12/10/2010 Physics 214 Fall 2010 1

Q15 A car just starting up from a stop sign has zero velocity at the instant that it starts. Must the acceleration of the car also be zero at this instant?

Q17 A racing sports car traveling with a constant velocity of 100 MPH due west startles a turtle by the side of the road who begins to move out of the way. Which of these two objects is likely to have the larger acceleration at that instant? Explain.

NO.There has to be acceleration otherwise the car would not move

The turtle. The car has zero acceleration

12/10/2010 Physics 214 Fall 2010 2

Q19 A car moves along a straight line so that its position (distance from some starting point) varies with time as described by the graph shown here.

A. Does the car ever go backward?

B. Is the instantaneous velocity at point A greater or less than that at point B?

Q20 For the car whose distance is plotted against time in Q19, is the velocity constant during any time interval shown in the graph?

d

t

A

B

Yes, at the end

It is greater at A

In each part of the journey the velocity is constant

12/10/2010 Physics 214 Fall 2010 3

Ch 2 #14

v0 = 5 m/s a = 1.2 m/s2 t = 2 secWhat is the final velocity?What distance is covered?

a) v = v0 +at = 7.4 m/s

b) d = v0t + ½ at2 = 12.4 m

- + xd

12/10/2010 Physics 214 Fall 2010 4

Ch 2 CP4

v0 = 14 m/s a = 2 m/s2 v = 24m/sWhat is the time?What is the distance?Computed at 1 second intervals.?

a) v = v0 + at t = 5s

b) d = v0t + ½ at2 = 95m

c) 1 sec = 15 2 sec = 32 3 sec = 51 m 4 sec = 72

- + xd

12/10/2010 Physics 214 Fall 2010 5

Q14 A ball is thrown straight upward. At the very top of its flight, the velocity of the ball is zero. Is its acceleration at this point also zero? Explain.

Q15 A ball rolls up an inclined plane, slows to a stop, and then rolls back down. Do you expect the acceleration to be constant during this process? Is the velocity constant?

Q19 Is it possible for an object to have a horizontal component of velocity that is constant at the same time that the object is accelerating in the vertical direction? Explain by giving an example, if possible.

No the acceleration is 9.8m/s2 down

The acceleration is constant the velocity is not

Yes a projectile

12/10/2010 Physics 214 Fall 2010 6

Ch 3 E8Ball thrown up at 15 m/sa) How high after 1 second?b)How high after 2 seconds?

After 1 sec d = v0t + ½ at2 = 15 – 4.9 = 10.1 mAfter 2 sec d = 15 x 2 – ½ 9.8 x 22 = 10.4 mTime to top v = v0 + at t = 15/9.8 = 1.53 s

Height at top d = 11.48m

+

g

15 m/s

t = 1.53 s

t = 2 s

12/10/2010 Physics 214 Fall 2010 7

Ch 3 E10

V0 = 18 m/s a = - 2 m/s2

a) What is v after 4 seconds?b)What is time to top?

a) v = v0 + at

b) v = 0 t = 18/2 = 9s

+

a=2m/s2

18 m/s

= 18 – 2 x 4 = 10m/s

12/10/2010 Physics 214 Fall 2010 8

Ch 3 E16

V0v = 30 m/s V0H = 30 m/s g = - 9.8m/s2

a) What is time to top?b)What is the range?

a) v = v0 + at t = 30/9.8 = 3.06s tR = 6.12s

b) d = 30 x tR = 183.6m

+30 m/s

30 m/s

g

12/10/2010 Physics 214 Fall 2010 9

Ch 3 CP2V01 = 0 m/s V02 = 12 m/sa) What are the velocities after 1.5s?b) How far has each ball dropped in 1.5s?c) Does the velocity difference change?

a) v1 = at = 9.8 x 1.5 = 14.7m/s

v2 = 12 + 9.8 x 1.5 = 26.7m/sb)d1 = ½at2 = 11.03m

d2 = v2t + ½at2 = 29.03mc) No

12 m/s

1 2

12/10/2010 Physics 214 Fall 2010 10

Questions Chapter 4Q8 A 3-kg block is observed to accelerate at a rate twice that of a 6-kg block. Is the net force acting on the 3-kg block therefore twice as large as that acting on the 6-kg block? Explain.

Q9 Two equal-magnitude horizontal forces act on a box as shown in the diagram. Is the object accelerated horizontally? Explain.

Q10 Is it possible that the object pictured in question 9 is moving, given the fact that the two forces acting on it are equal in size but opposite in direction? Explain.

The net force is the same

No the net force is zero

-F F

Yes, constant velocity

12/10/2010 Physics 214 Fall 2010 11

Q22 The engine of a car is part of the car and cannot push directly on the car in order to accelerate it. What external force acting on the car is responsible for the acceleration of the car on a level road surface? Explain.

Q23 It is difficult to stop a car on icy road surface. It is also difficult to accelerate a car on this same icy road? Explain.

Q18 The acceleration due to gravity on the moon is approximately one-sixth the gravitational acceleration near the earth’s surface. If a rock is transported from the earth to the moon, will either its mass or its weight change in the process? Explain.

It’s mass will not change but it’s weight wil be 6 times less

It’s the reaction force between the tires and the road

Because of a lack of friction the wheels will skid or spin

Ftire

Freaction

Fair

12/10/2010 Physics 214 Fall 2010 12

Q31 Two blocks with the same mass are connected by a string and are pulled across a frictionless surface by a constant force, F, exerted by a string (see diagram).

A. Will the two blocks move with constant velocity? Explain.

B. Will the tension in the connecting string be greater than, less than, or equal to the force F? Explain.

F

A. They will accelerate F = ma

B. The tension will be less

12/10/2010 Physics 214 Fall 2010 13

Q33 If you get into an elevator on the top floor of a large building and the elevator begins to accelerate downward, will the normal force pushing up on your feet be greater than, equal to, or less than the force of gravity pulling downward on you? Explain.

mg

N

g

+

a

N – mg = ma but a is negative so N is smaller than mgThe only force pulling you down is gravity so if you are accelerating down the force due to gravity must be larger than the reaction force N ( N is apparent weight)

12/10/2010 Physics 214 Fall 2010 14

Ch 4 E4

Net force = 75 N a = 75/2.5 = 30 m/s2

5 N2.5 kg 80 N

A 2.5kg block is pulled with a force of 80N and friction is 5Na) What is the acceleration?

