AP Physics: Volume 2 Chapters 9-17 Chapter 9: Solids and Fluids 9.1: States of Matter 3 states: Solid- Definite volume and shape Liquid- Definite volume; changing shape Gases- Neither definite volume nor shape Plasma- #4- A highly ionized substance containing equal amounts of + and – charges. *It exists inside stars. *Plasma is the most prevalent state of matter in our universe. *This is formed by matter being heated to ultra high temperatures resulting in a substance with free electrically charged particles. *Solids are either crystalline or amorphous. *Crystalline Solids- The atoms have an ordered structure. Ex. Salt (Pg. 257) *Amorphus Solids- Atoms are arranged randomly Ex. Glass *Liquids are always at a higher temperature then the solid form of the same substance. The high temperature causes the molecules to wander through the substance randomly. *Gases- Molecules are always in random movement and have only weak nuclear forces on each other. Their separation is quite large; much more so than liquids. They even collide occasionally. 9.2: The Deformation of Solids For this chapter, we will assume that all solids maintain their shapes, regardless of how much force is put upon them. *Stress- The force causing a deformation. *Strain- The degree of deformation. Elastic Modulus = stress strain Tensile Stress- The ratio of the magnitude of the external force (F) to the cross sectional area. The units are: 1N =1 Pa (pascals). m2 Tensile Strain- The ratio of ∆ length to original length L Lo Youngs Modulus- Elasticity in length. Uses stress . strain L FLo F Y = A Lo AL *This is usually used to characterize a rod or a wire under either tension or compression. *Pg. 260: Fig. 9.4 *Exceeded the stress/strain curve goes from straight to almost a square root graph until it reaches the breaking point *Exceeding the elastic limit results in deformation. Shear Modulus- Elasticity of shape S= shear _ stress F x Fh = shear _ strain A h Ax *∆x = horizontal distance the sheared face moves *h = height of object *F = force applied at a tangential direction *A = area of the face being sheared Bulk Modulus- Volume Elasticity B= *Since volume _ stress F V V = P volume _ strain A V V F = Pressure, use P instead A *The negative sign is used to make a positive reading possible because an increase in pressure causes a decrease in volume. Ex. +∆P causes -∆V *The reciprocal of bulk modulus is called the compressibility of the material. *Notice on the chart on Pg. 259 that solids and liquids have bulk modulii. However, they have no Youngs Modulus nor Shear Modulus; a liquid will not sustain a shearing stress or a tensile strength because it flows. **DO EXAMPLES ON PG. 261** 9.3: Density and Pressure *The density of a substance of uniform composition is defined as its mass per unit volume. p= m mass = V volume p = Rho (Greek abbreviation for density) *Pg. 262: Table 9.2 Density is kg g or 3 cm 3 m *The density of most liquid and solids varies slightly with temperature and pressure. *The density of gases varies greatly with temperature and pressure changes. *Under normal conditions, densities of liquids and solids are 1000 times that of gas *This implies that the average distance between molecules is 10 times in gases since 103=1000 The specific gravity of any substance is the ratio of its density to the density of water @ 4oC water @ 4oC = p = 1*103 kg m3 *Any force that applies stress to an object in a liquid tends to compress the object. The force exerted on an object in a fluid is always perpendicular to the surfaces of the object. Recall: P = F A P = pressure **DO TACK LAB ON PG. 263** *Place between thumb and forefinger and squeeze *Fthumb = Fforefinger per newtons 3rd law *Why does only sharp end hurt? P = F A *If force and pressure are equal, than area is the difference. *The same goes for baseball spikes or steal pointy bottom spikes. They have a small area on which to apply your weight onto the floor. This is why snowshoes keep you above snow. Ex. A waterbed is 2 m x 30 cm x 2 m a. Find its weight. Pwater = 1000 kg m3 Vbed = (2*2*0.3) m3 = 1.20 m3 M = pV = (1000 kg ) (1.20m3) = 1.2*103 kg m3 Fweight = Mg = (1.2*103 kg) (9.8 m ) = 1.18*104 N 2 s Note: 1.2*103 kg = 2640 lbs (Check for yourself.) b. Find the pressure the waterbed (full) exerts on the floor. From part “a”… w = 1.18*104 N and cross-sectional area (touching floor) = 2 m * 2 m = 4 m2 1.18 *10 4 N 2.95 *10 3 2 = 2.95*103 Pa P= 2 4.0m m c. Find the pressure if the bed is on its side. P= 1.18 *10 4 N = 3.9*104 Pa 2 * 0.3 d. Is the pressure the same if its stood on end as it is when its sat on its side? Why? 9.4: Variation of Pressure with Depth Normal atmospheric pressure at sea level is Po = 1.01*105 Pa = 14.7 lbs and the above in 2 equation tells us that at a depth (h) below the surface of a liquid (open to the atmosphere) is greater than Po by pgh. *Recall from last year Pascal’s Principle: 1. Pressure in a fluid depends on depth 2. Any increase in pressure at the surface is distributed evenly throughout the fluid. *Remember that P = F1 F2 A1 A2 *This is why a hydraulic jack the size of an air pump lifts a car. **SEE TOP OF PG. 266 – EGYPTIAN LEVEL** 9.5: Pressure Measurements Ex. Assume body area of 2000 in2 *Find atmospheric pressure (ap) on body. Answer is approximately 3*104 lbs Ex. Car Lift- Compressed air lift, exerts force on: *1st Piston (P1) Radius = 5 cm *2nd Piston (P2) Radius = 15 cm What force must be put on P1 to raise a car W = 1.33*103 N? A F1 = 1 A2 (5 *10 2 ) 2 (1.33 *10 4 N ) = 1.48*103 N F2 2 2 (15 *10 ) What air pressure produces this force? P= F1 1.48 *10 3 N = 1.88*105 Pa 2 A1 (5 *10 )m Note: Only 2 atmospheres. Ex. Ocean pressure @ 1000 m depth Pwater = 1*103 kg m3 Po = 1.01*105 Pa P = Po + Pgh P = 1.01*105 Pa + (1*103 kg m ) (9.8 2 ) (1.0*103 m) 3 s m P ~ 9.9*106 Pa What is the force on a 30 cm window at this depth? **ASSIGNMENT page #300 # 2, 3, 5, 18,19 9.6: Buoyant