Съдържание

Условие Part A: Plane pulses (1,3 т.)Part B: Waves pulses in fluid of varying depth (3,4 т.)Part C: Wave and magnetic effects (1,8 т.)Part D: Internal properties of ferrofluid within a strong magnetic field (3,5 т.)Appendix A: How to use Canon IXUS-185 Отговори Решение

APhO 2019, Експеримент — Задача 2. Wave pulses in a magnetically active fluid

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Всички материали от APhO 2019

Experiment · Asian Physics Olympiad Adelaide 2019 · Q2 · English (Official) · 10 т.

Условие

Introduction

In this question you will investigate the properties of wave pulses in ferrofluids. The surface wave pulses which can be generated in the fluids are due to the interaction of the effects of gravity, surface tension and magnetic forces.

For wave pulses, there is no single wavelength or frequency of the pulse, however, it is known that the speed of propagation of a pulse in the absence of a magnetic field is

v(d) = \kappa\sqrt{d}, \qquad (1)

where $d$ is the depth of the fluid and $\alpha$ is a constant of proportionality. This relationship only applies for smaller wave amplitudes. When the wave amplitude is large the changed fluid depth due to the pulse causes changes in speed of propagation resulting in nonlinear effects. When the nonlinear effects are not significant, wave pulses approximately maintain their profiles as they propagate. However, when nonlinear effects are significant the pulses can change their profile as they propagate and often develop ripples either ahead or behind of the main pulse.

Part A: Plane pulses (1,3 т.)

In order to generate plane waves in your ferrofluid, place the container with the ferrofluid on the wooden base, with a long side of the dish along the barrier at the edge of the base. The barrier is to ensure that your dish moves only in one direction when pushed. Ensure that the base is flat and level. To generate pulses, push the dish sharply with your hand a distance of around 2 cm. To ensure consistency in your pulse generation, you can use blu-tack and/or wooden spacers on the wooden base to mark starting and finishing positions for the ferrofluid container. It is recommended that you practice generating pulses at least a few times to be able to generate more consistent pulses.

The stand should then be set up over the dish. The lamp must be used to provide adequate lighting to be able to observe pulses in the fluid. Ensure that you can generate plane wave pulses and observe them by eye through the stand. Once you are able to do this, place the camera on the stand and make a video of a pulse. Note that the lighting and pulse amplitude may need to be adjusted to make a useful video. To adjust the lighting you can use blu-tack to place the light in an appropriate position, and may wish to use the aluminium foil to block or further direct the light.

When adjusting the lighting it is important to consider how to minimise light directly from the source into the camera, and also how to minimise reflections from the lid of the container. Reminder: you must NOT remove the lid from the container for safety reasons. Two options which can be useful in different situations are to have very diffuse light from all directions within the stand, or to direct light onto the fluid through the glass sides of the container.

See the description of the camera in Appendix A for how to make and playback videos. Note that the default automatic settings of the camera are suitable for making the required videos. If you are having difficulty clearly recording the observed effects, you need to adjust the lighting to be able to more clearly observe the wave pulses.

Three photos labelled (a), (b), (c): (a) a lidded rectangular glass dish of dark ferrofluid on a light wooden base; (b) the dish on the base under a white plastic stand (light shroud); (c) the white stand seen from above with a grid of round holes and a compact camera lying on top. — Photos of the required set up, (a) the glass dish with ferrofluid on the wooden base, (b) the glass dish on the wooden base, under the light shroud and stand, and (c) the camera on the light shroud and stand.

A.1 Draw a diagram of your set up, showing in particular how you positioned and directed the light. [0,3 т.]

A.2 Perform measurements and calculations from a video to find the speed of the waves in the ferrofluid. When performing measurements, also sketch diagrams of key frames of videos, showing important features, and marking any measurements made. [0,8 т.]

A.3 Make an estimate of the uncertainty of your measurement, showing any formulae you use to make the estimate. [0,2 т.]

Part B: Waves pulses in fluid of varying depth (3,4 т.)

