Class 5: Circuits, Magnetism, Waves & Sounds

Doppler equations

(+) = use top sign when the motion is toward v = 350 m/s speed of sound in air - remember that if the speeds of the source and detector are much smaller than the speed of the wave, the size of the shift (delta f) will be a fraction of the source frequency ex: source frequency = 1400 Hz vD = 10 m/s vS = 40 m/s *both of these speeds are much smaller than speed of sound (350 m/s) so the frequency of the detector will be close to 1400 Hz (i.e. 1540 Hz)

How are magnetic fields used to sort particles traveling at the same velocity by charge and mass as in mass spectrometry?

(+) vs (-) particles in a magnetic field will have opposite deflection - determine direction of particle based on RHR more mass = larger turning radius (qvB = mv^2r)

Wave Example 2: The speed of sound waves depends on the bulk modulus (the resistance to compression) of the medium and its density: v = sqrt(B/density). The general trend is for sound waves to travel slowest in gases, faster in liquids, and fastest in solids. Which is true? A. when you are under water and hear people talking on the boat above you, their voices are a higher pitch (frequency) from normal B. When you are under water and hear people talking on the boat above you, their voices are louder (more intense, i.e. more power per unit area) than normal C. when you are under water and hear people talking on the boat above you, the wavelength of their voices is longer than it would be in air D. there is no significant difference between the voices heard under water and how they would be heard in the air

(high bulk modulus is the harder it is to squeeze/compress something, material for sound to propagate) sound waves to travel slowest in gases, faster in liquids, and fastest in solids density of gas/air < water < solid but bulk modulus increases at a faster rate than density A. False, rule 2: frequency is constant moving from one medium (air) to another (water) B. violates conservation of energy, cannot get a larger amplitude or more energetic wave through transmission - voices would actually be quieter, much of the energy would be reflected off (most of sound energy is not transmitted from air into water) C. speed increases moving from air to water, so wavelength must increase because frequency is staying the same (v = wavelength x f so v : wavelength) D. yes there is a difference answer C

Power

*energy flux/dissipation (coming out) of the battery into the circuit per time Power output by a battery - electrical energy - P=IV Power dissipated across resistors - thermal energy (heat) - P=I^2R Each term equivalent according to Ohm's Law (V=IR); use whichever expression is most convenient in a problem, keeping in mind Kirchhoff's laws P = IV (amps x volts) = Watts P = I^2R (amps^2 x ohms) P=V^2/R *use if resistors are in parallel because voltage is constant Energy = P x time

Magnetic fields and forces

*when a charged particle moves in the presence of a magnetic field, it may experience a magnetic force* B: magnetic field, T = Tesla [B] = [Fb] / [q] [v] = N /C x m/s = N/C/s x m = N/ Am Newton per Amp meter = Tesla theta: angle between v and B Direction of force found using RHR for Fb: - flat hand - thumb in direction of v - fingers in direction of field = B - palm (positive charge) gives direction of Fb - back of hand gives direction of Fb for a negative charge *Fb is always perpendicular to both v and B Electric fields are created by - electric charges - if a charge is at rest, it produces an electric field in the surrounding space Magnetic fields are created by - moving electric charges - if this charge that was initially at rest and began to move, it would created an an additional force = magnetic field in the surrounding space - since charge in motion constitutes a current, magnetic fields are produced by electric currents - a magnetic field can only exert a force on a charge that is moving through the field - angle between v and B (if they are parallel or antiparallel, Fb=0) - magnetic forces DO NO WORK (W=Fdcos(90) = 0), making KE constant. Centripetal forces do no work - since magnetic forces cannot change the kinetic energy of a particle, they can't change the speed of a particle. ALL MAGNETIC FORCES CAN DO IS MAKE CHARGED PARTICLES CHANGE THEIR DIRECTION; THEY CAN'T MAKE THEM SPEED UP OR SLOW DOWN

Four particles traveling with the same initial velocity parallel to the ground pass through a region with a strong magnetic field pointing vertically down toward the ground. Which of the particles experiences the LEAST deflection from its initial velocity? A. A Cu2+ ion B. A lithium atom C. An electron D. A proton

- A lithium atom would experience the LEAST deflection from its initial velocity. - The magnitude of the magnetic force on a charge moving through a magnetic field is given by the equation F = qvBsinθ, which for cases like this where v is perpendicular to B yields F = qvB. - The velocity and the magnetic field strength are the same for all four particles, so the only factor differentiating the forces acting on each is charge. - Any non-ionized atom has zero net charge, and thus experiences zero magnetic force, so only the lithium atom experiences no deflection from its initial velocity.

