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Problem 1 (6 points)

A 0.2 kg block connected to a light spring for which the force constant is 5.0 N/m is free to oscillate on a

horizontal, frictionless surface. The block is displaced 5.0 cm from equilibrium and released from rest, as

in the figure below. (a) Find the period of its motion. (b) Determine the maximum speed of the block. (c)

What is the maximum acceleration of the block? (d) Using the given parameters, express the position,

speed, and acceleration of the block as functions of time.

Solution

Problem 2 (4 points)

An astronaut on the Moon wishes to measure the local value of the free-fall acceleration by timing pulses

traveling down a wire that has an object of large mass suspended from it. Assume a wire has a mass 0.004

kg and a length of 1.6 m, and that a 3.0-kg object is suspended from it. A pulse requires 36.1 ms to

traverse the length of the wire. Calculate gMoon from these data. (You may ignore the mass of the wire

when calculating the tension in it).

Solution

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Problem 3 (6 points)

A Doppler flow meter is used to measure the speed of blood flow. Transmitting and receiving elements

are placed on the skin, as shown in the figure. Typical sound-wave frequencies of about 5.0 MHz are

used, which have a reasonable chance of being reflected from red blood cells. By measuring the

frequency of the reflected waves, which are Doppler-shifted because the red blood cells are moving, the

speed of the blood flow can be deduced. Suppose that the receiver measured a Doppler shift of 900 Hz

(the difference between the transmitted and the received frequencies). What is the speed of blood flow?

The effective angle between the sound waves (both transmitted and reflected) and the direction of blood

flow is 45° as shown in the figure. Assume the velocity of sound in tissue is 1540 m/s.

Solution: The velocity component of the blood parallel to the sound

transmission is vblood cos 45° = 0.707vblood.

Using the Doppler equation the frequency received by the red blood cell is:

0.707 1540 / 0.7075

0 1540 /

blood bloodo

v v m s vf f MHz

v m s

Then the red blood cell becomes the source and the frequency received by the stationary receiver becomes:

1540 / 0.7070 1540 /5

0.707 1540 / 1540 / 0.707

blood

blood blood

m s vv m sf f MHz

v v m s m s v

1540 / 0.7075

1540 / 0.707

blood

blood

m s vf MHz

m s v

Since the red blood cell is moving away from the received frequency should become smaller than the transmitted frequency therefore:

2 0.707900 5 5

1540 / 0.707

blood

blood

vDoppler shift Hz MHz f MHz

m s v

Solving the equation for vblood:

0.1960 / 0.2 /bloodv m s m s

Problem 4 (4 points)

If the frequencies of two adjacent harmonics of an organ pipe are determined to be 550 Hz and 650 Hz,

calculate (a) the fundamental frequency of the pipe, (b) is the pipe open at both ends or open at only one

end? (c) What is the length of this pipe?

Solution: The resonant frequencies are equally spaced as numbers. The set of resonant frequencies then must be 650 Hz, 550 Hz, 450 Hz, 350 Hz, 250 Hz, 150 Hz, 50 Hz. These are odd-integer multipliers of the fundamental frequency of 50 Hz, so the pipe is closed at one end. Then the pipe length is:

1

343 /1.715

4 4(50 )

v m sL m

f Hz

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Problem 5 (7 points)

You are standing 2.5 m directly in front of one of the two loudspeakers as shown in the figure below.

They are 3.0 m apart and both are playing a 686 Hz tone in phase. As you begin to walk directly away

from the speaker, at what distances from that speaker do you hear a minimum sound intensity? Assume

the speed of sound in air v = 343 m/s.

Solution:

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Problem 6 (4 points)

A loudspeaker is placed between two observers who are 110 m apart, along the line connecting them. If

one observer records a sound level of 60.0 dB and the other records a sound level of 80.0 dB, how far is

the speaker from each observer?

