.





Simple Harmonic Motion

Simple Harmonic Oscillators and Differential Equations

F = m 2

2

dt

xd Insert forces into left hand side and compare to 0

2

2

2

xdt

xd

= I 2

2

dt

d Insert torque into left side, use small approx., compare to 02

2

2

dt

d

= 2 /T [rad/sec]

1. Mass on a Spring

-kx = m 2

2

dt

xd

2

2

dt

xd + (k/m) x = 0

So, = (k/m) which means = (k/m)½

= 2 /T

Tmass on spring = 2 [m/k]1/2

Remember that f = 1/T [Hz]

-Spring Period T does not depend on g or amplitude!

- position as a function of time is described by x(t)=Asin( t) or x(t)=Acos( t)



2. Pendulum

= I 2

2

dt

d

-rFsin = I 2

2

dt d

-(L)(mg)sin = I 2

2

dt

d now use small angle aprroximation sin ~

2

2

dt

d + (Lmg/I) = 0

So, = (Lmg/I) which for a mass on a string I= mL2 means, = (Lmg/mL

2)

½ = (g/L)

½

= 2 /T = (g/L) ½

Tpendulum = 2 [L/g]1/2

Note that pendulum period T does not depend on mass or Amplitude!

- angle as a function of time described by (t)=Asin( t) or (t)=Acos( t)



Torque and Center of Mass

Torque

rF rF sin

- r is from the axis to the point where the force is applied

I - net torque means tangential acceleration

- no net torque does not mean no rotation, just no tangential acc.

- I is the moment of inertia

Moment of inertia

...2

22

2

11

2

r mr mmr I

dr r dmr I 22

- r is distance from axis to m or dm, and is linear mass density m/L

- more mass farther from axis means larger moment of inertia

- larger moment of inertia means that it is harder to start rotating

Parallel Axis Theorem

2 MR I I cmtot M is total mass, R is from axis to CM

- use if you rotate object around an axis that is not at CM of object



Angular Momentum

L=I use for an extended object

L = rpsin = Rp = mvR use for a particle or point mass

See above diagram and note: the r in rpsin is the distance from the origin/axis to the particle

R is distance from origin to nearest line of approach to origin. Or if particle is going in a circle,

that means r=R and sin =1 so just use mvR.

=dL/dt [this is analogous to F=dp/dt]

Equilibrium

0 F 0

- no acceleration and no angular acceleration (usually means at rest)

- i.e. Torque in CW direction = Torque in CCW direction

Center of mass

...11

2211 r mr m M

mr M

CM tot tot

dmr M

rdm M

CM

tot tot

11

- r is distance from arbitrary point (side of object, middle of object, etc.) to mass

m or dm, but if there is an obvious axis it is usually a good idea to use it

- is the linear mass density m/L or m/r (so since m = r dm = dr)

- the CM tells you the position where an object could balance

- the CM is also where you could treat all the mass as residing



Electric Forces, Fields, Energy & Voltage

Fe is electric force [Newtons]+ charges feel force in same dir of the electric field (E), - charges feel force in opp. dir of electric field (E)

E is electric field [N/Coulomb or Volts/meter]+ charges create electric fields that point radially outward from charge, - charges create E pointing inward

Uelectric is electric potential energy [Joules]Electric Poetential Energy (U) is another form of energy that objects can have

V is electric potential [J/Coulomb or Volts]Electric Potential (V) at a point is the Electric Potential Energy (U) 1C of charge would have at that point

Note: + and - charges both feel a force toward lower PEelectric , also Electric fields E point toward lower V





Gauss’ Law

o

inQad E

Electric flux though closed surface is proportional to

charge inside surface.If E is constant over surface then…

o

inQ EA

For spherical symmetry (sphere of charge, point charge)

o

inQ

r E )4( 2

and if you only enclose some of the total charge, use )3

4( 3r V Q inin

For cylindrical symmetry (cylinder of charge, line of charge)

o

inQrh E )2( and if you only enclose some of the total charge in a cylinder, use

)( 2hr V Q inin or if there is a line of charge use h LQin

For a plane of charge (or if very near any surface of charge)

o

inQlw E )( and if you only enclose some of the total charge, use )(lw AQin

which means near the surface of any shape of surface charge o

E



Solving 2D electric force (vector) problems1. Draw the forces exerted on the charge you are concerned with(These are forces ON charge Q, not the forces charge Q is exerting on other charges)

