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Условие Objective Apparatus Experimental Procedure Part 1 (8 т.)Part 2 (2 т.)Note on uncertainty evaluation Отговори Решение

APhO 2012, Експеримент — Задача 1. FRICTION

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Всички материали от APhO 2012

13th Asian Physics Olympiad — New Delhi, India — Experimental Competition — Wednesday, 2nd May 2012 · 2 май 2012 г. · 10 т.

Внимание: Бележка към темата

The sample measurements and all printed official-key equations are reproduced as given. In Part 2, the signs in equations (6)–(7) conflict with the following positive logarithmic ratio and numerical tables. The key also calls μ the slope although the logarithmic graph has negative slope, and its half-value-method sentence is incomplete. These source inconsistencies have not been silently corrected. The marking scheme uses “and” between disjoint ranges and labels the mass relation without g as an incorrect equation; these criteria are retained as printed. The 8- and 2-point section totals are sums of its respective criterion maxima. No separate Experiment 1 answer sheet was found in the complete archived 2012 directory.

Please first read the following instructions carefully:

The time available is 2½ hours for each of the two experimental questions.
Use only the pen and equipment provided.
In the boxes at the top of each sheet of paper write down your student code.
There is a set of answersheets in which you have to enter your data and results. In addition blank sheets are provided which you can use.
Use only pencil to draw graphs on the graph sheets.
If you add any blank sheets to your main answersheets, write down the progressive number of each sheet (page number) on these additional sheets. You should enter the number of the question (Question No.) on such used blank sheets.
If you use some blank writing sheets for notes that you do not wish to be marked, put a large ‘X’ across the entire sheet and do not include it in your numbering.
Numerical results must be written with as many digits as appropriate; don’t forget the units.
You should use mainly equations, numbers, symbols, graphs, figures and as little text as possible in your answers.
At the end of the experiment arrange all sheets in the following order:
1. Main Answersheet
2. Used Graph sheets
3. Used writing sheets
4. The sheets which are marked with ‘X’
5. Unused writing sheets and graph sheets
6. The printed question paper
Put all the papers inside the envelope and leave the envelope on your desk.
You are not allowed to take any sheet of paper or any material used in the experiment out of the room.

Условие

If a cord is passed around a post or a beam and the tensions in the two segments are different, friction between the cord and the beam plays a role in controlling the motion of the cord (Fig.1). It is observed that to hold a body suspended at one end of the cord, the minimum force needed at other end of the cord is less than the weight of the body due to the presence of frictional forces. With the increase in the number of turns around the beam the decrease in force is spectacular. Sailors are known to use this idea for arresting the motion of ships by winding ropes tied to ships around posts at the docks.

A circle (the beam) with a cord passing over it; the left segment hangs down to a larger block labelled M_w g with tension arrow T_1, the right segment hangs down to a smaller block labelled M_p g with tension arrow T_2.

Objective

To explore the relationship among the three quantities: the load $W$ (= $M_\mathrm{w}\,g$ ), the minimum effort $P$ (= $M_\mathrm{p}\,g$ ) needed to keep the system in equilibrium and the angle $\theta$ , subtended by the segment of the cord in contact with one or more beams, by systematically varying these quantities and to express it in the form of an equation.

Apparatus

Sr. No.	Item	Quantity
1	An apparatus consisting of four pieces of steel pipe on four sides of an identical vertical piece at the centre, all fixed on a wooden platform	1
2	An acrylic sheet (with lines 1.5 mm apart) mounted in front of one of the horizontal pipes	1
3	A plastic pan (including supporting rods) with its mass ( $M_\mathrm{pan}$ ) written on its side	1
4	A magnifying glass with torch	1
5	White cord with blue markings (Dial cord)	2 pieces
6	Pink cord	1 piece
7	A weight box containing following weights
	500.0 g	1
	200.0 g	2
	100.0 g	1
	50.0 g	1
	20.0 g	2
	10.0 g	1
	5.0 g	1
	2.0 g	2
	1.0 g	1
8	A body of unknown mass, $M_\mathrm{u}$	1
9	A hanger of slotted weights with hook (each weight 100.0 g and total weight 800.0 g)	1
10	A piece of cleaning cloth	1

The blue switch can be moved to switch the torch ON/OFF. The acrylic plate with lines ruled on it is provided to detect the motion of the cord. The lines on the acrylic plate can be taken as reference against which the motion of the cord can be observed.