Measurement Devices Summary

- Ammeters and voltmeters obey Kirchoff's rules: the ammeter must be connected in series with the measured element to have the same current, and the voltmeter must be connected in parallel to have the same voltage - ideal meters don't affect the values they measure. Real meters do, but minimally

Simple Harmonic Motion: Dynamics

- Amplitude, +/- A: greatest displacement from equilibrium during oscillation - amplitude has a specific constant value for any given oscillation, while displacement is constantly changing as the oscillation moves from equilibrium to the -A back to equilibrium back to +A F = -kx - the spring exerts a force on the block that is proportional to its displacement - k tells us how strong the spring is, stiffer and stronger for larger values of k - k = N/m Restoring force: because the spring is always trying to restore the block to equilibrium, we say that the spring provides the restoring force that maintains the oscillations

Simple Harmonic Oscillation

- The tenth oscillation is identical to the first. - The restoring force of the oscillation is directly proportional to the displacement from equilibrium. - The period of the oscillation is independent of its amplitude. *when doing pendulum problems, find component of restoring force ie mgsintheta*

Circuit Measurement Devices

- Voltmeters and Ammeters Voltmeters - measure voltage - connect in parallel to circuit segment of interest (think Kirchoff) - high internal resistance Internal resistance: resistance of device which minimizes impact of measurement on circuit Ammeters - measure current (amps) - connect in series to the circuit segment of interest (think Kirchoff) - low internal resistance (so it minimally effects value you are measuring, elements in series add resistances *not that in figure, ammeter and voltmeter are not in parallel or in series with the battery. So if you wanted to measure voltage or current across the battery you would have to disconnect and place in parallel to battery or in series with battery, respectively - where you place ammeter/voltmeter is specific to what you are measuring like resistor 3

Permanent magnets

- all materials have some response to magnetic fields, but a few materials (iron, cobalt, and nickel) have a particularly strong response. Large magnetic fields are created by permanent magnets made of such materials as well, due to the alignments of their electron spins - magnet you stick to refrigerator - permanent magnets produce a magnetic field without having a current applied to it - all permanent magnets have a north and south pole. Magnetic field lines come out of the north pole and enter the south pole (all magnets are dipoles; have both N and S poles) -a permanent bar magnet is a source of a magnetic field because of the multitude of microscopic currents due to motions of the orbiting electrons within the metal; even a bar magnet's magnetic field is ultimately due to charges in motion *materials with a permanent magnet, do not require a current to be applied to it to generate a current, they will always have a magnetic field*

1. Currents into and out of an point in a circuit must equal each other

- another statement of continuity equation - conservation of charge - elements in series have equal current

Oscillations

- any motion that regularly repeats us referred to as periodic or harmonic motion ex: of an object undergoing uniform circular motion, a mass oscillating on a spring, and a pendulum - this type of motion can be characterized by its period or frequency

Sound waves

- are longitudinal waves in gas, liquid, or solid - regions of high pressure = compressions alternate with regions of low pressure = rarefactions - apply wave speed equation - apply 2 Big Wave Rules

When you think of capacitors

- charge

2. the sum of the voltages around any closed loop in a circuit must equal zero

- conservation of energy - elements in parallel have the same voltage (elements in parallel essentially make loops)

the closed end of a pipe

- corresponds to a displacement node (no motion) or a pressure antinode (maximum pressure fluctuations) *the pressure varies the most when there is no motion and the motion varies most when there is constant pressure

the open end of a pipe

- corresponds to an antinode - referred to as displacement antinodes (maximum displacement) also known as pressure nodes (constant pressure) wavelength-n = 2L/n f-n = nv/2L

AC generator (eq for V and Irms)

- creates a sinusoidally varying voltage - the voltage starts at zero, climbs to a peak value (max A), then falls back to zero; at this point the polarity reverses, and the voltage again rises to its peak value and then falls back to zero, and the cycle starts again - on a graph showing time-varying voltage, the first half of the cycle is positive and the second half is negative; the negative voltage means that it points in the opposite direction from the voltage in the first half of the cycle - when the voltage is positive, the current flows in one direction - when the voltage is negative, the current flows in the opposite direction DC and AC circuits use same power equations

When you think of resistors

- current

Two particles, one positively charged and one negatively charged, are traveling on the same path perpendicular to a constant magnetic field. Particles are moving with the same velocity. If the magnitudes of the charges are equal, the forces experienced by the two particles will:

- differ in direction but not in magnitude.

amplitude of a wave

- doesn't depend on frequency, period, wavelength, or speed - it is determined by how much energy we put into the wave to get its started - if we wiggle the rope up and down through a large distance (a large amplitude), this takes more energy on our part and as a result the wave carries more energy

Simple Harmonic Motion requiers:

- dynamics condition: restoring force is directly proportional to displacement from equilibrium (x = 0) and points towards that equilibrium point - kinematics condition: frequency and period are independent of the amplitude of oscillations

A motionless, positively-charged particle is situated in an unknown medium. The magnitude of the magnetic field generated by the particle will be:

- equal to zero. A magnetic field is created only by electric charges in motion. Since the charged particle described in this question is motionless, it generates no magnetic field.

standing waves obey the wave speed equation

- even though the standing wave pattern doesn't propagate, so wavelength and frequency are inversely proportional

Amplitude is independent of

- frequency, period, wavelength, wave speed - amplitude is determined by how much energy we put into the wave to get it started. If we wiggle the rope up and down through a large distance (a large amplitude), this takes more energy on our part, and as a result the wav carries more energy