Insert the nylon bolts into the holes in the corner of the wooden base. By adjusting the bolts you can set up the platform so that the fluid depth in the container varies linearly with the distance from the lower edge ( $y$ -direction), and does not vary in the perpendicular direction ( $x$ -direction). We chose $y = 0$ to be where the depth of the fluid $d = 0$ , so that the $xy$ plane is the plane of the fluid's surface.

As the speed of waves in a fluid varies with depth, if a plane pulse is generated at one end of the container it will become curved as it propagates.

Set up your glass dish with ferrofluid on the wooden base so that you will best be able to characterise the variation of pulse speed with depth in the ferrofluid.

Left: top-view diagram of the wooden base W with a barrier along its top edge, the rounded dish F on it, blue axes with origin 0 at the lower-left of the dish, x to the right and y upward, and two bolts marked B in the lower corners. Right: photo of the lidded dish of ferrofluid on the wooden base. — The set up of the glass dish with ferrofluid F on wooden base W so that the depth varies in the $y$ -direction.

B.1 i. Draw a diagram of the container with ferrofluid as you set it up. Mark measurements, with uncertainties, of all distances which are important for you to know depth as a function of $y$ in the container.

ii. Write an expression for $d(y)$ . [0,3 т.]

B.2 Generate plane wave pulses in your container, and qualitatively sketch a pulse you observe. Mark on your sketch the region where the pulse travels fastest with the letter A, and where it travels slowest with the letter B.

Note: You will need to adjust the position of the light to clearly observe the pulses. To adjust the lighting you can use blu-tack to place the light in an appropriate position, and may wish to use the aluminium foil to make light blockers or light reflectors to further direct the light. [0,3 т.]

To a first approximation, we consider the pulse which is initially travelling in the $x$ direction to remain travelling in the $x$ direction only. Note: $(x, y)$ are coordinates of points at time $t$ are coordinates of points on the pulse at time $t$ .

B.3 i. For a planar pulse which is at $x = 0$ when $t = 0$ , find a relationship between $x$ , $y$ and $t$ .

ii. Mark on your diagram from B2, with the letter V, where within the glass dish with ferrofluid as you have it set up, this approximation is more valid? [0,3 т.]

B.4 Draw a diagram of your set up, showing in particular the position and direction of the light.

Collect data to find $\kappa$ and record in the table in the answer sheet. Include estimates of uncertainty for all data. When performing measurements, also sketch diagrams of key frames of videos, showing important features, and marking any measurements made. [1,2 т.]

B.5 i. Use a graphical method to calculate a value for $\kappa$ . Include error bars for each measurement. Provide details of any formulae you use to calculate $v$ , its estimated uncertainty $\Delta v$ , $d$ and its estimated uncertainty $\Delta d$ . Record any additional calculated values for your graphs in the tables for B4.

ii. From your data, state conditions on $x$ , $y$ and $t$ for the relationship you developed in B3 to describe the observed wave pulses. [1,3 т.]

Part C: Wave and magnetic effects (1,8 т.)

Warning: if the magnets collide they are likely to break. Broken magnets are dangerous, and yours will not be replaced if you break it.

As you observed in Question 1, in the presence of a magnetic field, the ferrofluid moves into the region of the strongest field. Your aim in this part is to investigate the interaction of wave a magnetic effects qualitatively.

Set up the ferrofluid container so that the ferrofluid has constant depth, as when you generated plane pulses. Next, place two small magnets underneath, as shown below. The magnets should have their poles aligned so that they repel each other when they are pushed together sideways in the channel on the wooden board. They should be positioned so that, with the ferrofluid in place, they are as close together as possible without repelling.

A safe way to put the magnets in place is to put one in the channel, then place the glass dish with ferrofluid over the track. You can use a wooden spacer in the track to position your magnet. The second magnet can then be slid into position, starting next to the dish, and using the spacer to push it along the channel under the ferrofluid.

Generate wave pulses using the following three methods:

magnetically by rapidly withdrawing the large N42 magnet from near to the ferrofluid,
mechanically by sliding the container on the wooden base, and
mechanically by sliding the wooden base with the container fixed in place on top.