12/10/2010 Physics 214 Fall 2010 15

F = 4 x 9.8 – 15

F = ma = 24.2 N

a = 24.2/4 = 6.05m/s2

4 kg

Mg

15 N

A 4kg rock is dropped and experiencesair resistance of 15Na) What is the acceleration?

Ch 4 E14

12/10/2010 Physics 214 Fall 2010 16

Ch 4 CP4

a) Net Force = 60 x 9.8 – 500= 588 – 500

b) Will accelerate down

c) d = 1/2 at2 t = 1.38s

= 88 N

a = 88/60 = 1.47 m/s2

d) v = v0 + at v = 2.03 m/s

A 60kg crate is lowered from a height of 1.4m and the tension is 500Na) Will the crate accelerate?b) What is the acceleration?c) How long to reach the floor?d) How fast does the crate hit the floor?

500 N

60 kg

g

12/10/2010 Physics 214 Fall 2010 17

Ch 4 CP6

a) True weight = mg = 60 x 9.8 = 588 Nb) Net Force = Ma = 84 Nc) N = 588 – 84 = 504 N

Mg

1.4 m/s2N

d) 504 Ne) ↑1.4 m/s2

Net Force = 84 N↑ N = 588 + 84 = 672 N W = 672 N

A 60kg person accelerating down at 1.4m/s2

a) What is the true weight?b) What is the net force?c) What is N?d) What is the apparent weight?e) a, b, c, d with 1.4m/s2 up?

12/10/2010 Physics 214 Fall 2010 18

Questions Chapter 5Q6 A ball on the end of a string is whirled with constant speed in a counterclockwise horizontal circle. At point A in the circle, the string breaks. Which of the curves sketched below most accurately represents the path that the ball will take after the string breaks (as seen from above)? Explain.

•

4 3

2

1A

Path number 3

12/10/2010 Physics 214 Fall 2010 19

Q12 If a ball is whirled in a vertical circle with constant speed, at what point in the circle, if any, is the tension in the string the greatest? Explain. (Hint: Compare this situation to the Ferris wheel described in section 5.2).

Q19 Does a planet moving in an elliptical orbit about the sun move fastest when it is farthest from the sun or when it is nearest to the sun? Explain by referring to one of Kepler’s laws.

The tension is the greatest at the bottom because the string has to support the weight and provide the force for the centripetal acceleration.

When it is nearest

12/10/2010 Physics 214 Fall 2010 20

Q20 Does the sun exert a larger force on the earth than that exerted on the sun by the earth? Explain.

Q23 Two masses are separated by a distance r. If this distance is doubled, is the force of interaction between the two masses doubled, halved, or changed by some other amount? Explain.

The magnitude of the forces is the same they are a reaction/action pair

The force reduces by a factor of 4

12/10/2010 Physics 214 Fall 2010 21

A Ferris wheel with radius 12 m makes one complete rotation every 8 seconds.a) Rider travels distance 2πr every rotation. What speed do riders

move at?b) What is the magnitude of their centripetal acceleration?c) For a 40 kg rider, what is magnitude of centripetal force to keep him

moving in a circle? Is his weight large enough to provide this centripetal force at the top of the cycle?

d) What is the magnitude of the normal force exerted by the seat on the rider at the top?

e) What would happen if the Ferris wheel is going so fast the weight of the rider is not sufficient to provide the centripetal force at the top?

Ch 5 CP 2

12/10/2010 Physics 214 Fall 2010 22

a) S = d/t = 2πr/t = 2π(12m)/8s = 9.42 m/s

b) acent = v2/r = s2/r = (9.42m/s)2/12m = 7.40 m/s2

c) Fcent = m v2/r = m acent = (40 kg)(7.40 m/s2) = 296 NW = mg = (40 kg)(9.8 m/s2) = 392 NYes, his weight is larger than the centripetal force required.

d) W – Nf = 296 N = 96 newtonsN

W

e) rider is ejected

Ch 5 CP 2 (con’t)

Fcent

12/10/2010 Physics 214 Fall 2010 23

A passenger in a rollover accident turns through a radius of 3.0m in the seat of the vehicle making a complete turn in 1 sec.a) Circumference = 2πr, what is speed of passenger?b) What is centripetal acceleration? Compare it to gravity (9.8

m/s2)c) Passenger has mass = 60 kg, what is centripetal force

required to produce the acceleration? Compare it to passengers weight.

3 m

a) s = d/t = 2π(3.0m)/1 = 19m/s

c) F = ma = (60 kg)(118 m/s2) = 7080 NF = ma = m (12 g) = 12 mg = 12 weight

b) a = v2/r = s2/r = (19 m/s)2/3m = 118 m/s2 = 12g

Ch 5 CP 4

12/10/2010 Physics 214 Fall 2010 24

Questions Chapter 6Q1 Equal forces are used to move blocks A and B across the floor. Block A has twice the mass of block B, but block B moves twice the distance moved by block A. Which block, if either, has the greater amount of work done on it? Explain.

Q3 A string is used to pull a wooden block across the floor without accelerating the block. The string makes an angle to the horizontal.

A. Does the force applied via the string do work on the block?

B. Is the total force involved in doing work or just a portion of the force?

Work is Force times distance so the most work is done on B

F

d

A. Yes B. just the horizontal component

12/10/2010 Physics 214 Fall 2010 25

Q4 In the situation pictured in question 3, if there is a frictional force opposing the motion of the block, does this frictional force do work on the block? Explain.

Yes it does negative work since force is opposite the motion

Q8 A woman uses a pulley, arrangement to lift a heavy crate. She applies a force that is one-fourth the weight of the crate, but moves the rope a distance four times the height that the crate is lifted. Is the work done by the woman greater than, equal to, or less than the work done by the rope on the crate? Explain.