Direct current (DC)

- last class - circuits in which the current flows in one direction only - e- flow positive to negative terminal

Magnetic field in a magnet

- magnetic field lines created by a magnet will point north to south (use RHR for Fb and v on a proton or electron) - the north pole of a magnet wants to line up with the direction of an external magnetic field; the south pole wants to line up opposite to the field

Magnetic Fields

- magnetic fields are created by moving charges - direction of magnetic field found using RHR for Current-Generated B - point thumb in the direction of current - fingers curl in the direction of the magnetic field (CW, CCW) - current is defined by the motion of positive charges (so if an electron is moving to the left this means that the current is flowing right) - like electric fields, the magnetic field at a point in space is tangent to the magnetic field

Magnetic Fields video

- magnetic fields are created by moving charges (in contrast to electric fields, which are created by either a stationary or moving charges) - direction of magnetic field found using RHR Current: is defined by the motion of positive charges - direction of magnetic field using RHR for current-generated B - point thumb in direction of current (positive charge flow) - fingers curl in direction of the magnetic field B-long straight wire : 1/r *to get the actual vector at a point in space of the magnetic field, like electric fields, the magnetic field at a point in space is tangent to the magnetic field line there

Transverse Waves

- medium oscillates perpendicular to direction of wave propagation (rope exercise) - oscillations are perpendicular to propagation: particle of medium is moving up and down, where wave is propagating left to right - ex: ocean waves, wave on a string, electromagnetic (light) waves - v, speed is moving to the right

Magnetic Fields summary

- moving charges = currents or orbiting electrons in magnetic materials - generate magnetic fields - RHR (straight thumb, curled fingers) gives the direction of B generated by a current

A particle will move in a circle if...

- only true if there is only a magnetic force (no electric force, no gravity)

Longitudinal Waves

- oscillations are parallel to propagation - closer compression = greater pressure region - rarefaction = lower pressure - compression - rarefaction oscillation is analogous to crest - trough - wavelength is compression to compression or rarefaction to rarefaction - medium oscillates parallel to direction of wave propagation - ex: sound (longitudinal pressure wave), slinky

Kirchoff Summary

- parallel elements have same voltage - series elements have same current - batteries supply electrical power to circuits (usually by converting chemical potential energy to electrical) - resistors dissipate that power primarily as thermal energy

Simple Harmonic Motion: Dynamics and Kinematics summary

- springs exert a restoring force that is directly proportional to the displacement from equilibrium. This makes them (and all other spring analogs, such as chemical bonds) ideal examples of simple harmonic oscillators - double stretch, double compression, double force - period and frequency, inverses of one another, are the basic kinematic properties of oscillators. Both are constant over time for simple harmonic motion (77th oscillation takes the exact same amount of time as the first one did)

An alpha particle starts at rest in a region of space in which there are parallel electric and magnetic fields pointing vertically up. Which of the following best describes its subsequent motion?

- straight line vertically upward - An alpha particle has a charge of +2e. It therefore experiences a vertical force upward due to the electric field given by F = qE. Beginning at rest, the alpha particle initially experiences no magnetic force. As it begins to accelerate upward, its velocity remains parallel to the magnetic field, which means that, according to F = |q|vBsinθ, the magnetic force remains zero (because sin 0° = 0). Thus its motion is a straight line vertically upward. *an alpha particle is positively charged, and positively charged particles point in the same direction as the electric field

A battery, a resistor, a capacitor, and a switch are all connected in series, with the switch initially open. The switch is then closed, allowing the capacitor to charge. During this charging process, which of the following quantities DECREASES? I.The current through the resistor II.The energy stored in the capacitor III.The voltage of the battery

- the current through the resistor DECREASES. - As the capacitor charges, the voltage across its plates increases according to V = Q/C. (capacitors in series would increase the voltage) - Because the voltage drop across the resistor and the capacitor combined must equal the battery's voltage, VC + VR = V, as the voltage across the capacitor increases, the voltage across the resistor correspondingly decreases. Therefore, by V = IR, so does the current across the resistor, meaning I is true. - However, as the charge of the capacitor increases, so does the potential energy stored, according to PE = Q2/2C. Thus II is false. The important property of a battery is that it maintains a constant voltage, so III is false.

Wave properties summary

- the kinematic properties of oscillators apply to waves as well, with the addition of propagation speed (and wavelength, frequency, period) - remember that transverse waves are produced by oscillations perpendicular to the direction of propagation, and that there are infinitely many such directions (combinations of up and down or away and toward for a wave propagating left or right; both perpendicular to v) - longitudinal waves, with oscillations parallel to propagation, have a single oscillation direction (sound wave moves L to R and direction of propagation is also L to R)

Simple Harmonic Motion: Kinematics

- the middle, back, back to the middle, forth, and back to the middle = 1 period = one cycle = returning to the same position and velocity period = can be measured from equilibrium back to equilbrium OR - A to -A + A back to +A *all are complete cycles SHM important: independent of amplitude. - no matter if you compress the block 5 cm or 10cm it will take the same amount of time for one complete cycle to occur (unlike relationship between bouncing ball and amplitude)

Electric current will flow only: A. in the absence of resistance. B. through a potential difference. C. in a region of infinite resistance. D. in a perfect conductor.