Forces- Archimedes’s Principle *Any body partially or completely submerged in a fluid is buoyed up at a force with magnitude equal to the weight of the fluid that the body displaces. B = pfVg = Mg = Wf B = buoyant force pf = density of fluid V = volume of cube g = gravity M = mass of fluid Wf = weight of fluid displaced Weight of submerged object is given by: Wo = Mg = poVg But: Wf = pfVg And: Wo = poVg *These two formulas lead us to the conclusion that if density of the object is greater then the density of the fluid, it sinks. If the density is less than the fluid, it floats. *If the magnitude of the buoyant force equals the weight of the object, only part of the object will be submerged. *Floating Object: po VF p F Vo **TURN TO PG. 270 AND TALK ABOUT THE BRAIN FLOATING IN SPINAL FLUID** **First, do this problem: A raft (wood) with p = 600 kg , volume of 0.6 m3, and surface area of 5.7 m2, is m3 placed in fresh water. When pw = 1000 kg , how much of the raft is below the water m3 line? Weight of the Raft: Wr = pr g Vr = (600 kg m ) (9.8 ) (0.6 m3) = 3.5*103 N 2 3 s m Upward Force on the Raft Equal to Wr: B = pw g Vw = pw g Ah = 3.5*103 N *Solve for h: Nkgm 3.5 *10 3 B s2 h= = = 0.06 m kg m p w gA 2 1000 3 9.8 2 5.7m m s **NOW DO THE EXAMPLE ON PG. 272, #9.6** Assign #20, 21 on page 300 *Optional from here to end of chapter* 9.7: Fluid Flow pV1A1 = pV2A2 = Equation of Continuity *The fluid flow is constant through any cross section of the pipe. *So…if p = p: V1A1 = V2A2 tells us that if cross sectional volume is reduced, than V *Turn on sink with and without nozzle on to show example. Ex. A hose with a 1.0 cm radius fills a 20 liter bucket. If it takes 1.0 min. to fill, what is Vw as it leaves the hose. *1 liter = 103cm3 *Cross-sectional area of the hose is: A = π r2 = π (1)2 = π cm2 Flow rate = Av (A) Av = 20.0 liters 20 *10 3 cm 3 cm 20 *10 3 cm 3 106 = 3 min s 60 sec( A) 60 sec(cm ) Reduce the hose to 0.5 cm. V = 20 liters cm 20 *10 3 cm 3 = 424 2 s 60 sec(0.25cm ) *Why did it not only double? - Because 22 = 4 Bernoulli’s Equation- Word done by a fluid and work done on one end of a tube (on the fluid) is equal and opposite to work done on the other end. *As a fluid is moved along a pipe, part of the work is done changing KE, and part changing PE. P1 + 1 1 pM1V12 + pgy1 = P2 + pV22 + pgy2 2 2 OR P+ 1 pV2 + pgy = Constant 2 **DO EXAMPLE 9.19 ON PG. 277** *Surface Tension: = ﻻ F L **DO #41,42** *Surface tension is in J N Nm 2 m m2 m *Surface Tension is the energy content of the fluid at its surface per unit surface area =ﻻ F 2L *L = circumference of the wire *F = force on spring * = ﻻsurface tension Water Bug- Walks on Water Ex. An insect with a mass of 2*10-5 kg has 6 feet, each with a radius of 1.5*10-4 m. Find the angle θ of depression the legs cause the water to indent. *The length (L) along which the force acts is the distance around the insect’s foot, 2πR. (assume the pressure is equally distributed to all 6 legs). F = ﻻ2L = ﻻ2πR We want the vertical force: Fv = ﻻ2πRcosθ Distributed to 6 legs Fv = 1 w 6 *γ = surface tension of water = 0.073 N (chart on Pg. 281) m 1 1 w mg 2R cos 6 6 1 mg 6 cos 2R 2 1 m 2 * 10 5 kg 9.8 6 s N 2R 0.073 m cos 62 = θ *Cohesive Forces- forces between like molecules (water on water). *Adhesive Forces- forces between unlike molecules (glass on water). *Water climbs (clings) the glass on the edge because the adhesive forces between the class and water are greater than water on water. *Mercury curves down at the glass because the cohesive forces between the mercury atoms are greater than the adhesive forces between mercury and glass. Capillary Action: F=ﻻL=2πRﻻ Vertical component of this force is: Fv = ﻻ2 π R cosф In order to be in equilibrium: Fv = Wcylinder of water at height(h) W = Mg = pVg = p g π R2 h Therefore: Fv = W ﻻ2 π R cosф = p g π R2 h h = 2 R cos pgR 2 h If the angle ф = 0, then: h= 2 pgR Ex. Find the height the water rises up the tube with a radius of 5*10-5m. O ф = very small use ф = 0. N 2 0.073 m h= = 0.29 m m 3 kg 5 1 *10 3 9.8 2 5 *10 m m s **DO # 43, 45-47 ON PG. 302** Chapter 10: Thermodynamics 10.1-10.3 Recall- the exchange of energy between two objects that produces a change in temperature is called heat, i.e. Thermal Energy F = Fahrenheit C = Celsius γ = 2α (usually) γ = avg. coeff. of area expansion. Lo = original length α = alpha = coeff. of linear expansion β = beta = coeff. of volume expansion Vo = original volume β = 3α (usually) Chart on Pg. 319 K = Kelvin Conversions: K = C + 273.15 F = 9/5 C + 32 C = 5/9 (F – 32) 10.3 ΔL= α LO ΔT Therefore: new length For area of square objects use: L = L O + ά LO ΔT A = L2 = (Lo + α Lo ΔT)(Lo + α Lo ΔT) = Lo2 + 2 α Lo2ΔT + α2 Lo2ΔT2 But squaring the coefficient of linear expansion creates such a small value we can ignore the last term Therefore: A = L2 = Lo2 + 2 α Lo2ΔT = Ao + 2 α Ao ΔT ΔV = β Vo ΔT New: V = Vo + β Vo ΔT Ex. A cross sectional area of a hole in a piece of steel is 100cm2 at 20˚C. What is the area of the hole if heated to 100˚C? ΔA = γ Ao ΔT = ( 22x10-6 C-1)(100cm2)(80˚C) = .18cm2 Therefore: new hole = 100cm2 + .18cm2 = 100.18cm2 Now try the above using the whole equation for A = L2 including the last term with the coefficient of linear expansion squared to show why we disregard this term. Look at Pg. 322. It talks about the behavior of water from 0˚C → 4˚C Read pgs. 322-323. Do the thinking physics 2 **Assign Pg. 335 # 3,8,12,18,19 10.4 PV= nRT ←ideal gas equation Also PV= nRT can be used as Pf Vf Tf = R = universal Gas constant n = number of moles P = pressure V = volume T = Temperature in Kelvins R = 8.31 J/(mol)(K) = 0.0821 (L)(atm)/(mol)(K) ← in atmospheres Pi Vi Ti Do number 25 on the board ** Assign Page 336 # 21, 22, 24, 28 10.5-10.6 PV= nRT can be rewritten to include kb called Boltzmann’s constant Kb = R Na Na = 6.02 x 1023 molecules/mole (Avogadro’s number) N =N PV = N kb T Na R = 8.31 J/(mol)(K **SEE PAGE 