Standing Waves [93 marks] - Peda.net · Standing Waves [93 marks] 1. A student blows across the top...

Standing Waves [93 marks]

1. A student blows across the top of a cylinder that contains water. A first-harmonic standing sound wave is produced in the air of the cylinder. Morewater is then added to the cylinder. The student blows so that a first-harmonicstanding wave is produced with a different frequency.

What is the nature of the displacement in the air at the water surface and thechange in frequency when the water is added?

MarkschemeD

Examiners report[N/A]

[1 mark]

2. In an experiment to determine the speed of sound in air, a tube that isopen at the top is filled with water and a vibrating tuning fork is held overthe tube as the water is released through a valve.An increase in intensity in the sound is heard for the first time when the aircolumn length is . The next increase is heard when the air column length is .

Which expressions are approximately correct for the wavelength of the sound?I. 4II. 4

III. A. I and IIB. I and IIIC. II and IIID. I, II and III

MarkschemeB

Examiners reportThe question was well answered by students.

x y

x

y

4y

3

[1 mark]

3. A third-harmonic standing wave of wavelength 0.80 m is set up on astring fixed at both ends. Two points on the wave are separated by adistance of 0.60 m. What is a possible phase difference between the two points onthe wave?A.

B. C. D.

MarkschemeC

Examiners reportThis question had a low discrimination index with response D the most popularand an even spread between the other 3 answers. A third-harmonic standingwave of wavelength 0.8m must be on a string of length 1.2m giving 3 loops of0.4m each. Depending on where the initial point is chosen, two pointsseparated by 0.6m will either be in adjacent loops e.g. at 0.1m and 0.7 m witha phase difference of π or in the two end loops e.g at 0.3 m and 0.9m with aphase difference of 0. So for a standing wave there are only two possibleanswers, π (response C) or 0 (not included in these responses).

radπ4

radπ2

πrad

rad3π2

[1 mark]

4a.

The diagram shows the direction of a sound wave travelling in a metal sheet.

Particle P in the metal sheet performs simple harmonic oscillations.When the displacement of P is 3.2 μm the magnitude of its accelerationis 7.9 m s . Calculate the magnitude of the acceleration of P when itsdisplacement is 2.3 μm.

MarkschemeExpression or statement showing acceleration is proportional to displacement✔so « » ✔

Examiners reportThis was well answered at both levels.

-2

7.9 × » = 5.7«m s−22.33.2

[2 marks]

4b. The wave is incident at point Q on the metal–air boundary. The wavemakes an angle of 54° with the normal at Q. The speed of sound in themetal is 6010 m s and the speed of sound in air is 340 m s . Calculate the anglebetween the normal at Q and the direction of the wave in air.

Markscheme ✔

θ = 2.6° ✔

Examiners reportMany scored full marks on this question. Common errors were using thecalculator in radian mode or getting the equation upside down.

–1 –1

sin θ = × sin 54∘3406010

4c.

The frequency of the sound wave in the metal is 250 Hz.

State the frequency of the wave in air.

Markschemef = 250 «Hz» OR Same OR Unchanged ✔

[2 marks]

[1 mark]

Examiners reportMany used a ratio of the speeds to produce a new frequency of 14Hz (340 x250/6010). It would have helped candidates if they had been aware that thecommand term ‘state’ means ‘give a specific name, value or other briefanswer without explanation or calculation.’

4d. Determine the wavelength of the wave in air.

Markscheme» ✔

Examiners report[N/A]

λ = « = »1.36 ≈ 1.4«m340250

4e. The sound wave in air in (c) enters a pipe that is open at both ends. Thediagram shows the displacement, at a particular time T, of the standingwave that is set up in the pipe.

On the diagram, at time T, label with the letter C a point in the pipe that is at thecentre of a compression.

[1 mark]

[1 mark]

Markschemeany point labelled C on the vertical line shown below ✔eg:

Examiners reportThis was answered well at both levels.

5a.

The diagram shows the direction of a sound wave travelling in a metal sheet.