- through a potential difference. - only from a point of high potential to a point of lower potential - the direction of electric current is the direction that positive charges would move.

Kirchhoff's Laws

1. Currents into and out of an point in a circuit must equal each other 2. the sum of the voltages around any closed loop in a circuit must equal zero

Beats Frequency equation

- two waves with different frequencies combine - constructive interference followed by instance of destructive interference -because the frequencies don't match sometimes the waves are in phase and sometimes out of phase - crest and crest (additive = constructive) alignment then crest and trough alignment (subtractive effect = destructive) - cycles between constructive and destructive interference -> beats - max amplitude sounds loud, and minimum amplitude sounds soft beat frequency equal to the difference between the frequencies of the two original sound waves fB = I f1 - f2 I

When you think of batteries

- voltage

Standing Waves

- waves that are trapped - endpoints determine which wavelengths can be trapped - wave interferes with its own reflection, resulting in a wave that doesn't propagate (guitar string) - each point along the rope has its own amplitude - node = point of no displacement - antinodes = where amplitude is maximized - distance between two nodes = 1/2 wavelength - ex: standing wave in an organ pipe: pressure at end points is atmospheric pressure Open end of standing wave is a pressure node (displacement antinode) where atmosphere keeps pressure fixed, allows air to move freely Confined cap of closed end = pressure node *the node/antinode of displacement always correlates to the node/antinode of pressure

Simple Harmonic Motion: Pendulums

2 pi = 360 degrees - pendulums can be approximated as SH oscillators if the amplitude of oscillation (the maximum angle) is small - the period and frequency of a pendulum depend upon the local gravitational acceleration, unlike those of a mass-spring (spring block system) oscillator

Supposed two identical springs of constant k are set side by side and attached to a mass m. If the mass is pulled from equilibrium a distance A and then released, how do the maximum force and frequency of oscillation compare to those same values with a single spring? A. the values of F and f both increase by a factor of 2 B. the value of F increases by a factor of 2 and f increase by a factor of sqrt(2) C. the value of F stays the same and f increase by a factor of sqrt(2) D. both values stay the same

2 spring system Fnet = F1 + F2 = 2F (two forces because two restoring forces attached to each spring) F = -kx F : k (if force is doubled, spring constant doubles) frequency is proportional to sqrt(k) answer B

Two transverse waves travel in the same direction along a stretched string. Individually, they share the same amplitude and frequency. However, at t = 0, where Wave #1 has its maximum positive displacement, Wave #2 has zero displacement; and where Wave #1 has zero displacement, Wave #2 has its maximum positive displacement. What is the phase difference between them? A. 180° B. 45° C. 90° D. 135°

90°. The two waves are "off" by one-fourth of a wavelength. When discussing phase difference, a complete wavelength (one cycle) corresponds to 360°, so one-fourth of a wavelength corresponds to a phase difference of 1/4 = 90°. - think how top, middle, bottom, middle, top = 1 cycle

Simple Harmonic Motion: Dynamics (not high yield)

= back and forth motion (mass on a spring) that looks the same during each previous oscillation - restoring force = force attempting to move system back into its original equilibrium position is directly proportional to the displacement from that equilibrium - restoring force : displacement: F proportional to -d - compress spring twice as much, you are going to get twice as much push back force - larger k (N/m) = more difficult to compress spring - Amplitude, +/- A: greatest displacement from equilibrium during oscillation

Doppler Shift

= shift in the detected frequency of a wave due to the relative motion between the detector and the sound source (think horn increases as it approaches you due to higher frequency) - detector and source get closer = higher detected frequency - detector and source get father away = lower detected frequency *applies to both sound and light waves - "higher" and "lower" frequency do not imply "increasing" or "decreasing" - if the velocities are constant, so is the detected frequency (if frequency were to change then velocity and acceleration must be changing)

A basic circuit consists of a 60 V battery and a resistor. The current on the circuit is 12 A. If the circuit is modified so that two more equivalent resistors are added in series, what is the resulting current on one of the resistors? A. 4 A B. 6 A C. 24 A D. 36 A

A. Resistors in series can be added to determine equivalent resistance. In this case, the resistance goes from R to 3R. Using Ohm's law, V = IR, if R increases by 3 then I must decrease by 3, so the current becomes 12 / 3 = 4 A. Since the resistors are in series, each will have the same 4 A current. Note that the voltage on the battery is not needed.

A positively-charged particle is traveling on a path parallel to a straight wire that contains a current. Which of the following changes would increase the magnitude of the magnetic field at a point on the path of the particle?