328 FOR HOW AVOGADRO’S kb = 1.38 x 10-23 J/K NUMBER WAS FOUND Kinetic energy of a gas molecule is directly proportional to the absolute temperature of the gas. So we can say: KE = ½mv2 = 3/2 kb T And Vrms of a molecule of gas is ______ ______ V = √3 kb T = √3 RT m M ** Assign Page 336 # 33, 36, 38, 41 Chapter 11: Heat 11.1: Mechanical Equivalent of Heat Recall – heat flows from hot to cold until equilibrium is reached - Heat gained or lost is equal to the work done on or by the system - 1 J (Joule) is equal to the amount of work required to move one Newton one meter - 1 cal (calorie) is equal to the amount of energy (heat) needed to raise 1 gram of water to 1˚C (if the calorie is written as a capital C is understood as 1000cal or 1 kilocalorie (this is one calorie in food)) - Now we will use Joules in heat and the conversion is: 1 J = .239 cal or 1 cal = 4.186 J This is known as the mechanical equivalent of heat. *BTU (British thermal unit) = heat needed to raise 1 lb of water 1˚F. Ex. If Colin consumes 2000 calories of candy and Pluta finds out he makes him work it off lifting 50kg masses in the weight room. How many times must he lift the mass 2m? Remember 1C = 1000 cal So 2000 food calories is 2000C Converting to cal gives (2000C) (1000cal) = 2 x 106 cal 1C W = (2 x 106 cal)(4.186 cal) = 8.37 x 106J J Because W = mgh Total work will use N for the number of times he has to lift the weight therefore W = nmgh Solving for N n = W = 8.37 x 106J = 8.54 x 103 times mgh (50kg)(9.8 m/s2)(2m) Moral of the story, Colin don’t eat candy. Question: How many hours would this take if Colin lifts the weight once every 10 seconds. Answer: apx. 24 hours 11.2: Specific Heat Q = mcΔT You should recall this from last year. 11.3: Conservation of energy (Calorimentry) table of “C’s” on Pg. 342 Q`w + Q`m = Q``w + Q``m Heat of water plus heat of material before hand is equal to heat of the two substances after. This assumes NO heat loss to surroundings. (yes we are going to do a calorimeter lab) mwCwTiw + mmCmTim = mwCwTfw + mmCmTfm Ex. A blonde dear hunter fires a 2g silver bullet @ 200m/s from her 30-06 misses a 10pt buck and hits the pine walls of the outhouse. (Not to worry, her brother was all that was in the outhouse at the time.) Assuming that the bullet retains all the energy (heat) it generates on impact, find its temperature change. KE = ½mv2 = ½ (2 x 10-3kg)(200m/s) = 40 J = Q Q = mCΔT → ΔT = Q = 40 J mC (2 x 10-3kg)(234J/kg C) All this ΔT is due to friction = 85.5˚C Q: What would the ΔT be if Wendy used lead bullets instead of silver? A: 157˚C thus lead would cause twice as much friction due to its different specific heat. Assign Pg. 363 # 1, 2, 5, 8, 11, 15, 18 11.4: Latent Heat & Phase changes This book uses L (a capital L for heat of fusion and heat of vaporization rather than H as last years book did) i.e. L = latent means heat constant Thus Lf Latent heat of fusion Lv = Latent heat of Vaporization We do the problems the same way i.e. if ice @ -30˚C to steam @ 120˚C warm ice Q = mCΔT fuse ice Q= Lf m warm water Q = mCΔT vaporize water Q = Lv m warm steam Q = mCΔT Heat total = Σ (sum) of the heats Try this for 1g of ice A: 3110 J Cooling just yields a negative answer because it gave off heat. Ex. 11.5 Pg. 349 Liquid helium with a boiling point of 4.2 K and a Lv = 2.09 x 104 J/kg, and is supplied a constant power of 10w (watts) (1w = 1 J/s) from an immersed electric heater. At this rate, how long would it take to boil away 1kg of the liquid helium? Since Lv = 2.09 x 104 J/kg this is how much thermal energy it will take to boil away 1kg and since P = w/t and the power is 10w = 10 J/s T = w = 2.09 x 104 J = 2090 seconds P 10 J/s Try this for liquid nitrogen (same 1kg) and for water @ 100˚C (1kg) how long would it take? For nitrogen: apx. 5.6 hours For water: apx. 2.617 days Assign Pg. 363 20, 22, 26, 29a 11.5: Heat Transfer By Conduction (touching) The heat transfer rate (heat of current) is H (capital H) H= Q ΔT Note Q = Joules T = seconds so heat H is J/s watts For heat flow from one side of a piece of material to another use: H = k A (T2 – T1) L k = Constant of thermal Conductivity L = Thickness of material A = Cross-Sectional Area T2 = Higher Temperature T1 = Lower Temperature k = Large for metals lower for non heat conduction H = Heat Transfer Rate See chart on Pg. 351 Ex. For a concrete basement wall 2m high 3.65m long and 20cm thick find the amount of heat transferred through the wall if the inside temperature is 20˚C and the outside temperature is 41˚F in one hour. 41˚F = 8˚C T2 = 20˚C T1 = 8˚C k = 1.3 J/s · m· C˚ A = (2m)(3.65m) = 7.3m2 T = 1 hour = 3600 seconds Solving for Q Q = k A ΔT (T2 – T1) = L = (1.3 J/s · m · C˚)(7.3m2)(15˚C) = 2.6 x 106J .2m Heat loss through walls Q = A (T2 – T1) ΔT ΣR ΣR = Sum of the R values of the materials used to construct the walls values on page 352 Assign 30-33 11.6: Conduction – Heat transferred By the movement of a heated substance, i.e. forced air heat, hot water heat. 11.7: Radiation Warming w/o contact nor conduction, i.e. warming hands by a fire. No contact and conduction would be upward not outward (heat rises it does not move horizontally through conduction unless forced) Stefan’s Law P = θ AeT 4 Pnet = θ Ae (T4 – To4) P = Power (in watts) e = Constant of emissivity A = Surface area (in meters squared) θ = Constant = 5.6696 x 10-8 w/m2 · K4 T = Objects Temperature in Kelvins To = Temperature of Surroundings Anything that absorbs all the energy incident upon it is called an ideal absorber. Its emissivity is said to be equal to unity. It is also known as a Blake Body. By contrast anything with an emissivity of zero, absorbs no energy it, reflects all back off, is known as a perfect reflector. A person with surface area of 1.5m2 has a temperature of 98.6˚F in a 20˚C room. If they removed all clothing how much heat would be lost in 10mins. Pnet = θ Ae (T4 – To4) = = (5.67 x 10-8 w/m2 · K4)(1.5m2)(.9)(310K4 – 293K4) = 1.43 x 102 J/s In 10 mins. Qtotal = (Pnet)(Time) = (1.43 x 102 J/s)(600s) = 8.6 x 104 J .Assign 37-39 Test on Chapters 9, 10, 11 in 2 days eskin = .9 Chapter 12: The Laws of Thermodynamics 12.1 & 12.2: Heat and Internal Energy & Work and Heat *Internal Energy- All energy in a system that is stationary- includes heat, nuclear, chemical, and strain (stretched / compressed spring) energy. *Thermal Energy- The portion of internal energy that changes when temperature of system changes. *Heat Transfer- Caused by a temperature difference between the system and surroundings- Glass, Ice. *Recall the thermal energy of a monatomic gas is associated with the motion of its atoms. This thermal energy is just kinetic energy on a microscopic scale. Temperature ↑ Ice ↑ Thermal Energy ↑ *But… thermal energy also includes rotational, vibrational, and potential energy on the molecular level. **GO OVER * PARAGRAPH ON GAS/WORK ON PG. 374** W = F∆y = PA∆y *F = force *∆y = change in height *P = pressure *A = cross sectional area *W = work Remember if the piston moves: ∆V = A∆y *∆V = change in volume *Positive work is done by the system. *Negative work is done on the system. *Graphically, the work done from Vo to Vf is the area under the curve on a PV graph. *W = P (Vf – Vo) *If it’s a curved line, you must find the equation of the line and integrate from a = Vo to b = Vf Integral*dx *Adiabatic Free Expansion- NO heat is transferred through the insulated walls of a system. Thus the temperature of the system remains constant. No heat transfer = No work done. *Heat transfer and work depends on the path between the initial and final states. 12.3: The First Law of Thermodynamics *∆U = change in internal energy of a system *Isolated systems have no change in internal energy because they allow for no heat transfer, thus Q = W = O and ∆U = O Uo = Uf. *In a cyclic process (engine), the system begins and ends at the same state. Internal energy doesn’t change so +heat = work during ∆U = O but Q = W. Thermal Dynamics ∆U = Uf – Uo = Q – W *U = internal energy function *+Q = heat transferred into system *+W = work done by system *The First Law of Thermodynamics is essentially the principle of conservation of energy generalized to include heat as a mode of energy transfer. *Isobaric Process- Pressure remains constant and work done/heat transferred are both ≠0. Ex. A gas in a cylinder with a piston has a cross-sectional area of 0.10 m2, and a constant pressure of 8000 Pa. It has heat slowly added to it, resulting in the piston being pushed up 4.0 cm. If 42 J of heat is added to the system, what is the change in internal energy of the system? Work done by gas: W = P∆V = (8000 Pa) (0.10 m2) (4*10-2 m) = 32 J *According to the First Law of Thermodynamics, the change in internal energy is ∆U = Q – W = 42 J – 32 J = 10 J *If you added 42 J to a system without allowing the piston to move, then you would have W = 0 and ∆U = 42 – 0 = 42 J **LOOK AT EXAMPLES ON PG. 380-381 AND DO #2, 7, 11, 15, 20** 12.4: Second Law of Thermodynamics It is impossible to construct a heat engine that, operation in a cycle, produces no other effect than the absorption of heat from a reservoir and the performance of an equal amount of work. W = Qh – Qc = Qnet *Qh = heat from a hot reservoir *Qc = heat from a cold reservoir Qh > 0 Qc > 0 *Because it cycles ∆U = 0, therefore the net work done by a heat engine equals the net heat flowing into it. *Efficiency = e e=1– Qc Qh **DO EXAMPLE 12.6 ON PG. 383** Second part is e = 1 – Qc / Qh 0.20 = 1 – 3000 J Qh Qh (-0.80) = 3000 J 0.80 Qh = 3750 W = Qh – Qc = 3750 – 3000 = 750 J **GET OUT THERMO COIL AND PUT INTO CUP OF HOT WATER TO DESMONSTRATE CONVERSION OF HEAT (THERMAL ENGERGY) TO KE** **DO # 24, 26, 28, 36, ON PG. 398-399** Chapter 13: Vibrations and Waves 13.1: Hooke’s Law Fs Kx *It is negative because the force exerted by the spring is always directed opposite the displacement of the mass. *The direction of the spring’s force (restoring) is such that the mass is either being pushed or pulled towards the equilibrium position. *Simple harmonic motion occurs when the net force along the direction of a motion is a Hooke’s law type of force. (Ex. Net force is proportional to displacement and in the opposite direction). *Amplitude (A)- max distance traveled by an object from its equilibrium position *Period (T)- time (seconds) it takes to execute one complete cycle of motion *Frequency (F)- # of cycles per unit of time (or # of vibrations) Fs Kd mg K mg We will use in lab to measure spring constants d *Acceleration can be used instead of g for use on a horizontal surface. F Kx ma a K x m 13.2: Elastic Potential Energy *Elastic Potential Energy- energy stored in a stretched or compressed spring. (Ex. Cannon) PE 1 2 Kx 2 *Because we always square the distance x, the PE’s must always be positive. KE PE g PEs i KE PEg PEs F *If friction is present, then final energy does not equal initial energy, then work (heat) was done- this work is the result of the final energy minus the initial energy. NC- Non Conservative Forces: WNC KE PEg PEs F KE PEg PEs i 13.3: Velocity as a Function of Position *At maximum extension = A, the total energy of a spring system is *Anywhere between A and equilibrium Total Energy = KE PE 1 1 1 KA 2 mv 2 Kx 2 2 2 2 *x = present position *A = max position 1 KA 2 2 Solving for V: V K 2 A x2 m **GO OVER EXAMPLES ON PG. 410-413** **DO # 1, 3, 5, 6, 9, 12, 14, 16 ON PG. 434** 13.4: Comparing Simple Harmonic Motion with Uniform Circular Motion T 2 L m 2 g K V C A2 x 2 *V = velocity *C = constant **SEE PG. 414, FIGURE 13.9** A2 x 2 *A ball moving at a constant velocity (Vo), tangent to the circular path at this instant (Vball) in the x-direction is: V Vo sin sin sin V Vo V Vo A2 x 2 A A2 x 2 A Therefore: V Period: Vo A A 2 x 2 C A2 x 2 V0 C A 2A T Vo T 2A Vo 1 1 2 KA 2 mVo 2 2 A Vo T 2 f m K m K 1 1 T 2 K m *Where units