Photograph of the apparatus: a central vertical steel pipe and horizontal steel pipes mounted on a white wooden platform on four legs. — Fig.2(a) Steel pipes mounted on wooden platform

Photograph of a plastic box with brass weights of different sizes set in foam, including a row of slotted weights and hooks. — Fig.2(b) Set of weights

Photograph of a transparent plastic pan hung from three wire supporting rods joined at a hook. — Fig.2(c) Plastic pan

Photograph of a coiled white cord with blue markings in a plastic bag. — Fig.2(d) Dial cord

Photograph of a coiled pink braided cord. — Fig.2(e) Pink cord

Photograph of a hand-held magnifying glass with a blue slide switch marked ON and OFF. — Fig.2(f) Magnifying glass with torch

Photograph of a cylindrical brass weight with a hook on top. — Fig.2(g) Body of unknown Mass, $M_\mathrm{u}$

Photograph of a stack of brass slotted weights on a hanger rod with a hook. — Fig.2(h) Slotted weights with hook

Black-and-white photograph of the complete set-up: the steel pipes on the platform at the edge of a table with a cord hanging down. — Fig.2(i) Complete set-up

Experimental Procedure

Caution: Do not touch the surface of the pipes with which the cord would be in contact. Any greasy material can change the frictional properties of the surface (surface of the pipes as well as the cord). A cleaning cloth is provided if required.

Part 1 (8 т.)

(Note: Use the dial cord in this part.)

Use the hanger with slotted weights as load $M_\mathrm{w}$ . Attach the load at one end of the given piece of dial cord (whose mass is negligible) and a pan (with known mass) at the other. A weight box with weights is also provided. The angle $\theta$ subtended by the cord can be changed by passing it over/around two or more of the given pipes. (Refer Fig.3).

Left: a cord passing over two circles (pipes), each with a quarter-turn contact marked π/2; the left end carries the load W = M_w g with tension T_1, the right end the effort P = M_p g with tension T_2. Right: perspective drawing of the platform with four horizontal pipes around a central vertical post; the cord passes over one horizontal pipe (π/2), around the central post (π/2) and over another pipe (π/2), with the load W=M_w g hanging on the left and the pan P=M_p g on the right. — Fig.3 Schematic arrangements of the assembly for angles π and 3π/2 respectively

The minimum value of angle $\theta$ is obtained when the cord passes over two parallel rods without making any contact with the central post (Fig.3). By winding the cord around the vertical post and shifting the position of the effort the angle $\theta$ can be changed by steps of $\pi/2$ . The load $M_\mathrm{w}$ should be suspended from the pipe to which the ruled acrylic plate is mounted.

For increasing the angle $\theta$ the cord is to be turned around the central vertical post. To observe whether the cord suspending the load is slipping over the pipe, the transparent acrylic plate is provided. You can use the magnifying glass with torch to view whether the cord is slipping over the pipe. Ideally one should note the effort $M_\mathrm{p}$ when the load $M_\mathrm{w}$ is on the verge of moving down (overcoming static friction). But that is not possible. However, it is possible to determine the interval [ $M_{\mathrm{p}-}$ , $M_{\mathrm{p}+}$ ] within which this value lies. Since this interval is a measure of the uncertainty in the magnitude of $M_\mathrm{p}$ , it should be made as small as possible.

For $\theta = \pi$ , take observations over as wide range as possible with the weights provided. Make estimate of the uncertainties in the observations. Plot the necessary graphs and combine the results from the graphs to get the desired equation. On the basis of the analysis of your data write down the quantitative relation giving value of $P$ in terms of $W$ and $\theta$ . The value of $P$ is also dependent on friction between the cord and the pipe. Identify the term in your equation which accounts for friction and equate it to the coefficient of friction $\mu$ for the given system. Estimate the expanded uncertainty in its value.

Part 2 (2 т.)

(Note: Use the pink cord in this part.)

A body of unknown mass $M_\mathrm{u}$ and a pink cord is provided. Suspend the unknown mass from one end of the cord and the pan from the other end. Write down the relevant equations to determine $M_\mathrm{u}$ and $\mu_\mathrm{u}$ . With $\theta = \pi$ , take necessary observations to determine its mass and the coefficient of friction between the pink cord and the pipes using the relationship obtained in the experiment. Estimate the expanded uncertainty in your results.