An increase in the current in the wire would increase the magnitude of the magnetic field at a point on the path of the particle. Changing the test charge does not affect the field it is trying to measure, so the choices of "an increase in the charge of the particle" and "an increase in the velocity of the particle" are eliminated. - The field becomes weaker as we move away from the wire, so "an increase in the distance of the particle from the wire" is eliminated. This leaves "an increase in the current in the wire",

A mass connected to a spring is set in simple harmonic motion. Which of the following quantities reach their maximum value when the mass is passing through equilibrium? I.Potential energy II.Kinetic energy III.Acceleration

As the mass passes through equilibrium, the spring is unstretched, indicating that potential energy must be zero. Therefore, by energy balance, kinetic energy must take its maximum value at equilibrium. Similar to potential energy, spring force is also zero at equilibrium, indicating that acceleration must also be zero. Therefore, the correct answer is kinetic energy only.

Magnetic field created by long, straight current-carrying wire EQUATION

B : I / r

clockwise current

B field points into page inside the loop and points out of the page outside the loop

counterclockwise current

B field points out of the page inside the loop and points into the page outside the loop

Which of the following is true of the properties of a light wave that is traveling in vacuum? A. Increased frequency results in decreased speed. B. Increased frequency results in increased wavelength. C. Increased frequency results in increased speed. D. Increased frequency results in decreased wavelength.

D. Increased frequency results in decreased wavelength is true of the properties of a light wave that is traveling in vacuum. A light wave traveling through vacuum, whatever its frequency, has speed c. (This eliminates "Increased frequency results in increased speed" and "Increased frequency results in decreased speed".) Since λƒ = v, and c is constant, an increase in ƒ results in a decrease in λ.

Simple Harmonic Motion: Energy

Elastic Potential Energy= energy stored in a spring as it stretches/compresses x = displacement from equilibrium Spring force is a conservative force (like gravity) Wbyspring = -PEelastic In the absence of non-conservative forces (friction, gravity), conservation of ME applies: KEi + PEi = KEf + PEf

Sound: Intensity

Energy of a wave incident per unit area per unit time [I] = W/m^2 - can be tested on intensity with sound waves, but it can be applied to any type of a wave - intensity of a spherical wave is inversely proportional to r^2 (the distance from the source; the sun is brighter on Mercury than on Earth) - intensity is directly proportional to Amplitude (A^2) (a larger wave with greater amplitude has greater energy) - the Power-received is directly proportional to Intensity and Area-receiver

Intensity or Sound Level

Human ears respond to sound intensity logarithmically (enables larger sensitivity range) - I0 = threshold of hearing (10^-12 W/m^2), softest sound you can hear - [B] =dB (no conversion to SI units) Every time we multiply "I" by 10, we add 10 to B Every time we divide "i" by 10, we subtract 10 from B

Wave example I: The speed of a wave in a string is dictated by the tension and the linear density (mass per unit length) A wave transmits from a medium of less density to a medium of greater density. Which of the following MUST be the same before and after the wave crosses the boundary (noting that part of the wave reflects backward at the boundary)? I. wave speed II. transmitted frequency III. transmitted energy

II. frequency stays the same, rule 2 I. wave speed does change, depends on medium (rule 1) - tension is the same - linear density gets bigger so v will decrease and thus change III. reflection, not all energy is transmitted through the boundary with some energy bouncing off through the medium (total energy is the same), travels down hits boundary, travels through boundary but some bounces back - some of the energy did not transmit from the first medium (less dense) to the second medium (more dense) it had to bounce off (reflected E + transmitted E = initial E) answer: II only

A positive ion at rest is exposed to parallel electric and magnetic fields. What best describes its subsequent motion? A. The ion moves in a circular path. B. The ion moves in a straight line. C. The ion moves in a cycloid path. D. The ion remains stationary.

If a positive ion at rest is exposed to parallel electric and magnetic fields, the ion moves in a straight line. If an ion is exposed to an electric field, it will experience an electric force according to Felectric = qE. This eliminates the choice that the ion remains stationary. - As the ion accelerates due to this force, it gains velocity, but the angle it makes with the magnetic field remains either 0° or 180°, so it does not experience a magnetic force according to FB = |q|vB sin θ. - Thus its motion is a straight line. (Note that if you have no recollection of what a cycloid looks like, you should translate that answer choice to "none of the others": the MCAT will on occasion give you options you did not study in order to test your ability to do just that translation.)

A positive ion at rest is exposed to perpendicular electric and magnetic fields. What best describes its subsequent motion? A. The ion moves in a straight line. B. The ion moves in a cycloid path. C. The ion remains stationary. D. The ion moves in a circular path.

If a positive ion at rest is exposed to perpendicular electric and magnetic fields, the ion moves in a cycloid path. If an ion is exposed to an electric field, it will experience an electric force according to Felectric = qE. This eliminates the choice that the ion remains stationary. - As the ion accelerates due to this force, it gains velocity, and the angle it makes with the magnetic field is 90°, so it experiences a magnetic force according to FB = |q|vB sin θ. (sin90=1) - According to the right-hand rule for charges moving in magnetic fields, this force must be perpendicular to the electric force. Thus its motion cannot be a straight line. However, as the ion accelerates and gains speed due to the electric force, the magnitude of the magnetic force will change while the electric force remains constant in magnitude and direction. - This situation is fundamentally different from that of an ion moving with a constant speed entering a perpendicular magnetic field. This means the motion cannot possibly be circular. - The remaining choice of a cycloid motion must therefore be correct.