of frequency are s-1=Hertz angular frequency is omega ( ) 2f K m **DO EXAMPLE 13.6 ON PG. 417** **DO # 14, 16, 17, 19, 20, 25, 28 ON PG. 436** 13.5: Position, Velocity, and Acceleration as a Function of Time x A cos Because: cos x A *Since we will use a constant rotational velocity, we can say that t Therefore: x A cos(t ) *In one revolution the ball rotates 2π radians in the period (T). Therefore: 2 2f T T *f = frequency of rotational motion *But the angular speed of rotation on the reference circle = angular frequency of simple harmonic motion. Therefore: x A cos( 2 ft) *We will use this equation to represent the position of an object moving with simple harmonic motion as a function of time. The curve of this graph is a trig. function (sinusoidal). *In each case below, the object was released from rest at its maximum distance from equilibrium. *x = displacement *V = velocity *a = acceleration Pg. 436, #25 PROBLEM: A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 7.50 N is applied. A 0.500-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is pulled horizontally so that it stretches the spring 5.00 cm and is then released from rest at t = 0. (a.) What is the force constant of the spring? (b.) What are the period, frequency, and angular frequency ( ) of the motion? (c.) What is the total energy of the system? (d.) What is the amplitude of the motion? (e.) What are the maximum velocity and the maximum acceleration of the particle? (f.) Determine the displacement, x, of the particle from the equilibrium position at t = 0.500 s. SOLUTION: (a.) By Hooke’s Law: F Kx K F 7 .5 N 250 N/m x .03m (b.) Angular Frequency (omega) is: K m *K = spring constant *m = mass attached 250 N = 22.4 Rad/s 0.5kg And Period is: T 2 2 22.4 Rad = 0.281 sec. s And Frequency is: f 1 1 = 3.56 Hz T 0.281 (c.) Total Energy of System: E Eo E 0 1 1 2 2 mVo Kxo 2 2 1 250 N 0.05m 2 m 2 E = 0.313 J (d.) At maximum displacement from equilibrium, x = A = Amplitude. Plus, it is at rest (for and instant) implying all the energy is elastic potential energy. Therefore: 1 E 0 KA2 A 2 2(0.313 J ) 2E K 250 N m E = 0.05 m (e.) Vmax is at equilibrium (x = 0) where all the energy is kinetic Therefore: E 1 2E 2 mVmax 0 Vmax 2 m 2(0.313 J ) 1.12 m/s 0.5kg *Note that from Newton’s Second Law, the maximum acceleration of the particle occurs when the spring exerts maximum forces on it (when spring is stretched or compressed to its maximum). x A *Therefore: Qmax (f.) Since N Fmax KA 250 m 0.05m = 25 m/s2 m m 0.5kg K and our object starts at rest where x A and T 0 , the m displacement at any time is: K x A cos(T ) A cos T m *So, at T 0.5 sec onds : 250 N m = 0.92 cm x 0.05m cos 0.5 sec 0.5kg *Done in radians. **DO # 23, 24, 26 ON PG. 436** 13.6: Pendulums If 15 o , pendulums exhibit simple harmonic motion and the resultant force acting on the mass is equal to the component of the weight tangent to the circle. Its magnitude is mg sin . Since this force is always directed towards 0 , it is a restoring force. Ft mg sin restoring force *If we use radians with angles less than 15o, the radian measure is approximately equal to sin . *Therefore, for all 15 o , use: FT mg Hooke’s Law F KS , we can also say that S L FT mgS L Also use last year’s formulas for pendulums: T 2 K spring m T 2 L pendulums g **LOOK AT EXAMPLE 13.8 ON PG. 421** S L **DO PROBLEM # 31 ON PG. 437** a.) Period of pendulum: L gt 2 T 2 L g 4 2 Earth: L (9.8)(1) 2 0.25m 4 2 (3.7)(1) 2 Mars: L 0.094m 4 2 b.) Period of vibration of mass on spring: T 2 m KT 2 m K 4 2 10 N 1sec m 2 4 2 0.25kg Since this formula does not use gravity, the mass on Earth = mass on Mars. **DO #27-30 ON PG. 437** 13.7, 13.8, & 13.9: Damped Oscillations, Wave Motion, and Types of Waves *Damped Oscillations- This occurs in real systems where friction is present. They reduce amplitude, acceleration, velocity, period, frequency, etc. *Wave Motion- To produce mechanical waves, you must have: An elastic medium to disturb. An energy source to provide the disturbance. A physical mechanism that allows adjacent portions on the medium to influence on another (transfer the wave- slinky). *The 3 Parameters Important in Characterizing Waves: 1. Wave Length 2. Frequency 3. Wave Velocity *3 Types of Waves: 1. Transverse- Particles of disturbed medium move perpendicular to direction of wave velocity (slinky shook side to side). 2. Longitudinal- Particle’s displacement is parallel to direction of wave motion (slinky compressed (section) and released). 3. Surface Waves- A combination of both- circular motion. *There is a 4th type of wave which the book does not consider: 4. Electromagnetic- Never cross; always in same direction- more about these in February. 1310 & 13.11: Frequency, Amplitude, and Wavelength & The Speed of Waves on Strings *Wavelength is Lambda: *Amplitude: A *Hertz 1 sec ond *Period (1 vibration): T V x f t T **DO EXAMPLE PROBLEM #13.9 FROM PG.427** *Wave length of 0.4m *Amplitude of 0.15m A *Frequency of 8.0Hz *Find velocity 1 1 0.13 sec. f 8 Period: T Speed: 1 V f (8 Hz )(0.4m) (0.4m) 3.2 m/s 8 sec **DO #32-37 ON PG. 437** Also V F *F= Ftension on string * = linear density of string (mass per unit length) **DO EXAMPLE PROLEM #13.12 ON PG. 429** **DO #38-40 ON PG. 437** Chapter 14: Sound 14.1 & 14.2: Producing a Sound Wave & Characterizing a Sound Wave *Sound waves are longitudinal waves traveling through a medium, usually air, the motion of the wave particles is back and forth along the wave. *High Density- compression of condensation *Low Density- rare faction *Categories: - infrasonic waves < 20 Hz (long) - audible waves 20 20,000 Hz (long) - ultrasound waves > 20,000 Hz (long) *Ultrasound- Transforming electrical energy into mechanical energy is called the Piezoelectric effect. 