Particle P in the metal sheet performs simple harmonic oscillations.When the displacement of P is 3.2 μm the magnitude of its accelerationis 7.9 m s . Calculate the magnitude of the acceleration of P when itsdisplacement is 2.3 μm.

MarkschemeExpression or statement showing acceleration is proportional to displacement✔so « » = 5.7«ms » ✔

Examiners reportThis was well answered at both levels.

-2

7.9 × 2.33.2

–2

[2 marks]

5b. The wave is incident at point Q on the metal–air boundary. The wavemakes an angle of 54° with the normal at Q. The speed of sound in themetal is 6010 m s and the speed of sound in air is 340 m s . Calculate the anglebetween the normal at Q and the direction of the wave in air.

Markscheme ✔

✔

Examiners reportMany scored full marks on this question. Common errors were using thecalculator in radian mode or getting the equation upside down.

–1 –1

sin θ = × sin 5403406010

θ = 2.60

5c. The frequency of the sound wave in the metal is 250 Hz. Determine thewavelength of the wave in air.

[2 marks]

[1 mark]

Markscheme ✔

Examiners reportThis was very well answered.

λ = « = »1.36 ≈ 1.4«m»340250

5d.

The sound wave in air in (c) enters a pipe that is open at both ends. The diagramshows the displacement, at a particular time T, of the standing wave that is set upin the pipe.

A particular air molecule has its equilibrium position at the point labelled M.

On the diagram, at time T, draw an arrow to indicate the acceleration ofthis molecule.

Markschemehorizontal arrow «at M» pointing left ✔

[1 mark]

Examiners reportVery few candidates could interpret this situation and most arrows wereshown in a vertical plane.

5e. On the diagram, at time T, label with the letter C a point in the pipe thatis at the centre of a compression.

Markschemeany point labelled C on the vertical line shown below ✔eg:

Examiners reportThis was answered well at both levels.

[1 mark]

5f.

Sound of frequency f = 2500 Hz is emitted from an aircraft that moves with speedv = 280 m s away from a stationary observer. The speed of sound in still air is c= 340 m s .

Calculate the frequency heard by the observer.

Markscheme ✔

«Hz» ✔

Examiners reportThis was answered well with the most common mistake being to swap thespeed of sound and the speed of the aircraft.

–1 –1

f ′ = 2500 × 340340+280

f ′ = 1371 ≈ 1400

5g. Calculate the wavelength measured by the observer.

[2 marks]

[1 mark]

Markscheme«m» ✔

Examiners reportAnswered well with ECF often being awarded to those who answered theprevious part incorrectly.

λ′ = ≈ 0.24/0.253401371

6. Two strings of lengths L and L are fixed at both ends. The wavespeed isthe same for both strings. They both vibrate at the same frequency. Lvibrates at its first harmonic. L vibrates at its third harmonic.What is ?

A. B. 1C. 2D. 3

MarkschemeD

Examiners report[N/A]

1 21

2L1

L2

13

[1 mark]

7a.

A pipe is open at both ends. A first-harmonic standing wave is set up in the pipe.The diagram shows the variation of displacement of air molecules in the pipe withdistance along the pipe at time t = 0. The frequency of the first harmonic is f.

Sketch, on the diagram, the variation of displacement of the air moleculeswith distance along the pipe when t = .

Markschemehorizontal line shown in centre of pipe ✔

Examiners report[N/A]

34f

[1 mark]

7b. An air molecule is situated at point X in the pipe at t = 0. Describe themotion of this air molecule during one complete cycle of the standingwave beginning from t = 0.

Markscheme«air molecule» moves to the right and then back to the left ✔returns to X/original position ✔

Examiners report[N/A]

[2 marks]

7c. The speed of sound c for longitudinal waves in air is given by

where ρ is the density of the air and K is a constant.A student measures f to be 120 Hz when the length of the pipe is 1.4 m. Thedensity of the air in the pipe is 1.3 kg m . Determine the value of K for air. Stateyour answer with the appropriate fundamental (SI) unit.