Note on uncertainty evaluation

If it is observed that the magnitude $X$ of a quantity to be measured lies somewhere in an interval [ $X_1$ , $X_2$ ], and there is equal probability that it can have any value in this interval, then the probability distribution is said to be uniform or rectangular. The standard uncertainty for such distribution is given by $\dfrac{\left|X_1 - X_2\right|}{2\sqrt{3}}$ .
After evaluating the combined standard uncertainty in a measured quantity, the result is stated with an expanded uncertainty. If the expanded uncertainty is taken equal to twice the combined standard uncertainty, the confidence level is approximately 95 %. The number giving expanded uncertainty is rounded upwards to retain a single digit (generally) and the number giving the magnitude of the measured quantity is rounded to keep appropriate number of digits such that the last digit has the same decimal place as that of the expanded uncertainty.

Отговори

Решение

We will assume the relationship of the form:

P = f(W)\,h(\theta)

M_\mathrm{p} = f(M_\mathrm{w})\,h(\theta)

$M_\mathrm{w}$ represents the mass in the hanger (Load)

$M_\mathrm{p}$ represents the mass in the pan + the mass of the pan (i.e. $M'_p + M_{pan}$ ) (Effort)

The relation between these variables can be found in two parts:

Relation between $M_\mathrm{P}$ and $M_\mathrm{W}$
Relation between $M_\mathrm{P}$ and $\theta$

Part 1: Load dependence

Mass of the pan = 28.6 g

$\theta = \pi$ radians

Obs. No.	$M_\mathrm{w}/g$	$M'_{p-}$ /g	$M'_{p+}$ /g	$M'_{p(average)}$ /g	$M_{p(average)} = M'_{p(average)} + M_{pan}$ /g	$\Delta M_p = \dfrac{M'_{p+} - M'_{p-}}{2}$ /g
1	800.0	566	574	570	598.6	4
2	700.0	486	494	490	518.6	4
3	600.0	417	423	420	448.6	3
4	500.0	337	343	340	368.6	3
5	400.0	267	273	270	298.6	3
6	300.0	188	192	190	218.6	2
7	200.0	113	117	115	143.6	2
8	100.0	41	43	42	70.6	1

Graph of $M_\mathrm{P}$ v/s $M_\mathrm{W}$ :

Hand-drawn graph on graph paper titled 'Mp v/s Mw' (Experiment 1): M_P/g from 0 to 600 on the vertical axis against M_w/g from 100.0 to 800.0 on the horizontal axis; eight circled points lie on a straight line through near the origin; slope triangle between (150.0, 105) and (750.0, 560).

Slope of the graph = 0.7583

This shows that

P \propto W \qquad (1)

Part 1: Angle dependence

$M_\mathrm{w}$ = 800.0 g

$M_\mathrm{pan}$ = 28.6 g

Obs. No.	$\theta$ /rad	$M'_{p-}$ /g	$M'_{p+}$ /g	$M'_{p(average)}$ /g	$M_{p(average)} = M'_{p(average)} + M_{pan}$ /g	$\Delta M_p = \dfrac{M'_{p+} - M'_{p-}}{2}$ /g
1	$\pi$	565	575	570	598.6	5
2	$3\pi/2$	455	465	460	488.6	5
3	$2\pi$	396	404	400	428.6	4
4	$5\pi/2$	316	324	320	348.6	4
5	$3\pi$	276	284	280	308.6	4
6	$7\pi/2$	236	244	240	268.6	4
7	$4\pi$	182	188	185	213.6	3
8	$9\pi/2$	147	153	150	178.6	3
9	$5\pi$	133	137	135	163.6	2
10	$11\pi/2$	104	110	107	135.6	3
11	$6\pi$	88	92	90	118.6	2
12	$13\pi/2$	68	72	70	98.6	2
13	$7\pi$	54	56	55	83.6	1
14	$15\pi/2$	39	41	40	68.6	1
15	$8\pi$	29	31	30	58.6	1
16	$17\pi/2$	18	20	19	47.6	1

Hand-drawn graph on graph paper titled 'Mp v/s θ' (Experiment 1): M_p/g from 100 to 600 on the vertical axis against θ/rad from 2 to 24 on the horizontal axis; sixteen circled points on a smooth decreasing curve.