SHM: energy pt 2

In the absence of non-conservative forces (friction), conservation of ME applies: KEi + PEi = KEf + PEf - the system will have the greatest potential energy at -A and +A points, where KE = 0 (no motion because v=0); all ME of system is in PE - greatest KE? where velocity is maximum - what is the maximum speed of the block as a function of the amplitude? KEi + PEi = KEf + PEf 0 + 1/2k(-A)^2 = 1/2mv^2max + 0 vmax = sqrt(k/m) x A -maximum speed of oscillating block is proportional to amplitude - if pull or push the spring twice as far you are going to double that maximum speed **Springs store PE when compressed or extended from equilibrium (when work is done on them: pushing or pulling them), which can then be converted to KE

Magnetism

Magnetic fields exert forces on moving charges Direction of force found using Right Hand Rule for Fb - keep hand flat - point thumb in direction of v - fingers = field (direction of B) - palm gives the direction of Fb for a positive charge - back of hand gives the direction of Fb for a negative charge F is always perpendicular to both v and B - if v is east B is north - F cannot be parallel so south or west

Simple Harmonic Motion: Kinematics

Period, T - time is takes to complete one cycle - one cycle = returning to the same position and velocity - period is constant over time (the 1st oscillation takes the same time as the 100th oscillation; think of grandfather clock with pendulum) - period is independent of amplitude (it takes the same time for each cycle to occur) Frequency, f = the number of cycles that occur in one second [f] = 1/s = Hz **both frequency and period are independent of amplitude**

Waves

Propagating oscillations that transfer energy - medium is not propagating with energy (- A and +A averaged are mostly in equilibrium) - medium oscillates with a series of identical oscillations, slightly out of phase - same amplitude, same period (T), different in phases (parts of the wave are ahead of other parts)

A 2-meter-long organ pipe, closed at one end, is resonating at its fifth harmonic. How many times greater is this resonant frequency than the fundamental resonant frequency? (The speed of sound through the air in the pipe is 340 m/s.) A. 2.5 B. 10 C. 1.25 D. 5

Since fn = n(v/2L), then fn = nf1. Thus, the fifth harmonic frequency is five times higher than the fundamental. (This is a general fact that you will find worthwhile to simply know. Whether the pipe is open or closed, the nth harmonic frequency is n times higher than the fundamental frequency.)

period

T, the number of seconds per cycle

You are considering moving to the Moon and want to know if you should bring your grandfather clock (which keeps time using a simple pendulum) or your wind up clock (which keeps time using a torsional spring). Which of the following would correctly inform your choice? A. The acceleration of gravity is lower on the Moon than on Earth, so the spring clock would run slow. B. The acceleration of gravity is lower on the Moon than on Earth, so the grandfather clock would run fast. C. The frequency of any simple harmonic oscillator is independent of location, so either clock would run the same on the moon. D. The acceleration of gravity is lower on the Moon than on Earth, so the grandfather clock would run slow.

The acceleration of gravity is lower on the Moon than on Earth, so the grandfather clock would run slow The period of a simple pendulum is inversely proportional to the square root of g, so a decrease in g entails an increase in the period, meaning the grandfather clock would run slow on the Moon. The period of a spring oscillator depends on the mass being moved and the spring constant, neither of which changes moving from the Earth to the Moon.

At a particular instant in time, a pendulum has swung to the top of its arc and has not yet reversed its direction to swing downward. Which of the following is NOT true of this situation? A. The displacement of the pendulum from its equilibrium position is at a maximum. B. The kinetic energy of the pendulum is zero. C. The potential energy of the pendulum is at a maximum. D. The acceleration of the pendulum is zero

The acceleration of the pendulum is zero". At the top of its arc, the pendulum comes to rest momentarily; thus, its kinetic energy drops to zero (eliminating "The kinetic energy of the pendulum is zero."). Since its height above the bottom of its arc is at a maximum at this point, and since its (angular) displacement from the vertical equilibrium position is at its maximum also, choices "The potential energy of the pendulum is at a maximum" and "The displacement of the pendulum from its equilibrium position is at a maximum" are eliminated. The pendulum constantly feels the forces of gravity and tension and is thus continuously accelerating. (Note: If the pendulum's acceleration were zero at the moment that its velocity were zero, then its velocity would stay zero.)

At which point in a swinging pendulum's arc does it have the most energy?

The energy is the same at all points in a swinging pendulum's arc. Ignoring any dissipative forces due to friction, the total mechanical energy of the pendulum is conserved. While gravitational potential energy and kinetic energy are constantly being converted into one another, the total mechanical energy, KE + PE, remains constant.

Which is true about a charged particle moving perpendicular to a uniform magnetic field?

The magnetic force provides the centripetal force necessary for the particle to undergo uniform circular motion

For an ideal oscillating mass/spring system, which of the following is true? The speed is minimum at maximum displacement and the acceleration is minimum at equilibrium.