Top-view diagram of the wooden base W with a barrier along the top edge and two small holes in the lower corners; the rounded dish F sits on it, a channel C runs vertically under the dish, two small magnets m (marked with blue crosses) lie one above the other in the channel under the dish, and the large magnet M (green bar) is placed against the left side of the dish. — Diagram showing where to place the large magnet M, before rapidly removing it to generate a pulse travelling through the ferrofluid, where F is the glass dish with ferrofluid, m are the small magnets, W is the wooden base and C is the channel.

C.1 Sketch qualitative observations of the wave pulses for all three types of driving. Ensure that the diagrams clearly show how the presence of the magnets under the ferrofluid affects the wave front of the pulses. Identify all types of wave phenomena you observe. Mark with the corresponding number which effects cause which features of the observed pulse propagation.

reflection
refraction
doppler effect
beats
diffraction
interference
- (a) standing waves
- (b) from two slits or sources
- (c) from a diffraction grating
- (d) other [1,8 т.]

Part D: Internal properties of ferrofluid within a strong magnetic field (3,5 т.)

In this part your aim is to quantify the effect of a magnet on pulse propagation in a ferrofluid.

Set up the ferrofluid container so that the ferrofluid has constant depth, as when you generated plane pulses.

Generate pulses magnetically using the large N42 magnet, as in C1, and adjust the lighting to clearly observe the magnetically driven pulses.

Top-view diagram of the wooden base W with the rounded dish F in its upper part, the channel below the dish, two small holes in the lower corners, and the large magnet M (green bar) placed against the right side of the dish. — Diagram showing positioning of magnet in order to drive wave pulses in D1.

Top-view diagram of the wooden base W with the dish F; a single small magnet m (blue cross) lies under the lower-left part of the dish above the channel, and the large magnet M (green bar) is held against the upper right side of the dish. — The position of the small magnet under the glass dish with ferrofluid is marked with an x, and the position of the large magnet, which is held outside the dish and then rapidly removed to generate a wave pulse, is labelled M.

D.1 Qualitative sketch a magnetically driven pulse in the ferrofluid. [0,2 т.]

D.2 Determine the speed of the pulse with its uncertainty. Sketch and diagram of your set up, including light position and direction, and record all data and formulae you use. When performing measurements, also sketch diagrams of key frames of videos, showing important features, and marking any measurements made. [0,8 т.]

Now place a single small magnet into the channel in the wooden base, and then place the dish and ferrofluid over the magnet. Its position can be adjusted using the wooden spacers, both in the track and to push the magnet around under the glass dish.

Generate wave pulses magnetically, by starting with the large magnet in the position marked in the diagram above, and observe them travelling through the ferrofluid, including the region with the strong magnetic field.

D.3 Sketch qualitative observations of the wave pulses. Ensure that the diagrams clearly show how the presence of the magnet under the fluid affects the wave front. [0,4 т.]

D.4 Draw a diagram of your set up to determine the effect of the presence of the magnet under the fluid on the propagation time for a wave pulse. Clearly mark the positions of the magnets and also the light. [0,3 т.]

D.5 Make and record measurements with uncertainty estimates to determine the travel time of a pulse across the region with the magnet. When performing measurements, also sketch diagrams of key frames of videos, showing important features, and marking any measurements made. [1 т.]

D.6 Does the additional depth of the ferrofluid over the magnet explain the difference in speed of the wave pulses? Show calculations to support your answer. [0,8 т.]

Appendix A: How to use Canon IXUS-185

To turn the camera on or off press the button (O) labeled "ON/OFF" at the top of the camera. Note that the camera automatically turns off after a time.

To focus the camera, press the button (S) on the top of the camera part way.

To take a photo press the button (S) on the top of the camera.

To take a video press the button marked with a red dot (V) to start recording. Press it again to stop recording. Note that the default HD videos record at 25 frames per second.

To view photos or videos you have taken, press the review button (P). Use the right (R) and left (L) buttons on the ring button on the back of the camera to navigate through your photos and videos.