2 t *100 PR i i t * i = density of fluid * t = density of object (kid) 14.3: Speed of Sound *Speed of sound in air @ 0oC at sea level is 331 m/s (humidity can affect it too). *The sensation of sound is approximately logarithmic in ear. *The relative intensity of a sound is the intensity level called decibel level. *Threshold of human hearing is 1 * 10 12 w/m2 *Threshold of pain for humans is 1 w/m2 *Recall from Chapter 9 the formula for Bulk Modulus. stress P B V strain V * P = change in pressure * V = volume change * V = original volume *The speed of all mechanical waves is given by: V elastic ( properties ) inertial *In a solid rod, the speed of a longitudinal wave (go in the direction) of the particle (along rod). Solids: V y * y = Young’s Modulus of a solid * = density of material Liquids & Gases: B V * B = Bulk Modulus V 331m T s 1 273 * T = temp. in Celsius * V = velocity in air V f *Recall that: **DO #3 ON PG. 476** *Range of human hearing is from 20 20,000 Hz. Find the wavelengths at 37oC. *First find V: V @ 27 oC V 331 m 27 s 1 273 347 m/s *Now find wavelength: 347 m V s 0.01735m 17 cm s f 2 * 10 4 Hz L m V 347 s 17 m f 20 Hz **DO PROBLEMS #1-7 ON PG. 476** 14.4: Energy and Intensity of Sound Waves *The intensity of a spherical wave produced by a point source is proportional to the average power emitted and inversely proportional to the square of the distance from the source. I P A * P = power * A = area (d2) * I = intensity I I o 10 Log * = intensity level (in dB= decibels) * I = 1*10-12 w/m2 & reference & intensity level *Recall that 1*10-12 = min. for human hearing, and that 1 w/m2 = threshold for pain. Ex. A noisy grinder in a factory produced sound at 1*10-5 w/m2. Find the decimal level of this machine. Then calculate what would happen if a second machine was added. 1 *10 5 w 2 m 10 log 10 6 70 dB 10 log 1 * 10 12 w 2 m 2 * 10 5 w 2 m 73 dB 10 log 12 w 1 * 10 m2 **DO PROBLEMS #8,10,11 ON PG. 476** 14.5: Spherical and Plane Waves *When a small spherical object oscillates so that its radius changes periodically with time, it will produce a spherical sound wave. This wave moves outward from its source at a constant velocity (like dropping a stone in a pond) only in 3 dimensions. * PAV surface area of a sphere 4R 2 *I = intensity (this assumes no absorption into the medium) I P averagepower PAV AV 2 area A 4R *Since the intensity of the wave will decrease the further it gets from the source (called a point source), the intensity will be greater at 1 then at 2. A, B, and C are called wave fronts for spherical waves. The distance between A and B is the same as between B and C, or the point source and A. This distance is (lambda) waves. *The further the waves get from the point source, a small portion cut from the wave will have rays almost parallel to one another. This portion is called a plane wave. **LOOK AT EXAMPLE #13 ON PG. 476** *Train horn heard at 50 dB, 10 km away. Find: a) Average power of the horn b) Intensity level of horn’s sound heard by a person 50 m away *Treat horn as a point source with no absorption of the sound by the air. a) 10 log I Io From 14.4 on Pg. 449: *Io = 1*10-12 w/m2 * = 50 dB (given) I 50dB 10 log Io I 5 log Io I I o 10 5 I 1 * 10 12 w 10 m 5 2 I 1*10-7 w/m2 *The train horn is a point source, so A 4R 2 (spherical surface area). Average power emitted is P IA . *Use I and A as same distance from source (R). R 10km 1.0 *10 4 m I 1.0 * 10 7 w m2 Therefore: P 4 1 *10 4 m 1 * 10 7 w m2 1.3*10 w 2 b) at R 50m I P 1.3 *10 2 w 4.1*10-3 w/m2 A 4 50m 2 *Also at 50 m decibel level is 4 *10 3 w 2 I m 10 log 10 log I0 1.0 *10 12 w 2 m 10 log 4.1*10 9 96.13 dB **DO PROBLEMS #12-14 ON PG. 476** 14.6: Doppler Effect *To have a Doppler effect, the needs to be a relative motion between the sound wave source and the observer. Its effect is obvious, as it approaches you the sounds intense increases and vise versa. *Two cases: 1st- Sound stationary observer moving because you are moving through waves, you are hearing additional wave fronts. VoT *V0 = V of observer *T = time * = wave length *Because we are looking for frequency we can solve this for f. f ' f VBSmoving f Vo V f *V = velocity of sound *Vo = velocity of observer * f = frequency with observer *f = frequency of sound wave *Also sound stationary and observer moving away. V Vo f f V *2nd case: source moving- observer stationary Source moving away from person: V f f V Vs Source moving toward observer: V f f V V s *Vs = velocity of source *V = velocity of sound * f = frequency heard *f = frequency of sound in air **SEE SKETCH ON PG. 454** **SEE EXAMPLE FROM PG. 455-456** 14.7: Interference of Sound Waves *Interference of sound waves occurs when waves produced by two different sources overlap. *Constructive Interference- two or more sound waves that combine (waves are timed correctly) to produce maximum results. *Destructive Interference- when waves are ½ off, they will cancel each other out. *To find destructive interference, simply solve for V and move one source 1 2 f away from the observer to attain minimum sound intensity. 14.8: Standing Waves *When a string is vibrated so it oscillates up and down at a constant speed and one end is attached to a stationary object, the wave will invert and return to the oscillating source. *If the waves at one point are exactly opposite, they will cancel (destructive interference) and the line will be flat, a node occurs where two waves have equal, but opposite magnitude. Ex. Net displacement of this spot is zero. *Although there exists no movement at the nodes, halfway between nodes, there is a point where the maximum amplitude occurs. This point is called an antinode. **SEE FIGURE 14.16 ON PG. 459** **DO PROBLEMS #16-19, 24,26 ON PG. 477** *Hint on # 19: If both observer and source are moving, use: f V Vo f V Vs *Convert km/h to m/s first. **DO REVIEW PROBLEMS #49-51,53,54,59,60,63 OF CH. 14 ON PG. 479-480** **QUIZ TOMORROW** Chapter 15: Electric Forces and Electric Fields 15.1, 15.2, & 15.3: Properties of Electric Charges, Insulators and Conductors, & Coulomb’s Law *Recall that