PH12005 May2014 Solution

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Division of Physics

Examination in

PH12005 WAVES AND MECHANICS

May 2014 Two Hours Allowed

Answer EVERY question from Section AandTHREEquestions from Section B

Only calculators meeting the published Divisional regulations are permitted during

examinations

[The numbers given in square brackets indicate the mark allocated for that part of the

question. A formula sheet is attached at the end of the Examination paper]

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Section A

Answer every question in this section. These are multiple choice questions. Please write the

correct response to each question in your Examination book

1. A heavy boy and a lightweight girl are balanced on a massless seesaw. If they both

move forward so that they are one-half their original distance from the pivot point, what

will happen to the seesaw? Assume that both people are small enough compared to the

length of the seesaw to be thought of as point masses. "#$

(A) It is impossible to say without knowing the masses.

x (B) Nothing will happen; the seesaw will still be balanced.

(C) It is impossible to say without knowing the distances.(D) The side the girl is sitting on will tilt downward.

(E) The side the boy is sitting on will tilt downward.

2. Two compressible solids are formed into spheres of the same size. The bulk modulus

of sphere two is twice as large as the bulk modulus of sphere one. You now increase

the pressure on both spheres by the same amount. As a result of the increased pressure,

how is the change in volume of sphere two (!V2) related to the change in volume of

sphere one (!V1)? "#$

(A) !V2= 2!V1

(B) !V2= "!V1

x (C) !V2= #!V1

(D) !V2= !V1

(E) !V2= 4!V1

3. If you double the pressure on the surface of a can of water, the buoyant force on a stone

placed in that water will "#$

x(A) not change.

(B) double.

(C) decrease, but not by one-half.

(D) increase, but not double.

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4. You are driving a late model convertible car at the 60 m.p.h. speed limit with its soft

flexible roof closed up and the windows closed. You observe that the roof "#$

x(A) bows outward.

(B) bows inward.(C) bows inward only when you are driving downhill.

(D) is no different from when the car was at rest.

(E) bows inward only when you are driving uphill.

5. A mass M is attached to an ideal massless spring. When this system is set in motion

with amplitude A, it has a period T. What is the period if the amplitude of the motion is

increased to 2A? "#$

(A) T/2

x(B) T

(C) !!(D) 2T

(E) 4T

6. In designing buildings to be erected in an area prone to earthquakes, what relationship

should the designer try to achieve between the natural frequency of the building and the

typical earthquake frequencies? "#$

The natural frequency of the building should be

(A) almost the same as typical earthquake frequencies but slightly lower.

x(B) very different from typical earthquake frequencies.

(C) almost the same as typical earthquake frequencies but slightly higher.

(D) exactly the same as typical earthquake frequencies.

7. Four travelling waves are described by the following equations, where all quantities are

measured in SI units and y represents the displacement.

I: y= 0.12 cos(3x+ 2t)

II: y= 0.15 sin(6x- 3t)

III:y= 0.23 cos(3x+ 6t)

IV:y= -0.29 sin(1.5x- t)

Continued/..

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7. cont.

Which of these waves have the same speed? "#$

x(A) I and IV

(B) II and III(C) III and IV

(D) I and III

(E) I and II

8. A wave pulse travelling to the right along a thin cord reaches a discontinuity where the

rope becomes thicker and heavier. What is the orientation of the reflected and

transmitted pulses? "#$

(A) Both pulses are inverted.

(B) Both pulses are right side up.

x(C) The reflected pulse returns inverted while the transmitted pulse is right side

up.

(D) The reflected pulse returns right side up while the transmitted pulse is inverted.

9. The lowest-pitch tone to resonate in a pipe of length L that is closed at one end and

open at the other end is 200 Hz. Which one of the following frequencies will NOT

resonate in the same pipe? "#$

x(A) 400 Hz

(B) 600 Hz

(C) 1000 Hz

(D) 1400 Hz

(E) 1800 Hz

10. Shock waves occur when "#$

(A) the frequency of the waves is the resonant frequency of the system.

x(B) the wave source is traveling at a speed greater than the wave speed.

(C) the period of the waves matches the lifetime of the waves.

(D) the amplitude of waves exceeds the critical shock value.

(E) two waves from different sources collide with each other.

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!"#$%&' )

!"#$%& (") !"#$$*+%#,-."# -" ,/-# #%0,-."

%%& '() A solid uniform brick is placed on a sheet of wood. When one end of the sheet is

raised (see figure), you observe that the maximum that the angle $can be without

tipping over the brick is 49.6. There is enough friction to prevent the brick from

sliding.

'*) +,(- ( .,// 0123 2*(4,(5 1. 67/ 0,*89 :;/, 1? 67/ .*4;,/

(01>/& +/.*?/ (@@ 67/ .1,8/

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(b) A woman is riding a bicycle at 18.0 m/s along a straight road that runs parallel to

and right next to some railroad tracks. She hears the whistle of a train that is

behind. The frequency emitted by the train is 840 Hz, but the frequency the woman

hears is 778 Hz. Take the speed of sound to be 340 m s-1.

(i) What is the speed of the train? Is the train travelling away from or toward the

bicycle? [5]

The frequency the listener hears fLand the source frequency fSare related as follows.

fL= (cs- vL) / (cs+ vS) fS

=> vS= - cs+ (cs- vL) fS/ fL= 7.66 m s-1, moving away from the bicycle.