The product Fd is the same for both and the work is equal

12/10/2010 Physics 214 Fall 2010 26

5.0 kg box lifted (without acceleration) thru height of 2.0 ma) What is increase in potential energy?b) How much work was required to lift box?

a) PE = mgh ∆PE = PEfinal – PEinitial= mg(ho+2.0m) – mgho = mg(2.0m)= (5.0 kg)(9.8 m/s2)(2.0m) = 98J

b) F = ma = 0 = Flift – mgFlift = mg = (5.0kg)(9.8m/s2) = 49NW = Fd = (49N)(2.0m) = 98J

M

Mho+2.0m

M

mg

gFlift

Ch 6 E 8

12/10/2010 Physics 214 Fall 2010 27

100 kg crate accelerated by net force = 50 N applied for 4 s.a) Use Newton’s 2nd Law to find acceleration?b) If it starts from rest, how far does it travel in 4 s?c) How much work is done if the net force = 50 N?

a) F = ma a = F/m = 50N/100kg = 0/5 m/s2

b) d = v0t + ½at2 = ½(0.5)(4)2 4mc) W = Fd = (50N)(4m) = 200J

d) v = v0 + at = 0 + (0.5 m/s2)(4s) = 2m/se) KE = ½mv2 = ½(100kg)(2m/s)2 = 200 J

work done equals the kinetic energy.

M Fnet

Ch 6 CP 2

12/10/2010 Physics 214 Fall 2010 28

Questions Chapter 7

Q5 Are impulse and momentum the same thing? Explain.

Q6 If a ball bounces off a wall so that its velocity coming back has the same magnitude that it had prior to bouncing:

A. Is there a change in the momentum of the ball? Explain.

B. Is there an impulse acting on the ball during its collision with the wall? Explain.

A. Yes momentum is a vector

B. Yes a force acts for a short time

No impulse changes momentum

12/10/2010 Physics 214 Fall 2010 29

Q17 A compact car and a large truck have a head-on collision. During the collision, which vehicle, if either, experiences:

A. The greater force of impact? Explain.

B. The greater impulse? Explain.

C. The greater change in momentum? Explain.

D. The greater acceleration? Explain.

Q22 Is it possible for a rocket to function in empty space (in a vacuum) where there is nothing to push against except itself?

Yes. It ejects material at high velocity and momentum conservationmeans the rocket recoils

A. The forces are equal and oppositeB. The impulse for each is the sameC. The momentum changes are equal and oppositeD. F = ma so a is larger for the compact car

12/10/2010 Physics 214 Fall 2010 30

Q23 Suppose that you are standing on a surface that is so slick that you can get no traction at all in order to begin moving across this surface. Fortunately, you are carrying a bag of oranges. Explain how you can get yourself moving.

Q24 A railroad car collides and couples with a second railroad car that is standing still. If external forces acting on the system are ignored, is the velocity of the system after the collision equal to, greater than, or less than that of the first car before the collision?

Throw the oranges opposite to the direction you wish to move

The velocity after is exactly half

12/10/2010 Physics 214 Fall 2010 31

A ball has an initial momentum = 2.5 kg m/s, it bounces off a wall and comes back in opposite direction with momentum = -2.5 kg m/sa) What is the change in momentum of the ball?b) What is the impulse?

b) Impulse = Δp = - 5kgm/s

a) Δp = pf – pi = -2.5 – (+2.5) = - 5kgm/s

Ch 7 E 8

Pi = 2.5kgm/s

Pf = -2.5kgm/sF

+

12/10/2010 Physics 214 Fall 2010 32

M1 and M2 collide head ona) Find initial momentum of M1 and M2

b) What is the total momentum of the system before collision?

c) Ignore external forces, if they stick together after collision, which way do the masses travel?

a) p1 = -100 x 3.5 = 350kgm/s p2 = 80 x 6 = 480kgm/s

b) Total momentum = 480 – 350 = 130kgm/s east

c) The masses will travel east with p = 130kgm/sec

6.0m/s 3.5m/sM1 = 100kgM2 = 80kg eastwest

Ch 7 E 10

12/10/2010 Physics 214 Fall 2010 33

A bullet is fired into block sitting on ice. The bullet travels at500 m/s with mass 0.005 kg. The wooden block is at rest with amass of 1.2 kg. Afterwards the bullet is embedded in the block.a) Find the velocity of the block and bullet after the impact (assume

momentum is conserved).b) Find the magnitude of the impulse on the block of wood.c) Does the change in momentum of the bullet equal that of wood?

v

mM

No friction (ice)

b) Impulse = Δp = pfinal – pinitial

= (1.2 kg)(2.07 m/s) – 0 = 2.50 kg m/sc) Δp for bullet = (0.005 kg)(500 m/s) – (0.005 kg)(2.07 m/s)

= 2.50 kg m/sMomentum is conserved, so momentum lost by bullet is gained by wood.

Ch 7 CP 2

a) pfinal = pinitial = (0.005 kg)(500 m/s)pfinal = (Mbullet + Mwood)v = 2.5 kg m/s

v = (2.5 kg m/s)/(1.205 kg) = 2.07 m/s

12/10/2010 Physics 214 Fall 2010 34

Questions Chapter 8

Q6 Is the linear speed of a child sitting near the center of a rotating merry-go-round the same as that of another child sitting near the edge of the same merry-go-round? Explain.

Q11 The two forces in the diagram have the same magnitude. Which orientation will produce the greater torque on the wheel? Explain. F1

F2

The angular velocity is the same but v = ωr, so speed is greatest at the edge

F1 because it is the tangential component that produces the torque

12/10/2010 Physics 214 Fall 2010 35

Q13 Is it possible for the net force acting on an object to be zero, but the net torque to be greater than zero? Explain. (Hint: The forces contributing to the net force may not lie along the same line.)

Q20 Two objects have the same total mass, but object A has its mass concentrated closer to the axis of rotation than object B. Which object will be easier to set into rotational motion? Explain.

F

F

A has a smaller moment of inertia and torque = Iα so will accelerate faster

12/10/2010 Physics 214 Fall 2010 36

Q26 A child on a freely rotating merry-go-round moves from near the center to the edge. Will the rotational velocity of the merry-go-round increase, decrease, or not change at all? Explain.

Q29 Suppose you are rotating a ball attached to a string in a circle. If you allow the string to wrap around your finger, does the rotational velocity of the ball change as the string shortens? Explain.

L =Iω so I increases and ω will decrease. This requires work

L =Iω so I decreases and ω will increase. This requires work

12/10/2010 Physics 214 Fall 2010 37

Ch 8 E 6

Rotational velocity decreases from 6 rev/s to 3 rev/s in 12 s.What is rotational acceleration?