like charges repel and unlike charges attract. *All charges originate inside the atom. *Positive charges carried by protons and atomic particles bond permanently inside the nucleus with the neutrons. To change from positive to negative charge, an atom must gain electrons. Particles that revolve around the nucleus in paths called valence shells. The charge on a proton is equal and opposite that of an electron. *Electric charge is always conserved. Ex. Electric charges are transferred, not created. *Robert Millikan in 1909 found that charges were always in full units- not fractional. These units are said to be quanitized as “e” ±e, ±2e, ±3e, etc. *The value of “e” is 1.60219*10-19C. *C is a unit of electrical charge called a Coulomb. *Ways to change stuff: *Conduction- when two objects come in contact with each other and they did not have the same charge. *Induction- no contact- usually just passing close…Ex. Electric Vandegraff Generator. *The Earth is an infinite supply (reservoir) of electrons. *3 Classes: *Semiconductors- controlled magnitude of electrons. *Conductors- electrons move freely *Insulators- not chargeable- no electron movement *If there exists more positive charges on one side of a molecule, that molecule is said to be polarized. This happens because the negative electrons move to the other side due to an object passing near, or a grounding cable being attached. *1 Coulomb= current of one amp. amount of charge that flows through a wire point in one second. *Coulombs Law: F Ke q1 q 2 R2 *Ke = 8.99*109 Nm2/c2 *q1 = charge one *q2 = charge two *R2 = distance between charges e= charge mass electron -1.6*10-19 9.11*10-31 proton 1.6*10-19 1.67*10-27 neutron 0 1.67*10-27 1 6.3 * 1018 protons 1C e + proton e 1.6 *10 19 - electron e 1.6 * 10 19 1 6.3 * 1018 electrons 1C e * 10 6 C 1C mu coulomb *In 1cm3 of copper, there exists 1023 electrons. *F21 = force of q2 on q1 *F12 = force of q1 on q2 *Coulomb’s law is a vector quantity. * F21 F12 in this case- Newton’s 3rd Law Ex. In a hydrogen atom, an electron and a proton are separated by 5.3*10-11 m. What forces do they exert on each other? *Electromagnetic Force: Fe K e e R 2 2 8.99 *10 Nm 9 C2 15..63**1010 19 11 m C 2 2 8.2*10-8 N *Gravitational Force: Fg G m * m2 9.11*10 31 kg 1.67 *10 27 kg 11 Nm 2 6 . 67 * 10 3.6*10-47 N 2 2 2 11 k R 5.3 *10 m *Notice that electromagnetic force is 2*1039 greater, thus the gravitational force is negligible when compared to electromagnetic force- for an electron and a proton. **DO PROBLEMS #1-5 ON PG. 514** *Now for the superposition principle: finding the electromagnetic force on a charge when more than two charges are present. Ex. Three charges (points) at the corners of a triangle. F12 8.99 *10 9 5 *10 C 6 *10 C 3*10-6 N 9 9 0.3m2 *Q1 and Q2 are positive, therefore Q1 goes left. F13 8.99 *10 5 *10 3 *10 1.35*10-5 N 9 9 9 0.1m2 *Q1 = positive, Q3 = negative, thus charge goes down towards Q3 **FIND THE RESULTANT— YAY VECTORS!!!** FR tan 1 F12 2 F13 2 1.38 *10 5 N F13 77.5o 259.5o F12 FR = 1.38*10-5 N @ 257.5o **LOOK AT PROBLEM 15.3 ON PG. 493** Ex. Two positive and one negative change with all three charges along a line (on the axis) *Place Q2 at the origin. *Find where Q3 (a negative charge) must be placed so there exists a negative force of zero. q1 15C q 2 6C F31 K e F32 K e q3 1.5 *10 5 C 2 x 2 q3 6 *10 6 C x2 *Since the magnitude of the resultant on Q3 = 0, F31 = F3C, so set them equal to each other and cancel out the Ke and Q3. 1.5 *10 C 6 *10 C 5 2 x 2 6 x2 *Also multiply both sides by 106 15 2 x 2 6 2 15 x 2 62 x 2 x 15 x 6 2 x 15 x 2 6 6 x 21x 2 6 x 0.77 m **DO PROBLEMS #6-8,10,12,13 ON PG. 515** *A piece of aluminum foil with a mass of 5*10-2 kg is suspended in an electric field directly vertically. If the charge on the foil is 3C , find the strength of a field that will result in FT on the string of zero. ft m fe = qE fg = mg fy = T + fe – fg = 0 T = fg - fe = mg therefore… T = 0 mg = qE E = mg/q E = mg/q = (5x10-2)(9.8m/s2)/(3x10-6c) = 1.6x105 N/c Assignment:15, 17, 20, 22, 24 15.5. Electrical Field Lines *Relationship of the field and the field lines… 1) The electric field vector E is tangent to the electric field lines at each point 2) The number of lines per unit area through a surface perpendicular to the lines is proportionate to the strength of the electric field in as given region. Therefore, E is larger when the field lines are close together and small when they are far apart. Drawing field lines! 1) Lines must begin at the positive (+) charge (or infinity) and must terminate on negative charges or, in the case of excess positive charges, they end at infinity. 2) The number of lines drawn from one charge (+) to another(-) is proportionate to the magnitude of the charge. 3) No two field lines can ever cross one another!! + - Ex. #29 on page 517 a) Sketch the field lines around an isolated point charge q>0 (+). b) Now sketch around an isolated negative field when 2q<0. a) b) + - Notice… 1) The field lines in both cases exhibit radial symmetry. 2) Lines originate at positive or infinity and terminate at negative or infinity. 3) The number of lines in ‘a’ is ½ compared to ‘b’. Therefore, they are proportional for q and –2q. Look at pages 498-499 for change line pictures. 15.6. Conductors in Electrostatic Equilibrium For conductors in electrostatic equilibrium there are four rules… 1) The electric field is zero everywhere inside the conductor. 2) Any excess change resides entirely on its surface. 3) The electric field just outside the conductor is always perpendicular to its surface. 