The graph between $M_p$ and θ shows a curve.

There can be possibilities of different functional relationship.

The possible functions can be $\dfrac{1}{\theta}$ , $\dfrac{1}{\theta^2}$ , $e^{-k\theta}$

For the first two functions mentioned above, at θ = 0, $M_\mathrm{p}$ will reach infinite value which is not possible. For the third function we know that $M_\mathrm{p}$ will have some finite value.

If the function is anticipated as exponential one then it can be verified using half value technique whether at every half value of Mp, and then plotting ln $M_\mathrm{p}$ or better still $\ln\left(M_p/M_w\right)$ against $\theta$ .

If it is a straight line with slope – $k$ ,

M_p \propto e^{-k\theta}

P \propto e^{-k\theta} \qquad (2)

From (1) and (2),

P \propto We^{-k\theta} \qquad (3)

The constant $k$ in the above expression is equated to the coefficient of friction, $\mu$ .

P = CWe^{-\mu\theta} \qquad (4)

The constant of proportionality in equation (4) is 1. This is because at $\theta = 0$ rad, $P = W$ .

P = We^{-\mu\theta} \qquad (5)

Original logarithmic graph of ln(M_p/M_w) against theta, with measurement error bars, central and limiting slope lines and handwritten slope values.

From the graph,

\ln\left(\frac{M_p}{M_w}\right) = -\mu\theta

where $\mu$ is the slope of the graph.

From the graph, $\mu$ = 0.106

\frac{\Delta S}{S} = \frac{(S_1 \sim S_2)/2}{S} = 0.10049

\Delta S = 0.01067 = \Delta\mu

u_C(\mu) = 0.00616

U(\mu) = 0.0123 \approx 0.02

\mu = 0.11 \pm 0.02

Part 2

When the pan is moving up:

M_{p1} = M_u e^{-\mu\theta} \qquad (6)

When the pan is moving down:

M_{p2} = M_u e^{\mu\theta} \qquad (7)

$M'_{p1+}$	$M'_{p1-}$	$M'_{p1(average)}$	$\Delta M'_{p1}$	$M_{p1} = M'_{p1} + M_{pan}$
164	156	160	4	188.6

$M'_{p2+}$	$M'_{p2-}$	$M'_{p2(average)}$	$\Delta M'_{p2}$	$M_{p2} = M'_{p2} + M_{pan}$
49	45	47	2	75.6

For $M_\mathrm{u}$ :

Multiplying equation (6) and (7),

M_u = \sqrt{M_{p1} \cdot M_{p2}}

M_u = \sqrt{188.6 \times 75.6} = 119.4075g

For $\mu$ :

Dividing equation (6) by (7),

\frac{M_{p1}}{M_{p2}} = e^{2\mu\theta}

\mu = \frac{1}{2\theta}\ln\left(\frac{M_{p1}}{M_{p2}}\right)

For $\theta = \pi$

\mu = \frac{1}{2\pi}\ln\left(\frac{M_{p1}}{M_{p2}}\right)

\mu = \frac{1}{2\pi}\ln\left(\frac{188.6}{75.6}\right) = 0.1456

Uncertainty in $M_\mathrm{u}$ :

\frac{\Delta M_u}{M_u} = \sqrt{\left(\frac{1}{2}\frac{\Delta M_{p1}}{M_{p1}}\right)^2 + \left(\frac{1}{2}\frac{\Delta M_{p2}}{M_{p2}}\right)^2} = \sqrt{\left(\frac{1}{2}\frac{4}{188.6}\right)^2 + \left(\frac{1}{2}\frac{2}{75.6}\right)^2} = 0.0169

\Delta M_u = 0.0169 \times 119.4075 = 2.018

u_C(M_u) = \frac{1}{\sqrt{3}} \times 2.018 = 1.165

U(M_u) = 2 \times u_C(M_u) = 2 \times 1.165 = 2.33 \approx 3

M_u = 119 \pm 3\,\mathrm{g}

Uncertainty in $\mu_\mathrm{u}$ :

u_C(\mu_u) = \frac{1}{\sqrt{3}} \cdot \Delta\mu_u = \frac{1}{2\pi\sqrt{3}}\sqrt{\left(\frac{\Delta M_{p1}}{M_{p1}}\right)^2 + \left(\frac{\Delta M_{p2}}{M_{p2}}\right)^2} = \frac{1}{2\pi\sqrt{3}}\sqrt{\left(\frac{4}{188.6}\right)^2 + \left(\frac{2}{75.6}\right)^2} = 0.003117