Solution:

11

0

(10 ) logI

dBI

and 22

0

(10 ) logI

dBI

Therefore: 22 1

1

(10 ) logI

dBI

Also we know that: 1 22 21 24 4

s sP PI and I

r r Thus:

2

2 1

1 2

I r

I r

and:

2

1 12 1

2 2

(10 ) log (20 ) logr r

dB dBr r

2 1 80 60

1 20 20

2

10 10 10r

r

and 1 2 110r r m therefore: 1 2100 10r m and r m

Problem 7 (5 points)

(a) Show that the lens equation can be written in the Newtonian form: 2' fxx

where x is the distance of the object from the focal point on the front side of the lens, and x’ is the distance

of the image to the focal point on the other side of the lens. Calculate the location of the image if the

object is placed 45.0 cm in front of a converging lens with a focal length f = 32.0 cm using (b) the

standard form of the thin lens equation, and (c) the Newtonian form, stated above.

Solution. (do – object distance, di – image distance)

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Problem 8 (6 points)

In the figure, the three particles are fixed in place and have charges q1 = q2 = +e and q3 = +2e. The

distance a = 6.00 µm. What are the (a) magnitude and (b) direction of the net electric field at point P due

to the particles? (c) Find the electric potential at P, taking zero at infinity. (d) Find the work required to

move a 3 µC charge from infinity to P.

SOLUTION. By symmetry we see that the contributions from the two charges q1 = q2 = +e cancel each other, and we simply compute magnitude of the field due to q3 = +2e. (a) The magnitude of the net electric field is

CN

a

ek

r

ekE eenet /160

2/

2222

(b) This field points at 45.0°, counterclockwise from the x axis.

(c) total potential at P: mVa

ek

a

e

a

e

a

ekV ee 36.1

2/

4

2/

2

2/2/

(d) the work is JmVCVqWAB

96 101.436.1103

Problem 9 (6 points)

A light source emits visible light of two wavelengths: 430 nm and 510 nm . The source is used

in a double-slit interference experiment in which L= 1.50 m and d = 0.025 mm.

a) Find the distance between the locations of the third-order (m = 3) bright fringes corresponding to each

of the wavelengths.

b) Show that the overlapping bright fringes for two different wavelengths obey the following relationship

even for large values of the angle θ:

m

m

c) Find the value of y on the screen at which the fringes from the two wavelengths first coincide.

Solution: a) with m = 3, we find that the fringe positions corresponding to these two wavelengths are:

92

3 3

(430 10 )sin 3(1.5 ) 7.74 10

(0.025 10 )

brightbright

y mm y mL m m

d L d m

and 9

23 3

(510 10 )sin 3(1.5 ) 9.18 10

(0.025 10 )

brightbright

y mm y mL m m

d L d m

Hence, the separation distance between the two fringes is:

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2 2 29.18 10 7.74 10 1.40 10 1.4bright bright brighty y y m m m cm

b)

c)

Another version: ym = Lmλ/d for m=51 or m = 43 was also accepted. Problem 10 (6 points)

Find (a) the equivalent capacitance of the capacitors combination in the figure, (b) the charge on each

capacitor, and (c) the potential difference across each capacitor.

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Problem 11 (6 points)

In the circuit below, determine (a) the current in each resistor and (b) the potential difference across

200-Ω resistor.

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Problem 12 (20 marks, 1 mark for each question). Answer the following multiple choice questions.

Circle the correct answer.

1. D If a metallic wire of cross sectional area 3.0 106 m2 carries a current of 6.0 A and has a mobile charge

density of 4.24 1028 carriers/m3, what is the average drift velocity of the mobile charge carriers?

(charge value = 1.6 1019 C) a. 3.4 103 m/s b. 1.7 103 m/s c. 1.5 104 m/s d. 2.9 104 m/s

2.B A Nichrome wire has a radius of 0.50 mm and a resistivity of 1.5 106 m. If the wire carries a current of 0.50 A, what is the potential difference per unit length along this wire? a. 0.003 V/m b. 0.95 V/m c. 1.6 V/m d. 1.9 V/m

3.C The unit for rate of energy transformation, the watt, in an electric circuit is equivalent to which of the following? a. V/s b. A c. VA d. V/

4.B When a 24.0- resistor is connected across a 12.0-V battery, a current of 482 mA flows. What is the internal resistance of the battery? a. 0.02 b. 0.9 c. 25.0 d. 49.8