∆w = wf – wi = 3rev/s – 6rev/s = - 3rev/sα= ∆w/t = (-3rev/s)/12s = -1/4 rev/s2

α = -1/4 rev/s | 2πrad/rev = -π/2 rad/s2

12/10/2010 Physics 214 Fall 2010 38

Ch 8 CP 10

5N placed 10 cm from fulcrum of balance beam, what weight should be put 4 cm from fulcrum on other side to balance

τ = Fl = 0 = -(5N)(10cm) + x(4cm)x = 50Ncm/4cm = 12.5 N

? 5N

4cm 10cm

12/10/2010 Physics 214 Fall 2010 39

Ch 8 CP 4See Fig 8.23. A student sits on a stool with wheels with a bikewheel with Ib = 2kgm2 and wb = 5 rev/s. Bike wheel is spinning,student is not. The student on stool with bike wheel: Is = 6 kgm2.a) Initially, what is L?b) Student flips bike wheel. What is student’s L?c) Where does the torque come from that accelerates student?

a) Lb = Ib wb (just the bike wheel spinning).wb = 5 rev/s 2π/rev = 10π rad/sLb = 20π kg m2/s , upwards

b) L = Lb + Ls (flipped bike wheel plus student).-20π kg m2/s + Is ws = +20π kg m2/s Ls = +40π kg m2/s

c) Student supplies torque when he flips bike wheel.

12/10/2010 Physics 214 Fall 2010 40

Questions Chapter 9

Q1 Is it possible for a 100-lb woman to exert a greater pressure on the ground than a 250-lb man? Explain.

Q3 The same force is applied to two cylinders that contain air. One has a piston with a large area, and the other has a piston with a small area. In which cylinder will the pressure be greater?

Yes. The pressure will be mg/A so if A is small e.g. small heels the pressure will be very large

The pressure is F/A so the one with the smallest A

12/10/2010 Physics 214 Fall 2010 41

Q15 Is it possible for a solid metal ball to float in mercury?

Q16 A rectangular metal block is suspended by a string in a breaker of water so that the block is completely surrounded by water. Is the water pressure at the bottom of the block equal to, greater than, or less than the water pressure at the top of the block?

The upward force is the weight of liquid displaced and the downward force is the weight of the ball. If the density of the liquid is greater than that of the ball it will float.

The pressure is ρgh so the pressure is higher at the bottom. The difference in pressure provides the upward force on the block

12/10/2010 Physics 214 Fall 2010 42

Q19 A large bird lands on a rowboat that is floating in a swimming pool. Will the water level in the pool increase, decrease, or remain the same when the bird lands on the boat?

Q20 A rowboat is floating in a swimming pool when the anchor is dropped over the side. When the anchor is dropped, will the water level in the swimming pool increase, decrease, or remain the same?

The buoyant force is the weight of liquid displaced so to support a larger weight more liquid is displaced and the level rises

When the anchor is in the boat it’s whole weight is supported and the amount of water displaced balances that weight. When it is thrown overboard it sinks and only displaces it’s volume so the water level falls

12/10/2010 Physics 214 Fall 2010 43

Ch 9 E 4Pressure of gas in piston = 300 N/m2.Area of Piston = 0.2m2.What is force exerted by piston on gas?

P = F/A, F = PA = 300 N/m2 (0.2 m2) = 60 N

A = 0.2m2

p = 300 N/m2

12/10/2010 Physics 214 Fall 2010 44

Ch 9 E 6

Hydraulic system: A2 = 50 A1

F2 = 6000 N. What is F1?

Pressure is the same just underneath each pistonF1/A1 = F2/A2

F2/F1 = A2/A1 = 50A1/A1 = 50F1 = F2/50 = 6000/50 = 120N

A1 A2

F2 = 6000N.

12/10/2010 Physics 214 Fall 2010 45

Ch 9 E 12

Boat displaces 2.5 m3 of water. Density of water ρH2O = 1000 kg/m3.a) What is the mass of water displaced?b) What is the buoyant force?

a) Mass of fluid displaced (mFD) = volume x density of fluid.MFD = VFDρH2O = (2.5 m3)(1000 kg/m3) = 2500 kg

b) Buoyant force equals weight of fluid displaced.Fb = WFD = mFD g = (2500 kg)(9.8 m/s2) = 24500 N

Fb = WFD

12/10/2010 Physics 214 Fall 2010 46

Ch 9 CP 2Water density = ρH2O = 1000 kg/m3. Depth of swimming pool = 3m.a) What is the volume of a column of water 3m deep and cross

sectional area 0.5 m2?b) What is its mass?c) What is its weight?d) What is the excess pressure exerted on the pool bottom?e) Compare to atmospheric pressure.

a) V = Ad = (0.5 m2)(3m) = 1.5 m3

b) M = Vρ = (1.5 m3)(1000 kg/m3) = 1500 kgc) W = Mg = (1500 kg)(9.8 m/s2) = 14700 N

d) P = F/A = 14700N/0.5m2 = 29400 Pa

e) Atmospheric Pressure is about 100 kPaP is about 30 kPaP/Atm = (29400 Pa)/(1.013 x 105 Pa) = 0.29

3m

0.5m2

12/10/2010 Physics 214 Fall 2010 47

Ch 9 CP 4Wooden boat: 3m x 1.5m x 1m that carries five people.Total mass of boat and people equals 1200 kg.a) What is total weight?b) What is buoyant force required to float?c) What volume of water must be displaced to float?d) How much of the boat underwater?

a) W = Mg = 1200 kg (9.8 m/s2)W = 11760 N

W Fb

3m

1m

b) Fnet = Fb – W = 0Fb = 11760 N

c) Fb = ρH2O Vg (see Ch 9 E 12)Fb/ρH2Og = 11760N/(1000 kg/m3)(9.8 m/s2) = V = 1.2 m3

d) V = LWh = (3m)(1.5m)h = 1.2 m3 h = 0.27 m

12/10/2010 Physics 214 Fall 2010 48

Questions Chapter 10

Q1 Is an object that has a temperature of 0 C hotter than, colder than, or at the same temperature as one that has a temperature of 0 F?

Q2 Which spans a greater range in temperature, a change in temperature of 10 Fahrenheit degrees or a change of 10 Celsius degrees?

Water freezes at 0o C and 32o F so 0o F is colder

There is 100o C between water freezing and boiling and 180o F so 1oC = 1.8oF so a change of 10oC is larger

12/10/2010 Physics 214 Fall 2010 49

Q21 An ideal gas is compressed without allowing any heat to flow into or out of the gas. Will the temperature of the gas increase, decrease, or remain the same in this process?

Q23 Heat is added to an ideal gas, and the gas expands in the process. Is it possible for the temperature to remain constant in this situation?