U(\mu_u) = 2 \times u_C(\mu_u) = 2 \times 0.003117 = 0.00623 \approx 0.007

\mu_u = 0.146 \pm 0.007

Marking scheme

No.	Criterion	Marks	Maximum marks
Part 1
1	Observations of $M_P$ corresponding to values of $M_W$ for $\theta=\pi$		1.5
	4 readings	0.4
	5 readings	0.5
	6 readings	0.6
	7 readings	0.7
	8 readings	0.8
	5 readings with $\Delta M_P<5$	0.4
	6 readings with $\Delta M_P<5$	0.5
	7 readings with $\Delta M_P<5$	0.6
	8 readings with $\Delta M_P<5$	0.7
2	Graph of $M_p$ vs $M_w$		0.4
	Choice of Scale (to cover 70% or more space on graph sheet)	0.1
	3 points on straight line	0.1
	4 points on straight line	0.2
	Both Axis labeled with proper units	0.1
3	Value of slope $b:0.6\leq b\leq0.8$	0.3	0.3
	Value of slope $b:0.5\leq b<0.6$ or $0.8<b\leq0.9$	0.1
4	Observations of $M_P$ corresponding to values of $\theta$ for $M_W=800.0\,\mathrm{g}$		2.6
	6 to 7 readings	0.5
	8 readings	1
	9 readings	1.1
	10 readings	1.2
	11 readings	1.3
	12 readings	1.4
	8 readings with $\Delta M_P<5$	0.8
	9 readings with $\Delta M_P<5$	0.9
	10 readings with $\Delta M_P<5$	1
	11 readings with $\Delta M_P<5$	1.1
	12 readings with $\Delta M_P<5$	1.2
5	Graph of $M_P$ versus $\theta$		1
	Proper Choice of Scale (to cover 70% or more space on graph sheet)	0.2
	Both Axis labeled with proper units	0.2
	Smooth Curve	0.6
	Moderately smooth curve (atleast half the points on the curve)	0.4
6	Graph of $\ln M_P$ (or $\ln(M_P/M_W)$ ) vs $\theta$		0.9
	Choice of Scale (to cover 70% or more space on graph sheet)	0.2
	4 points on straight line	0.4
	5 points on straight line	0.5
	6 points on straight line	0.6
	Both Axis labeled with proper units	0.1
7	Value of slope, b: $0.09\leq k\leq0.13$	0.5	0.5
	Value of slope, b: $0.08<k<0.09$ and $0.13<k<0.14$	0.4
8	Valid method for estimation of uncertainty	0.1	0.2
	Uncertainty in $\mu\leq0.02$ (Expanded uncertainty)	0.1
9	Identifying coefficient of $\theta$ as $\mu$	0.1	0.1
10	Correct Equation: $P=We^{-\mu\theta}$ or $M_pg=M_wg e^{-\mu\theta}$	0.5	0.5
	Incorrect Equation: $M_p=M_w e^{-\mu\theta}$	0.2
	Incorrect Equation: $P=CWe^{-\mu\theta}$ or $M_pg=CM_wg e^{-\mu\theta}$	0.2
Part 2
11	Correct Equations for determining $M_u$ and $\mu_u$	0.4	1
	Correct method to obtain $M_u$ and $\mu_u$	0.6
12	Observations: $[M_{p1-},M_{p1+}]$ and $[M_{p2-},M_{p2+}]$	0.4	0.4
13	Value of $M_u$		0.2
	$115\leq M_u\leq121$	0.2
	$113\leq M_u<115$ and $121<M_u\leq123$	0.1
14	Uncertainty in $M_u\leq4\,\mathrm{g}$ (Expanded uncertainty)	0.1	0.1
15	Value of $\mu$		0.2
	$0.13\leq\mu\leq0.20$	0.2
16	Uncertainty in $\mu\leq0.01$ (Expanded uncertainty)	0.1	0.1
	Total		10

Оригинал в Архива: exp1.pdf · официални решения: exp1_sol.pdf · допълнителни данни: exp1_marking.pdf