5.A Three resistors connected in parallel each carry currents labeled I1, I2 and I3. Which of the following expresses the value of the total current IT in the combined system? a. IT = I1 + I2 + I3 b. IT = (1/I1 + 1/I2 + 1/I3) c. IT = I1 = I2 = I3 d. IT = (1/I1 + 1/I2 + 1/I3)

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6.C Resistors of values 6.0, 4.0, 10.0 and 7.0 are combined as shown. What is the equivalent resistance for this combination? a. 2.3 b. 3.0 c. 10.7 d. 27

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7.C What is the maximum number of 60-W light bulbs you can connect in parallel in a 120-V home circuit without tripping the 30-A circuit breaker? a. 11 b. 35 c. 59 d. 3 600

8.C Kirchhoff’s rules are the junction rule and the loop rule. Which of the following statements is true? a. Both rules are based on the conservation of charge. b. Both rules are based on the conservation of energy. c. The junction rule is based on the conservation of charge, and the loop rule is based on

the conservation of energy. d. The junction rule is based on the conservation of energy, and the loop rule is based on

the conservation of charge.

9.B The magnetic field of the Earth is believed responsible for which of the following? a. deflection of both charged and uncharged cosmic rays b. deflection of charged cosmic rays c. ozone in the upper atmosphere d. solar flares

10.B A proton and a deuteron are moving with equal velocities perpendicular to a uniform magnetic field. A deuteron has the same charge as the proton but has twice its mass. The ratio of the magnetic force on the proton to that on the deuteron is: a. 0.5. b. 1. c. 2. d. There is no magnetic force in this case.

11.D A 2.0-m wire segment carrying a current of 0.60 A oriented parallel to a uniform magnetic field of 0.50 T experiences a force of what magnitude? a. 6.7 N b. 0.30 N c. 0.15 N d. zero

12.D A proton, which moves perpendicular to a magnetic field of 1.2 T in a circular path of radius 0.080 m, has

what speed? (qp = 1.6 1019 C and mp = 1.67 1027 kg) a. 3.4 106 m/s b. 4.6 106 m/s c. 9.6 106 m/s d. 9.2 106 m/s

13.D If a charged particle is moving in a uniform magnetic field, its path can be:

a. a straight line. b. a circle. c. a helix. d. any of the above.

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14.C In a mass spectrometer, an ion will have a smaller radius for its circular path if: a. its speed is greater. b. its mass is greater. c. its charge is greater. d. the magnetic field is weaker.

15.B The current in a long wire creates a magnetic field in the region around the wire. How is the strength of the field at distance r from the wire center related to the magnitude of the field? a. field directly proportional to r b. field inversely proportional to r c. field directly proportional to r2 d. field inversely proportional to r2

16.A Two long parallel wires 40 cm apart are carrying currents of 10 A and 20 A in the same direction. What is the magnitude of the magnetic field halfway between the wires? a. 1.0 10-5 T b. 2.0 10-5 T c. 3.0 10-5 T d. 4.0 10-5 T

17.C Two parallel wires are separated by 0.25 m. Wire A carries 5.0 A and Wire B carries 10 A, both currents in the same direction. The force on 0.80 m of Wire A is: a. half that on 0.80 m of wire B. b. one-fourth that on 0.80 m of wire B. c. toward Wire B. d. away from Wire B.

18.B A current in a solenoid coil creates a magnetic field inside that coil. The field strength is directly proportional to: a. the coil area. b. the current. c. Both A and B are valid choices. d. None of the above choices are valid.

19.A Two parallel wires carry the a current I in the same direction. Midway between these wires is a third wire, also parallel to the other two, which carries a current 0.5 I, but in the direction opposite from the first two wires. In which direction are the net forces on the outer wires? a. Since the net forces are zero, there is no direction. b. Both forces are toward the center wire. c. Both forces are away from the center wire. d. Both forces on the two wires are in the same direction in space.

20.B A jet traveling at the speed of sound produces a fog pattern due to a. Increase in the air temperature. b. The large pressure variation. c. Changing speed. d. None of the above.