You can zoom in or out of photos using the ring (Z) on the top of the camera. Note: you cannot zoom after taking a video.
To play a video press the "FUNC. SET" button (F) in the middle of the ring button on the back of the camera twice.
To pause a video, press the "FUNC. SET" button (F) in the middle of the ring button on the back of the camera while the video is playing.
When a video is paused you have options on the bottom of the screen. From left to right the options are: Exit, Play, Slow Motion, Skip Backwards, Previous Frame, Next Frame, Skip Forward. Use the right (R) and left (L) buttons on the ring button on the back of the camera to navigate through options, and the "FUNC. SET" button (F) in the middle of the ring button on the back of the camera to change the volume of video playback.

To zoom in or out, push the ring (Z) to the left to zoom out and to the right to zoom in.

To enter the camera settings menu press the button (M) labeled "MENU", then use the right ring button (R) on the back of the camera to enter the camera settings menu. Note that the first option is to mute the camera sounds.

Use the right (R), left (L), up (U) and down (D) buttons on the ring button on the back of the camera to navigate through the menu.
Use the "FUNC. SET" button (F) in the middle of the ring button on the back of the camera to select options.
Use the "MENU" button (M) to exit the menu or to move back up through the menus.

To change the language of the camera, first enter the camera settings menu (see above). Press the down right button (D), labelled "INFO." on the back of the camera to scroll down the many page list to find "Language". Next press the right ring button (R) on the back of the camera to enter the list of language options. The the right (R), left (L), up (U) and down (D) buttons on the ring button on the back of the camera to navigate to the language of your choice. Use the "FUNC. SET" button (F) in the middle of the ring button on the back of the camera to select the language.

Note: You are not expected to change any settings on the camera, except for optionally changing the language, as the default automatic settings are suitable for the videos are you are required to make. You will benefit more from adjusting the lighting of your set up.

Two photos of the Canon IXUS-185 camera with blue labelled arrows. Back view: P review button, U up, V red record button, L left, F FUNC. SET centre button, R right, D down/INFO., M MENU button. Top view: O ON/OFF button, Z zoom ring, S shutter button.

Отговори

Решение

These quantitative values obtained depend strongly on the depth and other properties of the fluid, including how much oil has evaporated. The values mentioned in here correspond to the fluid as used at the APhO in Adelaide in May 2019.
Qualitative aspects of the solutions apply, even where values vary.

Throughout the light is used in two modes when taking videos. For parts A and D, the light is set up so that it is illuminating the ferrofluid surface through the sides of the box.

This direct lighting is used in Part A and Part F.

Top-view diagram of the wooden base W with the box F and the channel below it; the light L is a vertical bar to the right of the box with three arrows pointing left, into the side of the box (direct lighting).

F – box with fluid
L – light
W – wooden base

Diffuse lighting is used for other parts. In this case the light is direction up and away from the box with ferrofluid.

Top-view diagram of the wooden base W with the box F; the light L is a tilted bar in the lower right corner of the base with three arrows pointing down and to the right, away from the box (diffuse lighting).

Part A Plane pulses

A.1 Direct lighting was used, so see above for diagram.

A.2 Some example frames from videos are shown below.

Note that the scale on the graticule does not directly measure position.

A measurement with a ruler of the internal length of the container is $17.3 \pm 0.3$ cm.
The distance is equivalent to 13.4 graticule squares, so there is a factor of 17.0/13.4 to convert from graticule squares to cm.

Two video frames side by side, viewed from above through the stand: the bluish-grey ferrofluid dish with a bright vertical reflection band (the plane pulse) near the middle in the left frame and near the right end in the right frame.

The images above show two frames from a video of a plane pulse. The videos are recorded at 25 fps by default

Frames	Distance (graticule squares)	Distance (m)	Speed (m/s)
8	6.5	0.83	0.26
7	6.0	0.76	0.27
9	6.8	0.85	0.24

This gives $v = 0.26 \pm 0.02\ \mathrm{m/s}$

A.3 The uncertainty is estimated from the spread of values.

Part B Wave pulses in fluid of varying depth

B.1

Side-view sketch of the tilted set up: the wooden base on a bench line is raised at the right end by h=20 mm over a horizontal length l=195 mm; the dish with its lid rests on it, the horizontal fluid surface meets the sloping dish bottom, and the fluid depth d is marked with a double arrow.