The magnitude of the restoring force exerted by a spring with force constant k on a mass m is given by F = -kx, where x is the displacement from equilibrium. For an ideal mass/spring system (oscillating horizontally), this is the net force acting on the object. Newton's Second Law tells us that Fnet = -kx = ma. The maximum acceleration therefore occurs at maximum displacement. Similarly, the minimum acceleration occurs at x = 0 (i.e. equilibrium).

The displacement of an object executing simple harmonic motion is expressed by the equation x = 3 cos (ωt + π/3) [t in seconds, x in cm]. At which of the following positions x will the object have its greatest speed?

The object will have its greatest speed at position 0 cm. The speed is the greatest when all the energy of the object is in kinetic form, which happens when the potential energy is 0. The position where this occurs is the equilibrium position, x = 0.

Wave Properties

Wavelength: length of one cycle (crest to crest, trough to trough) Amplitude: maximum displacement of medium from equilibrium Wave speed (v): speed of wave (energy) propagation Period (T): time required for one cycle to go by, or for one particle of the medium to complete one oscillation FREQUENCY: number of crest passages per unit time

Wave Interference and Beats

When two waves combine, their displacements add Constructive Interference: - waves are in phase (crests and crests and troughs and troughs align, amplitude increases) - final amplitude is greater than the amplitude of the individual waves A = A1 + A2 Destructive Interference: - waves are out of phase (by a half cycle) - crests wave-1 and troughs wave-2 align - final amplitude is smaller than the amplitude of the individual waves A = (A1 - A2)

A sonic boom occurs when: A. a sound source emits a wave with large intensity. B. a sound source travels faster than the speed of sound. C. a light source travels faster than the speed of light. D. a sound source emits a wave with high frequency.

a sound source travels faster than the speed of sound. Since nothing can travel faster than the speed of light, that choice can be eliminated. If the source travels faster than the speed of sound, then the waves it creates are forced together, compressing a large amount of energy that trails behind the source. When this compressed energy reaches the detector, it is perceived as a loud boom.

loudness =

amplitude/intensity

Alternating current

an electric current that reverses its direction many times a second at regular intervals, typically used in power supplies (in homes). - changing current - changing voltage - the electrons that drift in the wires (and whose flow constitutes the current) are constantly forced to shuttle back and forth - producing an alternating voltage (and thus alternating current) on a scale that supplies electricity to entire cities is far easier than producing a steady, direct voltage

If two traveling waves, one with amplitude 4 cm and the other with amplitude 6 cm, interfere, which one of the following best describes the possible amplitudes of the resultant wave? A. Between 2 and 6 cm B. Between 2 and 10 cm C. Between 4 and 10 cm D. Between 4 and 6 cm

between 2 and 10 cm The greatest possible amplitude of the combined wave is the sum of the individual amplitudes: 4 + 6 = 10 cm (completely constructive interference). On the other hand, if the waves experience total destructive interference, then the amplitude of the resultant wave is the difference between the individual amplitudes: 6 — 4 = 2 cm. So the amplitude of the resultant wave is between 2 cm and 10 cm

Suppose that the railcar passes by a horn that is emitting a sound with frequency f. Which of the following describes the frequency f' that the person on the railcar hears?

f' > f before passing the horn, f' < f after passing it -due to the Doppler effect, the frequency that the person on the railcar hears before passing the horn is larger than the actual frequency of the sound emitted, while the person hears a frequency lower than the actual frequency after passing the horn.

frequency

f, the number of cycles per second

pitch =

frequency - think cochlea

If you move five times further from the source of a sound, how does the amplitude of the pressure wave change?

it decreases by a factor of 5 I : /r^2 I : A^2

As you speed away in your spaceship from an imploding star, the pitch of the sound you hear from the implosion: A. no sound can be detected. B. decreases. C. increases. D. stays the same.

no sound can be detected. If the Doppler effect was applicable, then the frequency of the sound you detect should be lower than the frequency of the sound emitted. Thus, "decreases" would be the correct answer except that the Doppler effect does not apply. One must remember that space is a vacuum, and sound cannot travel through a vacuum because there are no molecules to be compressed and rarefied.

What we call the earth's North Magnetic Pole, which lies close to the earth's geographic North Pole, is actually a south magnetic pole. If someone in the continental United States observes a permanently magnetized rod floating on the surface of a fluid and able to rotate freely, that rod will most likely align __________. A. parallel to the magnetic field with its south magnetic pole pointing north B. parallel to the magnetic field with its north magnetic pole pointing north C. perpendicular to the magnetic field with its south magnetic pole pointing east D. perpendicular to the magnetic field with its north magnetic pole pointing east

parallel to the magnetic field with its north magnetic pole pointing north. Opposite magnetic poles attract and like magnetic poles repel, so the north magnetic pole of the magnetized rod is attracted to the south magnetic pole that lies near the North Pole, while the south magnetic pole of the magnetize rod is repelled. This creates a torque that aligns the rod parallel to the field (eliminating the perpendicular alignment choices) with the rod's north magnetic pole pointing north.