Markschemewavelength = 2 × 1.4 «= 2.8 m» ✔c = «f λ =» 120 × 2.8 «= 340 m s » ✔K = «ρc = 1.3 × 340 =» 1.5 × 10 ✔kg m s ✔

Examiners report[N/A]

c = √ Kρ

–3

−1

2 2 5

–1  –2

[4 marks]

7d.

A transmitter of electromagnetic waves is next to a long straight vertical wall thatacts as a plane mirror to the waves. An observer on a boat detects the waves bothdirectly and as an image from the other side of the wall. The diagram shows oneray from the transmitter reflected at the wall and the position of the image.

Demonstrate, using a second ray, that the image appears to come fromthe position indicated.

Markschemeconstruction showing formation of image ✔Another straight line/ray from image through the wall with line/ray fromintersection at wall back to transmitter. Reflected ray must intersect boat.

Examiners report[N/A]

[1 mark]

7e. Outline why the observer detects a series of increases and decreases inthe intensity of the received signal as the boat moves along the line XY.

Markschemeinterference pattern is observedORinterference/superposition mentioned ✔

maximum when two waves occur in phase/path difference is nλORminimum when two waves occur 180° out of phase/path difference is (n + ½)λ✔

Examiners report[N/A]

[2 marks]

8. A first-harmonic standing wave is formed on a vertical string of length 3.0m using a vibration generator. The boundary conditions for this string arethat it is fixed at one boundary and free at the other boundary.

The generator vibrates at a frequency of 300 Hz.What is the speed of the wave on the string?A. 0.90 km sB. 1.2 km sC. 1.8 km sD. 3.6 km s

MarkschemeD

Examiners report[N/A]

–1

–1

–1

–1

9. A string stretched between two fixed points sounds its second harmonicat frequency f.

Which expression, where n is an integer, gives the frequencies of harmonics thathave a node at the centre of the string?A. B. nfC. 2nfD. (2n + 1)f

fn+12

[1 mark]

[1 mark]

MarkschemeB

Examiners report[N/A]

10a.

A loudspeaker emits sound towards the open end of a pipe. The other end isclosed. A standing wave is formed in the pipe. The diagram represents thedisplacement of molecules of air in the pipe at an instant of time.

Outline how the standing wave is formed.

Markschemethe incident wave «from the speaker» and the reflected wave «from the closedend»superpose/combine/interfere Allow superimpose/add upDo not allow meet/interact[1 mark]

Examiners report[N/A]

[1 mark]

10b.

X and Y represent the equilibrium positions of two air molecules in the pipe. Thearrow represents the velocity of the molecule at Y.

Draw an arrow on the diagram to represent the direction of motion ofthe molecule at X.

MarkschemeHorizontal arrow from X to the right MP2 is dependent on MP1Ignore length of arrow[1 mark]

Examiners report[N/A]

10c. Label a position N that is a node of the standing wave.

MarkschemeP at a node

[1 mark]

Examiners report[N/A]

–1

[1 mark]

[1 mark]

10d. The speed of sound is 340 m s and the length of the pipe is 0.30m. Calculate, in Hz, the frequency of the sound.

Markschemewavelength is λ = « =» 0.40 «m»

f = « » 850 «Hz» Award [2] for a bald correct answerAllow ECF from MP1[2 marks]

Examiners report[N/A]

–1

4×0.303

3400.40

[2 marks]

10e.

The loudspeaker in (a) now emits sound towards an air–water boundary. A, B andC are parallel wavefronts emitted by the loudspeaker. The parts of wavefronts Aand B in water are not shown. Wavefront C has not yet entered the water.

The speed of sound in air is 340 m s and in water it is 1500 m s .The wavefronts make an angle θ with the surface of the water. Determinethe maximum angle, θ , at which the sound can enter water. Give youranswer to the correct number of significant figures.

–1 –1

max

[2 marks]

Markscheme

θ = 13«°» Award [2] for a bald correct answerAward [2] for a bald answer of 13.1 Answer must be to 2/3 significant figures to award MP2Allow 0.23 radians[2 marks]

Examiners report[N/A]

=sin θc

3401

1500

c

10f. Draw lines on the diagram to complete wavefronts A and B in water forθ < θ .