2. Find the size of each force using F=kq1q2/r2

3. Break forces into vertical and horizontal components Fy and Fx (For completely vertical or horizontal forces one component will be zero, the other is + F)

(For diagonal forces you need to use sin and cos to break the vector into components)

(Up or Right is a positive component, Left or Down is a negative component)

4. Add up all the Fx and Fy to get the components of the total Force vector Fxtot and

Fytot

i.e. for the example shown Fxtot = Fa cos + 0 + (-Fc) Fytot = Fa sin + (-Fb) + 0

5. Find magnitude of total Force using the Pythagorean Theorem i.e. Ftot2 = Fxtot

2

+ Fytot2

6. Find the angle from horizontal by using = tan-1

(Fytot/Fxtot) (Note: This angle is always the angle from the “nearest” x-axis to the total Force vector)



Circuits

Current

dt

dQ I

[Coulomb/Second = Amperes]

- defined to be in the direction of positive charge flow (or opposite direction of e-)

- is directed out of the + terminal of a battery, and into the - terminal

ResistanceThe resistance of a length L of cylinder made with resistivity , and cross sectional area A is,

R= L/A [Ohms]

Ohm’s Law

V = IR (V is voltage drop across resistor, I is current through the resistor, R is resistance)

- V is not necessarily the voltage of the battery!

- Ohmic materials have constant resistance (slope on V vs. I), regardless of what the current is

Electrical Power

P = IV = I2

R = V2

/R and to find energy/heat delivered… IVdt Pdt E

Capacitors

C = Q/V (C is capacitance, Q is charge on + plate, V is voltage across capacitor)

- Capacitance tells you how well a capacitor can store charge

- Inserting a Dielectric between a capacitor always increases capacitance by a factor of k

- Capacitors store energy as well, which is given by

Ecapacitor = ½ QV = ½ CV

2

- For a parallel plate capacitor with plates of area A separated by a distance d, capacitance is,

C = oA/d [Farads]

Inductors

= -L dI/dt Einductor= ½ LI2 [L is measured in Henry]

- Inductors make it so you can not instantly change current.

- Current through inductor right before you flip switch = Current though inductor right after you flip switch



Combining Resistors

Combining Capacitors



Kirchoff’s Rules

Junction Rule: Iin = Iout - Total current flowing into junction equals total current flowing out of junction

Loop Rule: V = 0

- The sum of the changes in voltage around any closed loop always equals zero

V = -IR (if you pass through resistor in the same direction as current)

V = IR (if you pass through resistor in the opp. direction as current)

V = + battery (if you pass through the battery from – terminal to + terminal)

V = - battery (if you pass through the battery from + terminal to - terminal)

Terminal Voltage

Vab = - Ir (Vab is the terminal voltage, is the emf of battery, r is internal resistance)

-Every battery has an internal resistance r which will lower the terminal voltage when current flows

- A 9V battery will not necessarily have a measured terminal voltage of 9V, unless no current flows

- The of a 9V battery is 9V even when no current flows, but the measured terminal voltage will be less

- Slope of Vab vs. I graph is negative the internal resistance. The y intercept is the emf .

Electrical Meters

Voltmeter- Measures voltage change across circuit element (resistor, battery, etc.)

- Ideally has infinite resistance so it does not draw any current away from circuit

- Needs to be hooked up in parallel with circuit element

Ammeter- Measures current through a circuit element (resistor, battery, etc.)

- Ideally has no resistance so it does not change or affect the current it is trying to measure

- Needs to be hooked up in series with circuit element



Differential equations and R-L-C circuitsWhen you are asked to find something (current, voltage, charge, etc.) as a function of time, use the loop rule

to write and solve a differential equation.