$h = 20 \pm 1\ \mathrm{mm}$ , $l = 195 \pm 2\ \mathrm{mm}$ . The latter uncertainty is larger due to difficulty lining up ruler with projection of end of the board.

ii.
Since the depth varies linearly, $d = y\ tan\theta$ .
Here $\tan\theta = \frac{h}{l} = 0.103 \pm 0.006$ , so $d = 0.103\ y$ .

B.2

A – fastest at largest y.

Video frame of the tilted dish of ferrofluid seen from above in blue light, with a curved pulse front visible on the right-hand side.

B – slowest at y=0.

Sketch of the dish outline with dashed axes from the origin (0,0): x to the right, y upward. A thick curved pulse front starts near the x-axis in the middle and bends upward to the right, reaching near the top right. Double arrows mark A along the top (largest y) and B along the x-axis.

B.3 If x=0 when t=0, then $x = vt$ , however, $v = \alpha\sqrt{d}$ , so $x = \alpha\sqrt{d}t$ .
Hence, $x = \alpha\sqrt{y\tan\theta}\,t$ , or equivalently, $x^2 = \alpha^2\tan\theta\, t^2 y$ .
Here $\tan\theta = \frac{h}{l} = 0.103 \pm 0.006$ , so $x = \alpha\sqrt{0.103y}\cdot t$ .

This approximation will be least valid when the direction of propagation varies the most from the x-direction, and when the depth is least as there are nonlinear effects. Both of these happen for small y, in other words, $y < k$ for some constant $k$ . This is shown as the grey shaded region marked V on the diagram below.

The same sketch as for B.2 (curved pulse front, arrows A at top and B along the x-axis, origin (0,0), axes x and y), with the upper band of the dish shaded grey and marked V.

B.4 Diffuse light is more appropriate as the pulses are curved, so it is not possible to get direct reflections from the available source from enough of the pulse. See the second diagram on p. 1 for light position.

Videos of the curved pulses are recorded, with adjustments to the lighting, until a clearly visible pulse is observed near where the fastest travelling sections have reached the end. For that single frame the coordinates of points along the pulse are recorded.

Sketch of the dish outline with dashed axes from the origin (0,0), x to the right and y upward; a grey curved pulse front rising from the x-axis to the upper right, with one point on it marked by a cross and labelled (x,y).

The 8th frame after the generation of the pulse was used so $t = 8/25\ \mathrm{s} = 0.28\ s$ .
Here $x_0$ is the x coordinate of the position where the pulse front is at y=0. This is treated as the new origin, and the offset in the $x = 0$ position is $x_0 = 4.5$ gr sq.

$x$ (gr sq)	$y$ (gr sq)	$(x - x_0)^2$ (gr sq) $^2$	$\Delta[(x - x_0)^2]$
5.0	0.2	0.25	0.02
6.0	0.3	2.25	0.15
7.0	0.6	6.3	0.4
7.8	1.4	10.9	0.6
8.5	2.2	16.0	0.8
8.7	2.8	17.6	0.8
9.0	4.0	20.3	0.9
9.3	5.0	23.0	1.0
9.8	6.0	28.1	1.1
10.0	8.0	30.3	1.2
9.9	7.0	29.2	1.2

B.5

Graph of y (gr sq), 0 to 10, against (x − x0)^2 (gr sq)^2, 0 to 30, with orange data points and horizontal and vertical error bars; a dotted blue fit line through the points with y > 2 is labelled y = 0.3665x - 3.6081 and crosses zero near 10. — $y$ vs $x^2$ , $y > 2$ , in the 8th frame

Converting to SI units, the slope m = 0.280 /m.

m = \alpha^2\tan\theta\, t^2

Hence $\alpha = 5.9 \pm 0.5\ m^{1/2}s^{-1}$ , and $v(d) = 5.9 \times\sqrt{d}$ .

Uncertainty was calculated from the slope of the line of worst fit m = 0.331/m.