Markschemecorrect orientationgreater separation Do not penalize the lengths of A and B in the waterDo not penalize a wavefront for C if it is consistent with A and BMP1 must be awarded for MP2 to be awarded

[2 marks]

max[2 marks]

Examiners report[N/A]

11a. Outline how the standing wave is formed.

Markschemethe incident wave «from the speaker» and the reflected wave «from the closedend»superpose/combine/interfere Allow superimpose/add upDo not allow meet/interact[1 mark]

Examiners report[N/A]

11b. Draw an arrow on the diagram to represent the direction of motion ofthe molecule at X.

MarkschemeHorizontal arrow from X to the right MP2 is dependent on MP1Ignore length of arrow[1 mark]

Examiners report[N/A]

[1 mark]

[1 mark]

11c. Label a position N that is a node of the standing wave.

MarkschemeP at a node

[1 mark]

Examiners report[N/A]

11d. The speed of sound is 340 m s and the length of the pipe is 0.30m. Calculate, in Hz, the frequency of the sound.

Markschemewavelength is λ = « =» 0.40 «m»

f = « » 850 «Hz» Award [2] for a bald correct answerAllow ECF from MP1[2 marks]

–1

4×0.303

3400.40

[1 mark]

[2 marks]

Examiners report[N/A]

11e. The speed of sound in air is 340 m s and in water it is 1500 m s .The wavefronts make an angle θ with the surface of the water. Determinethe maximum angle, θ , at which the sound can enter water. Give youranswer to the correct number of significant figures.

Markscheme

θ = 13«°» Award [2] for a bald correct answerAward [2] for a bald answer of 13.1 Answer must be to 2/3 significant figures to award MP2Allow 0.23 radians[2 marks]

Examiners report[N/A]

–1 –1

max

=sin θc

3401

1500

c

[2 marks]

11f. Draw lines on the diagram to complete wavefronts A and B in water forθ < θ .

Markschemecorrect orientationgreater separation Do not penalize the lengths of A and B in the waterDo not penalize a wavefront for C if it is consistent with A and BMP1 must be awarded for MP2 to be awarded

[2 marks]

Examiners report[N/A]

max

12. A pipe of fixed length is closed at one end. What is ?

A.

B. C. 3D. 5

MarkschemeC

third harmonic frequency of pipefirst harmonic frequency of pipe

15

13

[2 marks]

[1 mark]

Examiners report[N/A]

13. The diagram shows a second harmonic standing wave on a string fixed atboth ends.

What is the phase difference, in rad, between the particle at X and the particle atY?A. 0B.

C.

D.

MarkschemeA

Examiners report[N/A]

π4π2

3π4

[1 mark]

14. Two pulses are travelling towards each other.

What is a possible pulse shape when the pulses overlap?

MarkschemeA

Examiners report[N/A]

15. The frequency of the first harmonic standing wave in a pipe that is openat both ends is 200 Hz. What is the frequency of the first harmonic in apipe of the same length that is open at one end and closed at the other?A. 50 HzB. 75 HzC. 100 HzD. 400 Hz

[1 mark]

[1 mark]

MarkschemeC

Examiners report[N/A]

16. Water is draining from a vertical tube that was initially full. A vibratingtuning fork is held near the top of the tube. For two positions of the watersurface only, the sound is at its maximum loudness.

The distance between the two positions of maximum loudness is x.What is the wavelength of the sound emitted by the tuning fork?A. B. xC. D. 2x

MarkschemeD

Examiners report[N/A]

x2

3x2

[1 mark]

17a.

A student investigates how light can be used to measure the speed of a toy train.

Light from a laser is incident on a double slit. The light from the slits is detectedby a light sensor attached to the train.The graph shows the variation with time of the output voltage from the lightsensor as the train moves parallel to the slits. The output voltage is proportionalto the intensity of light incident on the sensor.

Explain, with reference to the light passing through the slits, why aseries of voltage peaks occurs.