4) On the irregularly shaped conductor, the electrical charge per unit area is always greatest at the location where the curvature of its surface is greatest. (i.e. sharp points) take a look ( ) at the figures on page 501 Assignment: pg. 517 #27, 28, 30-37 all Give chapter 15 test in groups of three (all 15 problems from supplemental problems book) Chapter 16 16.1. Electric Potential and Potential Difference When we did W= Fd we learned that the distance was all that counted, not the path taken. This was because of the conservation of energy we learned about later. In the last chapter, we applied this to Coulomb’s Law. As you’ve probably guessed, electrostatic force is also conservative. Work done by a charge by the electric force… W = Fd = qEd q = charge d = distance -a b E = electric field The following is only valid for a uniform electric field… PE = -w = -qEd …because the change in potential energy is equal to the negative of work. E = uniform electric field from a b The potential difference between the two points a and b (Va – Vb ) is defined as the change in potential energy ( PE) divided by the charge (q) that is moved through the field (E) from a b. V = Va – Vb = PE/q Because PE is a scalar (direction doesn’t count) electric potential is also a scalar. The units (which you must use!!!) are joules over Coulombs, and together they are a volt. 1 V = 1 J/C In other words, 1J of work must be done to move a 1C charge through a potential difference of 1V. When this happens, the 1C charge gains or loses 1J of energy. so… PE/q = Va – Vb = -Ed Remember that these formulas are only valid for a uniform electric field! For a positive charge q: It would want to move in the direction of the field so we must apply a positive (upward) external force on the charge to move it from a to b. Therefore, work is done on the charge so it gains electrical potential energy when moved in the direction opposite the electric field. (This is like a boulder gaining PE when raised up from the floor.) Likewise, if the positive charge is released at a and goes towards b, it accelerates downward and loses PE, while gaining equal amounts of kinetic energy. Lastly, as the positive charge moves from a to b, its electric potential decreases. Conversely, when the charge q is negative, it moves opposite the field direction from a point of low to high PE and, in the process, undergoes a decrease in electrical potential energy and an increase in kinetic energy. Therefore, they act the same, but go in opposite directions in the field. Assignment: 1-9 all 16.2. Electric Potential and Potential Energy due to Point Charges In electric circuits, a point of zero electric potential is defined by grounding some point in the circuit. In the case of a point charge, the point of zero potential is taken to be an infinite distance from the charge. The potential at a given point in space depends on the quantity of charge on the object setting up the potential and the distance (R) from the object to the specific point. V = Ke(q/R) A potential can exist in space at a point whether or not there is a charge at that point. PE = q2V1 = Ke(q1q2/R) Superposition principle: a scalar (not a vector) for a group of charges. The total energy in the system is the sum of the energy between each pair. Ex. A 5 c point charge is located at the origin and a -2 c point charge is at (3,0). a) Find the electric potential at point P. b) If the electric potential is taken to be zero at infinity, Find the total electric potential at point P, WITH COORDINATES (0,4). (0,4) P R1 R2 5 c + q1 (0,0) - -2 c (3,0) q2 a) V1 = Ke(q1/R1) = (8.99x109 Nm2/c2 )(5x10-6 c/4m) = 1.12x104 V V2 = Ke(q2/R2) = (8.99x109 Nm2/c2 )(-2x10-6 c/5m) = -.36x104 V Vp = V1 + V2 = 7.6x103 V How much work is done bringing a 4 micro coulomb charge from infinity to point P? W = q3Vp = (4x10-6 c)(7.6x103 V) = 3.1x10-2 J Assignment: pg 550 #10-13,15 16.3. Potentials and Charged Conductors 16.4. Equipotential Surfaces We know that w = - PE and PE = q V Thus W= -q V=-q(V1 -V2) But if V1 =V2 (both charges have the same potential then W=0) *All points on the surface of a charged conductor in electrostatic equilibrium are at the same potential. *The electric potential is constant everywhere inside a conductor. Because the field inside a conductor is zero, it takes work to move a charge from one spot to another in the conductor. The potential inside is therefore equal to its value on the conductor’s surface. Electron volt = the energy that an electron/proton gains when accelerated through a potential difference of 1 volt 1Ev =1.6x10-19J recall V= Ke(q/R) So a change in voltage between two points is defined by the equations V1 = Ke(q1/R1) V2 = Ke(q2/R2) You saw these yesterday! Assignment: problems #16 & #17 16.6. Capacitance C = capacitance Q = magnitude of charge V = potential difference Q = CV C = Q/ V = coulombs/volts = farads (F) A capacitor is a device consisting of a pair of conductors separated by an insulating metal. A charged capacitor stores a charge (energy) that I can use later for a specific need. 16.7. Combination of Capacitors parallel: The potential difference across each capacitor in a parallel combination of capacitors is the same across each one. This potential difference is always larger than any one individual capacitor. series: The magnitude of the charge is the same on all plates. In general, the potential difference across capacitors in series ( or any other circuit elements) is equal to the sum of potential differences across the individual capacitors. Equivalent capacitor = Ceq = 1/Ceq = 1/C1 + 1/C2 + 1/C3 +… add them as reciprocals Ex. 3 F 6 F 12 F 24 F + all in a series so… 1/Ceq = 1/3 + 1/6 + 1/12 + 1/24 = 5/8 inverse of 5/8 = 1.6μF 18v Find the charge on a 12 F capacitor. Q = Ceq V = (1.6x10-6F)(18v) = 29 C Ex. Find the charge from a to b. parallel so 1+3=4 1F 4F 3F a b 6F 8F 2F parallel so 6+2=8 4 4 ¼+¼=½ a inverse = 2 b 8 8 1/8 + 1/8 = ¼ inverse = 4 2 a b 4 a 6 b Therefore, the equivalent capacitor of the circuit is 6 F Now if the charge was a 12V circuit the charge on the 8μF capacitor would be Q=CeqΔV = (6 X 10-6 F )(12V) = 72μC Assignment page 552 29-33 all END OF CHAPTER 16 ASSIGN THE SUPPLEMENTAL PROBLEMS FROM OTHER BOOK AS A TEST, IN GROUPS OF THREE.