(ii) What frequency is heard by a stationary observer located between the train and

the bicycle? [5]

fL= cs/ (cs+ vS) fS= 821 Hz

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12. (a) A 20.0 kg uniform door has a width of 1.20 m and a height of 2.50 m. The door is

mounted on a post by a pair of hinges, marked 1 and 2 in the figure, at the top and

bottom of the door. An external force of 60.0 N, at an angle of 30.0 above the

horizontal, is applied to the small doorknob, as shown in the figure. The doorknob is

1.00 m above the bottom of the door.

'*) +,(- ( .,// 0123 2*(4,(5 1. 67/ 211, 2*,/86@3 1? 67/ .*4;,/ (01>/& +/.*?/

(@@ 67/ .1,8/

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(b) A string 40.0 cm long of mass 8.50 g is fixed at both ends and is under a tension of

425 N. When this string is vibrating in its fourth harmonic (i.e., third overtone), you

observe that it causes a nearby pipe, open at both ends, to resonate in its third

harmonic (i.e., second overtone). The speed of sound in the room is 344 m s-1.

(i) How long is the pipe? [5]

The fundamental frequency for the string is f1= (T/)1/2

/(2L1) where = m/L1.

The third overtone frequency for the string is f1= 4 f1.

The wavelength in the pipe is "2= cs / f1.

The length of the pipe is written as L2= 3 "2/2 = 3/4 csL1/ (T/)1/2

= 0.730 m.

(ii) What is the fundamental frequency of the pipe? [5]

The wavelength is written as "2= 2 L2.

The corresponding fundamental frequency is f2= cs/ "2= 236 Hz.

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13. (a) The two water reservoirs shown in the figure are open to the atmosphere, and the

water has density #1 =1000 kg m-3. The manometer contains incompressible

mercury with a density of #2 = 13,600 kg m-3. If necessary, you can use the

atmospheric pressureP0and the gravitational acceleration on Earth gto answer thefollowing questions.

(i) Write down the expression for the water pressures at A and B, and define them as

PAandPB, respectively. You can use the parameters defined in the problem. [4]

PA= P0+ #1g y1

PB= P0+ #1g y2

(ii) What is the difference in elevation h1? First, write down the expression for h1in

terms of #1, #2, and h2. Then calculate h1assuming that the manometer reading is

h2= 25 cm. [6]

The pressure at the opposite side from A (Ain the figure) is the same as the pressure at

A. The pressure at Ais also written as follows.

PA = PB+ #2g h2

=> PA= PB+ #2g h2

#1g (y1- y2) = #2g h2

#1g (h1+ h2) = #2g h2

=> h1= h2 (#2- #1)/ #1= 3.15 m

A

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(b) A tiny vibrating source sends waves uniformly in all directions. An area of 3.25

cm2on a sphere of radius 2.50 m centred on the source receives energy at a rate of

4.20 J s-1.

(i) What is the intensity of the waves at 2.50 m from the source and at 10.0 m from the

source? [5]

I1= 4.20/(3.25 x 10-4) = 1.29 x 104W m-2 (at 2.50 m)

I2= I1(r1/r2)2= 8.08 x 102W m-2 (at 10.0 m)

(ii) At what rate is energy leaving the vibrating source of the waves? [5]

P = I14$r12= I24$r2

2= 1.01 x 10

6W

14. (a) A large cylindrical water tank is mounted on a platform with its central axis vertical.

The water level is 3.75 m above the base of the tank, and base is 6.50 m above the

ground. A small hole 2.22 mm in diameter has formed in the base of the tank. Both the

hole and the top of the tank are open to the air. Ignore air resistance and treat water as

an ideal fluid with a density of 1000 kg m-3.

(i) How many cubic meters of water per second is this tank losing? [5]

The speed of water at the surface can be neglected compared to that at the bottom.

Assuming that the atmospheric pressure is P0, the Bernoullis equation at the surface

and at the bottom of the tank is written as follows.

Po+ #g L1+ 0 = Po+ 0 + #/2 v12 (L1= 3.75 m, and #= 1000 kg m

-3)

=> v1= (2 g L1)1/2

=> v1 $(d/2)2= 3.3 x 10-5m3 (d=2.22 mm)

(ii) How fast is the water from the hole moving just as it reaches the ground? [5]

Using the energy conservation equation, we can determine the water speed at the

ground as follows.

v2= (v12+ 2 g L2)

1/2= 14.2 m s-1 (L2= 6.50 m)

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(b) A 12.0 N object is oscillating in simple harmonic motion at the end of an ideal

vertical spring. Its vertical position y as a function of time tis given as follows.

y(t) = A cos(%t- &/8)

Here, A = 4.50 cm, and %= 19.5 rad s-1.

(i) What is the maximum speed that the object reaches? [2]

v(t) = dy/dt = - 19.5*4.50 sin[(19.5 rad s-1) t - &/8] => 0.878 m s-1

(ii) What is the maximum acceleration of the object? [2]

a(t) = dv/dt = - 19.5*19.5*4.50 cos[(19.5 rad s-1) t - &/8] => 17.1 m s-2

(iii) How long does it take the object to go from its highest point to its lowest point? [2]

T = 2$/19.5 => T/2 = 0.161 s

(iv) What is the spring constant of the spring? [2]

%= (k/m)1/2=> k = m*19.5*19.5 = 466 N m-1

(v) How much did the spring stretch from its original length at its equilibrium position?

[2]

k x0= mg => x0= mg/k = 2.58 cm

[END OF PAPER]

PH12005 May2014 Solution

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