[3 marks]

Markscheme«light» superposes/interferespattern consists of «intensity» maxima and minimaORconsisting of constructive and destructive «interference»voltage peaks correspond to interference maxima

Examiners report[N/A]

17b. The slits are separated by 1.5 mm and the laser light has awavelength of 6.3 x 10 m. The slits are 5.0 m from the train track.

Calculate the separation between two adjacent positions of the train when theoutput voltage is at a maximum.

–7[1 mark]

Markscheme« » 2.1 x 10 «m»

If no unit assume m.Correct answer only.

Examiners report[N/A]

s = = =λD

d

6.3×10−7×5.01.5×10−3

–3

17c. Estimate the speed of the train.

Markschemecorrect read-off from graph of 25 m s

v = « » 8.4 x 10 «m s »

Allow ECF from (b)(i)

Examiners report[N/A]

= =xt

2.1×10−3

25×10−3–2 –1

[2 marks]

17d. In another experiment the student replaces the light sensor with asound sensor. The train travels away from a loudspeaker that is

emitting sound waves of constant amplitude and frequency towards a reflectingbarrier.

The sound sensor gives a graph of the variation of output voltage with time alongthe track that is similar in shape to the graph shown in the resource. Explain howthis effect arises.

MarkschemeALTERNATIVE 1«reflection at barrier» leads to two waves travelling in opposite directionsmention of formation of standing wavemaximum corresponds to antinode/maximum displacement «of air molecules»ORcomplete cancellation at node position

Examiners report[N/A]

[2 marks]

18a. Explain, with reference to the light passing through the slits, why aseries of voltage peaks occurs.

Markscheme«light» superposes/interferespattern consists of «intensity» maxima and minimaORconsisting of constructive and destructive «interference»voltage peaks correspond to interference maxima

Examiners report[N/A]

18b. The slits are separated by 1.5 mm and the laser light has awavelength of 6.3 x 10 m. The slits are 5.0 m from the train track.

Calculate the separation between two adjacent positions of the train when theoutput voltage is at a maximum.

–7

[3 marks]

[1 mark]

Markscheme« » 2.1 x 10 «m»

If no unit assume m.Correct answer only.

Examiners report[N/A]

s = = =λD

d

6.3×10−7×5.01.5×10−3

–3

18c. Estimate the speed of the train.

Markschemecorrect read-off from graph of 25 m s

v = « » 8.4 x 10 «m s »

Allow ECF from (b)(i)

Examiners report[N/A]

= =xt

2.1×10−3

25×10−3–2 –1

[2 marks]

A student investigates how light can be used to measure the speed of a toy train.

Light from a laser is incident on a double slit. The light from the slits is detectedby a light sensor attached to the train.The graph shows the variation with time of the output voltage from the lightsensor as the train moves parallel to the slits. The output voltage is proportionalto the intensity of light incident on the sensor.

As the train continues to move, the first diffraction minimum is observed when thelight sensor is at a distance of 0.13 m from the centre of the fringe pattern.

18d. Determine the width of one of the slits.

Markschemeangular width of diffraction minimum = «= 0.026 rad»

slit width = « » 2.4 x 10 «m» Award [1 max] for solution using 1.22 factor.

Examiners report[N/A]

0.135.0

= =λ

d

6.3×10−7

0.026–5

[2 marks]

18e. Suggest the variation in the output voltage from the light sensor thatwill be observed as the train moves beyond the first diffraction

minimum.

Markscheme«beyond the first diffraction minimum» average voltage is smaller

«voltage minimum» spacing is «approximately» sameORrate of variation of voltage is unchanged OWTTE

Examiners report[N/A]

[2 marks]

18f. In another experiment the student replaces the light sensor with asound sensor. The train travels away from a loudspeaker that is emittingsound waves of constant amplitude and frequency towards a reflecting barrier.

The graph shows the variation with time of the output voltage from the soundssensor.

Explain how this effect arises.