Standing wave equations 2 for two open ends

standing-wave frequencies for two open ends

Standing wave equations for two fixed ends

standing-wave wavelengths for two fixed ends

A resistor, a fully charged capacitor, and a switch are all connected in series, with the switch initially open. The switch is then closed, allowing the capacitor to discharge. During this discharging process, which of the following quantities DECREASES? I. The current through the resistor II. The energy stored in the capacitor III. The power dissipated by the resistor

the current through the resistor, the energy stored in the capacitor, and the power dissipated by the resistor DECREASES. As the capacitor discharges, the voltage across its plates decreases according to V = Q/C. Because the voltage drop across the resistor must equal the capacitor's voltage, VC - VR = 0, the voltage across the resistor correspondingly decreases. Therefore, by V = IR, so does the current across the resistor, meaning I is true. The power dissipated by a resistor is given by P = I2R, so as the current decreases, so does the power dissipated across the resistor, meaning III is true. Moreover, as the charge of the capacitor decreases, so does the potential energy stored, according to PE = Q2/2C. Thus II is true.

if both ends of a pipe are open

the length of the pipe, L, must be a whole number of half-wavelengths: L=1(wavelength/2), 2(wavelength/2), 3(wavelength/2)

if one end of a pipe is closed

the length of the pipe, L, must be an odd number of quarter wavelengths: L1(wavelength/4), L3(wavelength/4), L5(wavelength/4)

Which of the following is true about a charged particle moving perpendicular to a uniform magnetic field? A. The magnetic force causes the particle to move in the direction of the field. B. The particle moves through the magnetic field with a constant velocity. C. The magnetic force provides the centripetal force necessary for the particle to undergo uniform circular motion. D. The magnetic force provides the centrifugal force necessary for the particle to undergo uniform circular motion.

the magnetic force provides the centripetal force necessary for the particle to undergo uniform circular motion. This requires an understanding of the basic concepts that charged particles behave under the influence of the magnetic field. The right hand rule tells us that the direction of the magnetic force, the velocity of the particle, and the force exerted on the particle are all perpendicular to each other. Thus, the velocity of the charged particles undergoes constant change under a constant perpendicular force, resulting in a perfect circular motion in the magnetic field. Comparing centrifugal and centripetal forces, keep in mind that the centrifugal force is a phenomenon created through the presence of a non-inertial frame of reference, not a real force. Therefore, the magnetic force must act through the centripetal force for charged particles to undergo uniform circular motion.

Wave rule 1: the speed of a wave is determined by

the speed of a wave is determined by the type of wave and the characteristics of the medium NOT by the frequency - waves v is constant in a medium f : 1/wavelength - exception to this rule is dispersion

Nine capacitors of capacitance C are connected in series with a battery of voltage V and resistor of resistance R. After the capacitors have been allowed to charge fully, a tenth capacitor of capacitance C is added in series. After this capacitor too is allowed to charge fully, which of the following quantities has decreased? A. The capacitance of the first capacitor B. The total voltage drop across all of the capacitors combined C. The total charge stored in the circuit D. The equilibrium current through the resistor

the total charge stored in the circuit has decreased. Capacitors in series add reciprocally, just like resistors in parallel, so the equivalent capacitance of the circuit decreases from 1/(9C) to 1/(10C) when the tenth capacitor is added. Qtotal = CeqV, so the total charge stored in the circuit has decreased. The other three quantities remain the same (at equilibrium, the current flowing through the resistor is zero).

Nine capacitors of capacitance C are connected in parallel with a battery of voltage V and resistor of resistance R in series with each other. After the capacitors have been allowed to charge fully, a tenth capacitor of capacitance C is added in parallel. After this capacitor too is allowed to charge fully, which of the following quantities has increased? A. The voltage across each capacitor B. The energy stored by the first capacitor C. The total charge stored in the circuit D. The equilibrium current flowing through the resistor

the total charge stored in the circuit has increased. Capacitors in parallel add linearly, just like resistors in series, so the equivalent capacitance of the circuit increases from 9C to 10C when the tenth capacitor is added. Qtotal = CeqV, so the total charge stored in the circuit has increased. The other three quantities remain the same (at equilibrium, the current flowing through the resistor is zero).

wave rule 2 applies (where frequency is constant)

to a single wave in different media

wave rule 1 applies (where v is constant in a medium)

to different waves in one medium

Wave Speed Equation

v (m/s) = wavelength (m) x f (Hz or 1/s) 2 BIG RULES FOR WAVES: 1. speed of a wave in a medium depends on the type of wave (L vs T) and the physical properties of the medium (air, water, steel) NOT BY THE FREQUENCY - v is constant in a medium, regardless of frequency or wavelength (if v is constant; wavelength and f are inversely proportional) 2. When single a wave passes into another medium, its speed changes, BUT ITS FREQUENCY DOES NOT - f is constant between the medium, v and wavelength are directly proportional (frequency is constant as a wave moves from air to water)

Wave rule 2: when a wave passes into another medium

when a wave passes into another medium, its speed changes, but its frequency does not - 1 wave moving from one medium to another, will maintain same frequency v : wavelength