Part C Wave and magnetic effects

Mechanically driven by pushing glass

Dish outline with two grey circles (lumps over the magnets) on the left and four vertical pulse fronts to their right: fronts at t1 and t2 are wavy near the lumps and marked 2 (refraction), with velocities v2, v1 and v pointing right; one front marked 6 (interference); a straight front at t3 marked 1 travels left with velocity v (reflection).

Refraction observed by perturbation of front over magnets,
Interference when the fronts meet,
Reflections from end.
$t_1 < t_2 < t_3$

Mechanically driven by pushing base

Dish outline with two grey circles: a straight front at t1 left of them moving right with v; a front at t2 right of them bent near the circles and marked 2 moving right with v; a front at t3 near the right end marked 1 moving left with v.

Refraction as wave is perturbed over the fronts,
Reflections from ends,
Maybe some diffraction after passing between the lumps over the magnets.
$t_1 < t_2 < t_3$

Magnetically driven

Dish outline with two grey circles one above the other joined by a short bar; curved pulse fronts spread out from the left, with fronts behind the gap marked 5 (diffraction), crossing fronts marked 6 (interference) and fronts reflected from the walls marked 1.

Reflections from side walls and ends
Diffraction after passing between the lumps.
Interference from many waves crossing.
Each line represents the position of a pulse front in a particular frame.

Part D Internal properties of ferrofluid within a strong magnetic field

D.1

Dish outline with a sequence of concentric circular-arc pulse fronts expanding from the left side toward the right, and short arcs near the upper and lower right showing reflections from the sides.

Magnetically driven pulses are not planar, so there is resulting reflection from the sides of the container. Note the multiple lines demonstrate the same pulse at different times.

D.2 Direct lighting, as in the first diagram on p. 1.

Speed measurements of the magnetically driven waves are around the same, to somewhat faster than small amplitude mechanically driven waves. Students should find that pulses are around 0.1-0.4 (gr sq)/m faster than in A2/3.

Data and sketches should look similar to A2/3 otherwise.

Two dish outlines side by side, each with a horizontal y axis starting from a tick mark: on the left a slightly curved pulse front near the start labelled t1; on the right the front near the far end labelled t2.

Measurements of the y position at two times, and the difference in the times allow calculation of the speed.

The height of the lump can be estimated by observing from the side, or more accurately by using the limiting value from E1 Part C.

D.3 The light was placed in the diffuse light set up. See the second figure on p.1.

Sketch of a video frame: the dish edge on the left with the magnet M as a black bar at the lower left; three black curved pulse fronts moving to the upper right, the last one past a point marked x (the small magnet), and a blue curve just behind it showing the expected pulse position.

The black represents observed pulses over the magnet x. The blue represents the expected positioning of the pulse after travelled that distance.

D.4 The diffuse lighting set up was used again.

Top-view diagram of the dish with a grey circle marked m (the small magnet) near the centre and the large magnet M as a black bar against the lower left side of the dish.

D.5
A wave front was observed to travel as additional 0.8 gr sq over 8 frames. The extent of the lump is assumed to be 3cm in diameter and the additional distance in the time all occurs over the lump. The wave crossed the lump in around 2 frames This means that the wave travels at a speed of around 0.35 m/s over the lump.

Top-view diagram of the dish with the large magnet M at the lower left side and a grey circle m in the centre, with two pulse fronts, one just before and one just after the circle.

The two wavefronts shown above are only 2 frames apart.
Measurements of position were taken when the wavefronts where further apart.

D.6 The height of the lump can be estimated by observing from the side, or more accurately by using the limiting value from E1 Part C.
The best estimate of the height of the lump is found to be 5mm. The expected speed of fluid of depth 9mm, compared to fluid of depth 4mm is $\sqrt{\frac{9}{4}} = \frac{3}{2}$ times larger, which is $\frac{3}{2}\times 0.26\frac{m}{s} = 0.39\ m/s$ .

As these are estimates they are close enough that we cannot conclude that the change in wave speed is due to anything other than the additional depth of the fluid.

Оригинал в Архива: exp2.pdf · официални решения: exp2_sol.pdf