[2 marks]

Markscheme«reflection at barrier» leads to two waves travelling in opposite directions mention of formation of standing wavemaximum corresponds to antinode/maximum displacement «of air molecules»ORcomplete cancellation at node position

Examiners report[N/A]

19. A pipe of length L has two open ends. Another pipe of length L′ has oneopen end and one closed end.The frequency of the first harmonic of both pipes is the same. What is ?A. 2B. C. 1D.

MarkschemeD

Examiners report[N/A]

L′

L

32

12

[1 mark]

20a.

A longitudinal wave is travelling in a medium from left to right. The graph showsthe variation with distance x of the displacement y of the particles in the medium.The solid line and the dotted line show the displacement at t=0 and t=0.882 ms,respectively.

The period of the wave is greater than 0.882 ms. A displacement to the right ofthe equilibrium position is positive.

(i) Calculate the speed of this wave.(ii) Show that the angular frequency of oscillations of a particle in the mediumis ω=1.3×10 rads .3 −1

[4 marks]

Markscheme(i)ALTERNATIVE 1«distance travelled by wave =» 0.30 m

ALTERNATIVE 2evaluates «=4.7ms» to give f = 210 or 212 Hzuses λ=1.6 m with v=fλ to give 340ms(ii)ALTERNATIVE 1λ=1.60m

or 1.34×10 radsALTERNATIVE 2«0.882 ms is of cycle so whole cycle is» 1.35×10 radsAllow ECF from (b)(i).

Examiners report[N/A]

v =≪ =≫ 340ms−1distancetime

T = 0.882×10−3×1.60.3

–1

ω =≪ 2πf =≫ 2π × = 1.3 × 1033401.60

3 –1

0.31.6

2π×316×0.882×10−3

3 –1

20b. One particle in the medium has its equilibrium position at x=1.00 m.(i) State and explain the direction of motion for this particle at t=0.(ii) Show that the speed of this particle at t=0.882 ms is 4.9ms .−1

[4 marks]

Markscheme(i)the displacement of the particle decreases OR «on the graph» displacement isgoing in a negative direction OR on the graph the particle goes down OR onthe graph displacement moves towards equilibrium/0to the leftDo not allow “moving downwards”.(ii)y=–1.5mm

«v=4.939≈4.9ms »Allow ECF from (b)(ii).Do not allow .

Examiners report[N/A]

v = 2π × 212 × √(4.0 × 10−3)2− (1.5 × 10−3)2

-1

= 4.87ms−14.3mm0.882ms

20c. The travelling wave in (b) is directed at the open end of a tube of length1.20 m. The other end of the tube is closed.

(i) Describe how a standing wave is formed.(ii) Demonstrate, using a calculation, that a standing wave will be established inthis tube.

[3 marks]

Markscheme(i)the superposition/interference of two oppositely moving/reflected «identicaltravelling» waves(ii)the allowed wavelengths in the tube are , n = 1, 3, 5,…ORdiagram showing of a standing wavelength in the tube

ORjustification that

Allow diagram showing of a wavelength for MP1.

Examiners report[N/A]

λ = =4Ln

480n

34

1.6 = ⇒ n = 34.80n

× 1.6 = 1.2m34

34

21a. Outline what is meant by a black hole.

Markschemeregion of space with extreme/very large curvature of spacetimesuch that light cannot escape the region OR escape speed within region is > cDo not allow “large” or omission of degree of curvature.

Examiners report[N/A]

[2 marks]

Printed for Jyvaskylan Lyseonlukio

© International Baccalaureate Organization 2020 International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

21b. An observer views a distant spacecraft that is 23.0 km from the centreof a black hole. The spacecraft contains a clock that ticks once every

second and the ticks can be detected by the distant observer. In 2.00 minutes theobserver counts 112 ticks of the clock.Determine the mass of the black hole.

Markschemetime for 1 second spacecraft tick in observer frame =1.07s

OR R =2.96×10 m

kg

Examiners report[N/A]

1.07 = 1.00

√1−RS

2.3×104

S 3

M =≪ =≫ 2.0 × 1030c2×2.96×103

2×6.67×10−11

[3 marks]