The temperature will increase ΔU = W

Yes. This is an isothermal expansion and W = Q

12/10/2010 Physics 214 Fall 2010 50

Q25 Heat is added to a hot-air balloon causing the air to expand. Will this increased volume of air cause the balloon to fall?

Q27 A block of wood and a block of metal have been sitting on a table for a long time. The block of metal feels colder to the touch than the block of wood. Does this mean that the metal is actually at a lower temperature than the wood?

Archimedes principle states that the buoyant force is equal to the weight of liquid displaced. So if the balloon stays the same size and as the air expands it leaves the balloon it will rise faster because the weight of the air inside will be less. If no air escapes but the balloon increases in size it will also rise because the buoyant force is larger

No. What you feel is heat flow and the thermal conductivity of metal is much bigger than that of wood. This is also why you feel much colder when the air is damp and why trapped dry air in fiberglass is used for insulation. It is also because room temperature is lower than body temperature

12/10/2010 Physics 214 Fall 2010 51

Ch 10 E 2

Temperature is 14 F. What is the temperature in Celsius?

Tc= 5/9 (TF – 32) = 5/9 (14 - 32)=5/9 (-18) = -10 C

12/10/2010 Physics 214 Fall 2010 52

Ch 10 E6

How much heat does it take to raise the temperature of 70 g of H2O from 20 C to 80 C? C= 1 cal/gram/oC.

Q = mc∆T=(70)(1)(80-20) = 4200 cal

12/10/2010 Physics 214 Fall 2010 53

Ch 10 E 12

Add 600 J to 50 g of H2O initially at 20 Ca) How many calories?b) What is the final temperature of the H2O

a) 600 /4.186 = 143.3 cal = Q

b) Q=mc∆T∆T=Q/mc=143.3/(50)(1)=2.87 CTF=22.87 C

12/10/2010 Physics 214 Fall 2010 54

Ch 10 E 16

Add 500 cal of heat to gas. Gas does 500 J of work on surroundings.What is the change in internal energy of gas?

∆U = Q-W

Q = 500/ 4.186 = 2093 J

W=500 J

∆U = 2093 – 500 = 1593 J

12/10/2010 Physics 214 Fall 2010 55

Ch 10 CP 4

150 g of metal at 120 C is dropped in a beaker containing 100 g H2Oat 20 C. (Ignore the beaker). The final temperature of metal and water

is 35 C.a) How much heat is transferred to the H2O?b) What is the specific heat capacity of metal?c) Use the same experimental setup – how much metal at 120°C to have final

temperature of metal and water = 70°C?

a) Q = mc∆T = (100)(1)(35-20) = 1500 cal

b) Q = mc∆T → c = Q/m∆T =c = -1500/(150)(35-120) = 0.12 cal/gram/ C

c) Q H2O = mc∆T = (100)(1)(70-20) = 5000 calQ metal = - Q H2O = - 5000Q metal = mc∆T; m = Q metal/c∆T = -5000/(0.12)(70-120)m = 833 g

12/10/2010 Physics 214 Fall 2010 56

Ch 11 E 6

A Carnot engine takes in heat at 650 K and releases heat to a reservoir at 350K.What is the efficiency?

ec = (TH – Tc) / TH = 650-350 / 650 = 0.46

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Ch 11 CP 2Carnot engine operates b/w 500 C and 150 C and does

30 J of work in each cycle. a) What is the efficiency?b) How much heat is taking in from the high-temp reservoir each cycle?c) How much heat is released to low-temp reservoir each cycle?d) What is the change, if any, in the internal energy of gas each cycle?

a) εc= (TH-Tc)/ TH= (500+273.3)-(150+273.3)/(500+273.3) = .045

b) ε = W/QH , QH = W/ε = 30/0.45 = 66.3 J

c) W = QH-Qc, Qc = QH-W = 66.3J - 30J = 36.3J

d) ∆U = Q-W, W = 30JQ = QH - Qc = 66.3 – 36.3 = 30 J∆U = 30-30 = 0

12/10/2010 Physics 214 Fall 2010 58

Q20 Is it possible for an electric field to exist at some point in space at which there is no charge?

Q21 Two charges, of equal magnitude but opposite sign, lie along a line as shown in the diagram. Using arrows, indicate the directions of the electric field at points A, B, C, and D shown on the diagram.

• • •- +

•

C-q

Aq

B

D

Yes. Electric fields are created by charge but extend to infinity

12/10/2010 Physics 214 Fall 2010 59

Q25 If we move a positive charge toward a negative charge, does the potential energy of the positive charge increase or decrease?

Q23 Three equal positive charges are located at the corners of a square, as in the diagram. Using arrows, indicate the direction of the electric field at points A and B on the diagram.

•

•

+

++

B

q qA

q

•

The potential energy decreases

12/10/2010 Physics 214 Fall 2010 60

Q32 Would you be more likely to be struck by lightning if you stood on a platform made from a good electrical insulator than if you stood on the ground?

Q26 If we move a negative charge toward a second negative charge, does the potential energy of the first charge increase or decrease?

It increases because they repel each other it means as you get closer the force gets larger. You have to do work to bring them closer

Standing on an insulator is the safest which is why the safest place in inside a car because the tires are insulators. It’s also the same with fallen power lines

12/10/2010 Physics 214 Fall 2010 61

Ch 12 E 8

Electron and proton have charges of equal magnitude, 1.6 10-19C, but opposite signs. If the electron and proton are separated by R=5×10-11m, what is the electrostatic (vector) force b/w them?

F = kq1q2 / r2 = -(9×109)(1.6×10-19)2 / (5×10-11m)2

F = 9.22 × 10-8 N towards proton

r

e-

P+

12/10/2010 Physics 214 Fall 2010 62

CH 12 E14Charge q = -4 10-6 c placed in Electric field of E = 8.5 104 N/CTowards right. What is the electrostatic (vector) force on charge q?

F = qE = (4 10-6)(8.5 104)F = 0.34 N

Since positive charge moves with field lines and q isnegative, q moves to the left

F = 0.34 N to left

E9m

12/10/2010 Physics 214 Fall 2010 63

Ch 12 CP 4

4 equal positive charges located at corners of the square(see diagram)a) Use small arrows to indicate direction of electric field at

each labeled pointb) Would the magnitude of electric field be equal to zero at any

labeled point?