RC circuit

-for the voltage drop across a resistor use – I R = -(dQ/dt)R

-for the voltage drop across a capacitor use –Q/C

-for most charging RC circuits you can use

0C

Q R

dt

dQ

And you separate Q and t, then integrate to get,

)1()( RC

t

o eQt Q which means that RC

t

o RC

t

o e I e RC Q

dt dQt I )(

Note: = RC gives you an idea of how long it takes to charge/discharge

-for most discharging RC circuits you can use

0C

Q R

dt

dQ Note: You often keep the dQ/dt term negative for a discharging

capacitor since the Q flows through R= Q o –Q still on capacitor

And you separate Q and t and integrate to get,

RC

t

oeQt Q )( which means that RC

t

o RC

t

o e I e RC

Q

dt

dQt I )(

Note: Q o = CVo for charging and discharging capacitors (Vo is the maximum voltage across capacitor)



RL circuit

-for the voltage drop across a resistor use – I R

-for the voltage drop across an inductor use – LdI/dt

-for most RL circuits with a battery being connected in the loop you can use

0dt dI L IR

And you separate I and t, then integrate to get,

)1()( / R L

t

o e I t I

Note: = L/R gives you an idea of how long it takes for inductor to allow current to change

-for most RL circuits with a battery being disconnected in the loop you can use

0dt

dI L IR

And you separate I and t, then integrate to get,

R L

t

oe I t I /)(

Note: Io = Vo/R Io is the maximum current that flows



Magnetism

Magnetic forces

Bvq F B

FB = qvBsin vB (q is charge, v is speed, B is magnetic field, is angle between v and B)- The direction of force on + charge is given by the Right hand rule (Very-Bad-Finger)

- If the charge is negative the force is in the opposite direction

- Magnetic forces never do Work (since FB is always perpendicular to motion W=Fdcos90=0)

- Magnetic forces often make charges (q) of mass m travel in circles of radius r given by,

r = mv/qBNote: If you want a charged particle to travel in a straight line (“velocity selector”), create an electric field E so

that the forces cancel, i.e. speed is ratio of E to B

v=E/B (since FB = FE or qvB = qE)Note: The forces have to be of equal size, not the fields! (i.e. FB = FE , but E does not equal B)

Magnetic force on wire

FB = ILB sin LB -to find direction of FB use the same right hand rule (except v is now direction of I)

Magnetic fields

Biot Savart Law 2

ˆ

4 r

r l d I B

o

(true for all cases, but rarely used)

Long straight wire B = oI/2 r [Tesla] o = x 10-7

T m/A

- The magnetic field from a long straight wire is directed along a circle centered at wire with direction given by

right hand rule (Thumb in direction of current, fingers curl in direction of B)

- Note: Wires with I in same direction will attract, Wires with I in opp. Direction will repel

- Also for a solenoid or a toroid,

Bsolenoid = o(N/L)I Btoroid = o(N/2 r)I



Magnetic flux and induced voltage

ad B M

and if B is uniform in space then… co BA M

dt

d M

- You will induce a voltage/current in a loop of wire only if you change either B, A, or in time

For a piece of wire or a conducting bar of length L the induced voltage will be

= LvB (remember there is voltage on Las Vegas Boulevard)

Lenz’s Law - Induced current always opposes the change in flux

If flux increases, induced current creates a magnetic field B in opp direction of external B

If flux decreases, induced current creates a magnetic field B in same direction of external B



Ampere’s Law

enclosed o I l d B

Integral around a closed loop is proportional

to the current inside loop.

If B is constant along loop then…

encloo I BL

For a long straight wire just integrate in a circle around wire to get,

B(2 r) = oI

B= oI/(2 r)

For a solenoid integrate in rectangle (part inside solenoid, part outside) which encloses N turns of wire to get,

B(L)= oNI

B= o(N/L)I

For a toroid integrate in a circle around inside of toroid which encloses all N turns of wire to get,

B(2 r) = oNI

B= o(N/2 r) I

Documents

AP C Concepts