+ +

+ +

B

A CE

D

a)++

++

b) Electric Field is equal to zero at the center of the square (A).

12/10/2010 Physics 214 Fall 2010 64

Questions Chapter 13

Q3 In a simple battery-and-bulb circuit, is the electric current that enters the bulb on the side nearer to the positive terminal of the battery larger than the current that leaves the bulb on the opposite side?

Q4 Are electric current and electric charge the same thing?

The current is the same

No. electric current is the flow of charge I = dq/dt

12/10/2010 Physics 214 Fall 2010 65

Q11 A dead battery will still indicate a voltage when a good voltmeter is connected across the terminals. Can the battery still be used to light a bulb?

Q12 When a battery is being used in a circuit, will the voltage across its terminals be less than that measured when there is no current being drawn from the battery? Explain.

A battery has internal resistance so although one can measure an open circuit voltage when connected to a circuit the voltage will drop and the current flow will be very low.

The voltage will be less because of the voltage drop due to the internal resistance

12/10/2010 Physics 214 Fall 2010 66

Q13 Two resistors are connected in a series with a battery as shown in the diagram. R1 is less than R2.

A. Which of the two resistors, if either, has the greater current flowing through it? Explain.

B. Which of the two resistors, if either, has the greatest voltage difference across it? Explain.

Є

R1

R2

The current is the same in both. Since V = IR the greatest voltage drop will be across R2

12/10/2010 Physics 214 Fall 2010 67

Q14 In the circuit shown below, R1, R2, and R3 are three resistors of different values. R3 is greater than R2, and R2 is greater than R1. Єis the electromotive force of the battery whose internal resistance is negligible. Which of the three resistors has the greatest current flowing through it?

I3 = I1 + I2 so I3 is the largest

Є

R3

R1R2

12/10/2010 Physics 214 Fall 2010 68

Q15 In the circuit shown in question 14, which of the three resistors, if any, has the largest voltage difference across it?

Q16 If we disconnect R2 from the rest of the circuit shown in the diagram for question 14, will the current through R3 increase, decrease, or remain the same?

V = IR and both I and R are the largest for R3

Є

R3

R1R2

The resistance of the circuit will increase so the current through R3 will decrease.

12/10/2010 Physics 214 Fall 2010 69

Q21 Is electric energy the same as electric power?

Q22 If the current through a certain resistance is doubled, does the power dissipated in that resistor also double?

Q23 Does the power being delivered by a battery depend on the resistance of the circuit connected to the battery?

Power is the rate at which energy is used. Your electrical bill is for the total energy you use.

P = I2R so the power increases by a factor of 4

Yes because increasing the resistance lowers the current

12/10/2010 Physics 214 Fall 2010 70

Ch 13 E 14

24 Ω resistor has voltage difference 3V across leads.a) What is the current through the resistor?b) What is the power dissipated in resistor?

a) V = IRI = V/R = 3/24 = 0.125A

b) P = IV = V2/R = (3)2/24 = 0.375W

24 Ω3V

12/10/2010 Physics 214 Fall 2010 71

Ch 13 E 16

A toaster draws current = 7A on a 100-V AC linea) What is the power consumption of the toaster?b) What is the resistance of the heating element

in the toaster?

R110 V

→ I = 7Aa) P = IV = 7.110 = 770 W

b) V = IR, R = V/I = 110/7 = 15.7 Ω

12/10/2010 Physics 214 Fall 2010 72

Ch 13 CP 2

Three 30-Ω light bulbs connected in PARALLEL to 1.5 V battery with negligible internal resistance.

a) What is the current through the battery?b) What is the current through each bulb?c) If one bulb burns out, does the brightness of the

other bulbs change?

1.5 V R R R R = 30 Ω

12/10/2010 Physics 214 Fall 2010 73

Ch 13 CP2 cont.

a) 1/Rp = 1/R1 + 1/R2 + 1/R3 = 1/30 + 1/30 + 1/30 = 1/10Rp = 10 ΩV = ItRp , It = V/Rp = 1.5/10 = 0.15 A

b) V = IR, I = V/R = 1.5/30 = 0.05 ANotice that total current, It, through the battery is the sum ofcurrents through each bulb.

It = 3(I) = 1.5 A

c) Brightness of remaining two bulbs do not change. Instead, load on the battery is reduced. Each remaining bulb still feels 1.5 V.So, each remaining bulb still draws 0.05 A of current. Since,P=IV, each remaining bulb still outputs same power. This is benefit of hooking circuits in parallel.

12/10/2010 Physics 214 Fall 2010 74

Q6 If we regard the earth as magnet, does its magnetic north pole coincide with its geographical north pole? What defines the position of the geographical north pole?

Q7 We visualized the magnetic field of the earth by imagining that there is a bar magnet inside the earth (fig. 14.7). Why did we draw this magnet with its south pole pointing north?

The geographical north pole is defined by the axis of rotation. The magnetic north pole is determined by the currents and fields in the iron core of the earth. About every 250,000 years the field of the earth reverese.

The definition of the North pole is the point at which the North pole of a magnet would point. This means the North pole is a physical magnetic south pole.

12/10/2010 Physics 214 Fall 2010 75

Q9 A horizontal wire is oriented along an east-west line, and a compass is placed above it. Will the needle of the compass deflect when a current flows through the wire from east to west, and if so, in what direction?

Q11 A uniform magnetic field is directed horizontally toward the north, and a positive charge is moving west through this field. Is there a magnetic force on this charge, and if so, in what direction?

The current will produce a field that appears clockwise looking west. This means the compass will point north/south

Point index finger along the velocity, the middle finger in the direction of B and then the thumb points in the direction of the force. The force points up.

12/10/2010 Physics 214 Fall 2010 76

Q15 If we look down at the top of a circular loop of wire whose plane is horizontal and that carries a current in the clockwise direction, what is the direction of the magnetic field at the center of the circle?

Q17 A current-carrying rectangular loop of wire is placed in an external magnetic field with the directions of the current and field as shown in the diagram. In what direction will this loop tend to rotate as a result of the magnetic torque exerted on it?

The field is perpendicular to the plane in the direction that if you look in that direction the current is clockwise. So the answer is down.

B F

F

B

12/10/2010 Physics 214 Fall 2010 77

Q24 A horizontal loop of wire has a magnetic field passing upward through the plane of the loop. If this magnetic field increases with time, is the direction of the induced current clockwise or counterclockwise (viewed from above) as predicted by Lenz’s law?

The current induced produces a magnetic field that opposes the increase so the induced magnetic field points down so the current must be clockwise viewed from above.

Q25 Two coils of wire are identical except that coil A has twice as many turns of wire as coil B. If a magnetic field increases with time at the same rate through both coils, which coil (if either) has the larger induced voltage?

The flux in A is twice that in B so the induced voltage is twice as large.

Q28 Does a simple generator produce a steady direct current? No. As the coil turns at constant angular velocity the rate of change of flux depends on the angle of the coil to the field so the current is AC

12/10/2010 Physics 214 Fall 2010 78

Q30 Can a transformer be used, as shown in the diagram below, to step up voltage of a battery? Explain.

Q31 By stepping up the voltage of an alternating current source using a transformer, can we increase the amount of electrical energy drawn from the source?

V2/V1 = N2/N1So if N2 > N1 the voltage is stepped up.

No. For an ideal transformer the input power = output power. In a real transformer energy is lost due to heat. Feel the transformer for your laptop.

12/10/2010 Physics 214 Fall 2010 79

Ch 14 E 10Loop of wire enclosing Area, A = 0.03m2, has magnetic field passing thru its plane at an angle. Component of magnetic field perpendicular to plane = 0.4T, while component parallel to plane = 0.6T. What is magnetic flux thru coil?

I = B1A = 0.4(0.03)

= 0.012Tm2 A

12/10/2010 Physics 214 Fall 2010 80

Ch 14 E 12Coil of wire with 60 turns and cross-sectional area, A = 0.02m2, lies with it’s plane perpendicular to B = 1.5T magnetic field. Coil is rapidly removed B-field in time t=0.2s.a) What is initial magnetic flux thru coil?b) What is average voltage induced in coil?

a) Φ = NB1A = (60)(1.5)(0.02) = 1.8Tm2

b) ε = ΔΦ/t = (1.8 Tm2 – 0)/0.2s = 9V

12/10/2010 Physics 214 Fall 2010 81

Ch 14 CP 4Transformer is designed to step down line voltage of110V to 22V. Primary coil has 400 turns of wire.a) How many turns of wire on secondary coil?b) Current in primary I1 = 5A. What is max current in

second coil?c) If transformer gets warm during operation, will current

in secondary coil equal that computed in previous question (b)?

a) ΔV2/ ΔV1 = N2/N1 , N2 = N1(ΔV2/ ΔV1) = 400(22/110) = 80 turns

b) ΔV2I2 ≤ ΔV1I1 I1 = 110/22 (5) = 25AMax current in second coil = 25A.

c) No, heat that warms up transformer is power dissipated in the formP = I2R. Power is lost to heat.

12/10/2010 Physics 214 Fall 2010 82

Q24 Suppose that we increase the temperature of the air through which a sound wave is traveling.

A. What effect does this have on the speed of the sound wave? Explain.

B. For a given frequency, what effect does increasing the temperature have on the wavelength of the sound wave? Explain.

Q25 If the temperature in a organ pipe increases above room temperature, thereby increasing the speed of sound waves in the pipe but not affecting the length of pipe significantly, what effect does this have on the frequency of the standing waves produced by this pipe?

The speed increases and the wavelength increases

The length is unchanged so the wavelength is unchanged so the frequency increases

12/10/2010 Physics 214 Fall 2010 83

Q26 Is the wavelength of the fundamental standing wave in a tube open at both ends greater than, equal to, or less than the wavelength for the fundamental wave in a tube open at just one end?

Q27 A band playing on a flat-bed truck is approaching you rapidly near the end of a parade. Will you hear the same pitch for the various instruments as someone down the street who has already been passed by the truck?

A tube open at both ends has half a wavelength whereas the tube open at one end has one quarter of a wavelength.

The pitch increases as the band approaches and decreases as it is going away

12/10/2010 Physics 214 Fall 2010 84

Ch 15 E 2Water waves have a wavelength λ = 1.4 m and Period T = 0.8s . What is the velocity of the waves?

v = λ / T = 1.4/0.8 = 1.75 m/s

12/10/2010 Physics 214 Fall 2010 85

Ch 15 E 8String of length 0.8 m is fixed at both ends. The story is plucked so that there are two nodes along the string in addition to those at either end. What is the wavelength of the interfering waves?

At each node wave T and wave R cancel. From the picture a total of 3/2 λ fit on 0.8 m string. 3/2 λ = 0.8 m → λ = (2/3) 0.8 = 0.53 m

fixed head

node node

snapshot v

v

T

R

λ ½ λ

12/10/2010 Physics 214 Fall 2010 86

Ch 15 E 10What is the frequency of a sound wave with wavelength λ = 0.68 m travelling in room temperature air (v=340m/s)?

v = λf → f = v/λ = 340/0.68 m = 500 Hz

12/10/2010 Physics 214 Fall 2010 87

Ch 15 CP 2A guitar string has length 1.25 m and mass 40 g. After stringing the guitar, string has 64 cm between fixed points. It is tightened to tension = 720 N.

a) What is mass per unit length of string?b) What is wavespeed on tightened string?c) What is wavelength of traveling waves that interfere to

form the fundamental standing wave?d) What is frequency of the fundamental standing wave?e) What are wavelength and frequency of the next

harmonic?

12/10/2010 Physics 214 Fall 2010 88

Ch 15 CP2 cont.a) µ=M/L=0.04/1.25=0.032 kg/m

b) ν = √F/µ = √720/0.032 = √22500 = 150 m/s

c)

0.64 m = ½ λλ = 1.28 m

Snapshot of fundamental standing wave

d) λ=1.28 m ; f = ν/λ = 150/1.28 = 117 Hz

e)

node node nodeanti-node anti-node

Snapshot of second harmonic

λ = 0.64 mf = ν/λ = 150/0.64 = 234 Hz

vR

Tv

node antinode node

=

12/10/2010 Physics 214 Fall 2010 89

Questions Chapter 16

Q3 Is it possible for an electromagnetic wave to travel through a vacuum?

Q11 Skylight is produced by scattering of light from the direct beam coming from the sun. Why is the color of the sky different from the color of the light of the sun itself?

Yes

Blue light is scattered more strongly than red light

12/10/2010 Physics 214 Fall 2010 90

Q12 Two waves interfere to form fringes in Young’s double-slit experiment. Do these two waves come from the same light source?

Q13 If two waves start out in phase with one another, but one wave travels half a wavelength farther than the other before they come together, will the waves be in phase or out of phase when they combine?

Yes. They must be coherent

They will be exactly out of phase

12/10/2010 Physics 214 Fall 2010 91

Q15 When light is reflected from a thin film of oil on a water puddle, the colors we see are produced by interference. What two waves are interfering in this situation?

One ray is reflected by the top of the oil, the second from the oil/water interface

12/10/2010 Physics 214 Fall 2010 92

Ch 16 E4

X-rays often have wavelength of about λ=10-10m. What is the frequency of such waves?

f = c/λ = 3×1018 Hz

12/10/2010 Physics 214 Fall 2010 93

Ch 16 E 8Light of 500 nm is reflected from a thin film of air between two glass plates. The thickness of film: d=1µm=1000 nm.a) How much farther does light reflected from bottom plate travel than that reflected from top plate?b) How many wavelengths of light does this represent?

a) Path difference = 2d= 2000 nm

b) 2000 nm/λ = 44 wavelengths of light

d

top bottom

12/10/2010 Physics 214 Fall 2010 94

Ch 16 CP 2Light of wavelength λ=600 nm passes through a double slit with d = 0.03 mm. The resulting fringe pattern is observed on a screen 1.2 m from the double slit.a) How far from the center of the screen is the first bright fringe?b) Second bright fringe?c) First dark fringe?d) Sketch a picture of the central seven bright fringes (central fringe & 3 on

either side). Clearly indicate the distance from each fringe to the center of the screen.

y1

x = 1.2 md = 0.03 mm

y2z1

z1- first dark fringe

y1, y2 - first and secondbright fringes

12/10/2010 Physics 214 Fall 2010 95

Ch 16 CP 2 cont.a) d y/x=nλ or yn= n x/d λ = n (1.2)/(0.03×10-3) (600×10-9)

yn=n(0.024)my1 = 1(0.024) = 0.024m = 2.4 cm

b) y2 = 2 (0.024) m = 4.8 cm

c) z is spacing nth dark fringedz/x = (n-1/2) λ

or zn = (n-1/2) x/d λzn = (n-1/2) (0.024m)z1 = (1-1/2)(0.024m) = 1.2 cm

d) y1 = 2.4 cmy2 = 4.8 cmy3 = 3(0.024m) = 7.2 cm

y3 = +7.2 cm

y2 = 4.8 cm

y1 = +2.4 cm

y = 0

y-1= -2.4 cm

y-2 = -4.8 cm

y-3 = -7.2

+4

-4

12/10/2010 Physics 214 Fall 2010 96

Questions Chapter 18

Q11 Assuming that cathode rays are a beam of charged particles, how could you demonstrate that these particles are negatively charged?

Q13 Would you expect X rays to be produced by a television picture tube?

By deflection in a magnetic field

Yes the accelerated electrons hitting the screen do produce X rays

12/10/2010 Physics 214 Fall 2010 97

Ch 18 E 6How many electrons are required to produce 1 microcoulomb of negative charge?

e = 1.6 x 10-19C1 microcoulomb = 10-6C = nen = 10-6/(1.6 x 10-19) = 6.25 x 1012 electrons

12/10/2010 Physics 214 Fall 2010 98

Ch 18 E 10Suppose a photon has wavelength λ = 520 nma) What is frequency of photon?b) What is photon’s energy in Joules?

a) c = fλ , f = c/λ = (3 x 108)/(520 x 10-19) = 5.769 x 1014Hz

b) E = hf = (6.626 x 10-34)(5.769 x 1014) = 3.823 x 10-19J

12/10/2010 Physics 214 Fall 2010 99

Q4 Is it possible for atoms of the same chemical element to have different chemical properties?

Q5 Which number, the mass number or the atomic number, determines the chemical properties of an element?

Q7 In a nuclear reaction, can the total mass of the products of the reaction be less than the total mass of the reactants?

No the chemical properties are determined by the electrons

The atomic number gives the number of electrons

Yes. E = mc2 and mass can be turned into energy

12/10/2010 Physics 214 Fall 2010 100

Q13 In a time equal to two half-lives of a radioactive isotope, would you expect all of that isotope to have decayed?

Q15 Chemical reactions and nuclear reactions can both release energy. On the average, would you expect the energy released per unit of mass in a chemical reaction to be greater than, equal to, or less than what is released in nuclear reaction?

NO. In one half life ½ decay so in two half lives ¼ would be left

Chemical reactions involve changes in the electron energy levels and these are very low energy compared to nuclear reactions

12/10/2010 Physics 214 Fall 2010 101

Q17 Suppose that you light a match to a mixture of oxygen and hydrogen, which then reacts explosively to form water. Is this a chemical reaction or a nuclear reaction?

Q18 The most common isotope of uranium is uranium-238. Is this the isotope that is most likely to undergo fission?

It is a chemical reaction in which two hydrogen atoms and one oxygen atom join.

NO U235

12/10/2010 Physics 214 Fall 2010 102

Q19 What property of the fission reaction leads to the possibility of a chain reaction?

Q21 Do the control rods in a nuclear reactor absorb or emit neutrons?

The emittance of more than one neutron.

They absorb neutrons

12/10/2010 Physics 214 Fall 2010 103

Q22 If you wanted to slow down the chain reaction in a nuclear reactor, would you remove or insert the control rods?

Q28 How does nuclear fusion differ from nuclear fission?

Insert

Fission is breaking a very heavy element into two lighter elementsFusion is joining two very light elements into on heavier elementBoth produce energy. Fusion produces the most and is cleaner but much more difficult

12/10/2010 Physics 214 Fall 2010 104

Q29 Is nuclear fission the main process involved in the energy generated in the sun?

Q31 Which can produce larger yields of energy, a fission weapon or a fusion weapon?

No, the process is fusion.

A fusion weapon. The original bomb was a fission bomb. The hydrogen bomb uses a fission bomb to trigger a fusion bomb.

12/10/2010 Physics 214 Fall 2010 105

Homework problemsAssignment Problems

1 12 2,43 3,54 1,35 3,46 2,47 3,48 29 2,410 1,411 1,2,312 2,3,413 414 1,4,515 1,3,416 1,417 1

This list of homework problems is a guide with typical problems from each of the chapters 2 - 19

Assignment Problems18 1,3,519 1,520 1,321 1,422 323 1,324 2,425 1,326 227 3,428 2