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Experimental determination ofunknown masses and theirpositions in a mechanical blackboxBhupati Chakrabarti1, Shirish Pathare2, Saurabhee Huli2 andMadhura Nachane2

1 Department of Physics, City College, Kolkata-700 009, India2 Homi Bhabha Centre for Science Education (TIFR), V N Purav Marg, Mankhurd,Mumbai-400088, India

E-mail: bhupati2005@yahoo.co.in and shirish@hbcse.tifr.res.in

AbstractAn experiment with a mechanical black box containing unknown masses ispresented. The experiment involves the determination of these masses andtheir locations by performing some nondestructive tests. The set-ups areinexpensive and easy to fabricate. They are very useful to gain anunderstanding of some well-known principles of mechanics.

Introduction

In different experimental set-ups occurring inphysics, a ‘black box’ refers to a system thatacts as a single unit but may be comprisedof components that cannot be seen or accessedindividually. This unit may be part of a largersystem, and can contain electronic, electrical,magnetic, optical or mechanical devices orcomponents, or a combination of them. Theidea behind a black box experiment refers to anapproach where instead of going into every detailof a particular part, the whole thing is treated asa single unit whose overall performance can beassessed.

Black boxes that contain essentially me-chanical components may be explored throughmechanics experiments that are normally referredto as mechanical black box (MBB) experiments.The basic idea behind any black box experimentremains essentially the same, i.e. by performing

some nondestructive experiments one tries to findout what items are inside and how they arearranged. These types of box are quite interestingand have been introduced into the experimen-tal sections of international competitions suchas the International Physics Olympiad (IPhO)[1, 2]. MBBs may contain springs and massesat particular locations or other suitable elementswhose characteristics, such as the spring constant,mass and location, can be determined. Here weshall describe an experiment with a particular typeof MBB containing two masses.

Design of a black box containing a pair ofunknown masses

The MBB may have different shapes and sizesdepending on the experimental requirements. TheMBB that has been devised and used to performseveral experiments is described below.

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Figure 1. Different parts of the empty box.

Figure 2. Black box with one mass (m1) placed at adistance L from the midpoint of the box.

Figure 1 describes different parts of an emptyblack box. This is essentially a long (∼40 cm)narrow metallic box (breadth ∼ 2.5 cm anddepth ∼ 2 cm), usually made from a U-shapedaluminum channel (sheets 1 and 2 make analuminum channel). The upper surface is coveredwith a plastic sheet (sheet 3) that behaves like alid. The box is sealed at the ends with two verylight pieces of plastic sheet (sheet 4) of negligiblemass, after the placement of the masses at knowndistances.

This box (figure 5) contains two masses ofsmall size and regular shape fixed symmetricallyon the axis of the box. The main objective of theexperiment is to determine the unknown massesand their positions within the box. The massof the empty box (M) normally needs to besupplied. Figures 2–4 show sketches of boxesthat contain different combinations of massesand distances from the midpoint of the uniformempty box.

In this experiment, two unequal masses areplaced at a distance L from the midpoint of therectangular box (figure 3).

First the box is used like a beam of a balanceand moments about some suitable points are takenusing external known weights by balancing it overa knife edge (figure 6). This helps to develop some

Figure 3. Black box with two unequal masses (m1 andm2) placed at equal distances L from the midpoint ofthe box.

Figure 4. Black box with two equal masses (m1) at twounequal distances (L1 and L2) from the midpoint of thebox.

Figure 5. The black box.

relationships involving the unknown masses anddistances.

Two different relationships emerge.

(1) The geometric midpoint of the box is madethe fulcrum. In this case, the weight of theempty box cannot contribute towards themoment about that point.

(2) Any other point is chosen as the fulcrum.Here the weight of the empty box doescontribute to the moments.

The second type of operation involves thebifilar oscillation of the box about its centre ofmass. This point is normally not the geometricmidpoint after being internally unevenly loadedwith unknown weights. Here we shall describethe principle of the experiment with two unequalmasses m1 and m2 that have been placed at anequal distance L on both sides of the midpoint ofthe empty box. Here the mass of the empty box isalso supplied, reducing the unknown quantities tom1, m2 and L.

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Figure 6. Knife edge with a marker on the front side.

Figure 7. Moment of MBB about geometrical centre.

ExperimentKnown quantities.

M—mass of the empty box = 115.0 g.m—mass in the hanger + mass of the plastic

holder (mass of the plastic holder = 1.0 g)(figure 8).

z—distance of mass m from the midpoint ofthe box.

Step 1: taking moments about the midpoint ofthe box by balancing it on the knife edge.

As per figure 7:

m1L− m2L− mz = 0

(m1 − m2)L− mz = 0

mz = (m1 − m2)L

∴ z =1m(m1 − m2)L.

(1)

From the graph of z versus 1m we obtain

slope = (m1 − m2)L.Observations. See table 1 and figure 9.

Slope = (m1 − m2)L = 449.28 g cm.

Step 2: taking moments about a pointbetween the unknown mass and the midpoint ofthe box by balancing it on the knife edge.

Figure 8. Hanger attached to the rectangular plasticholder.

Figure 9. Graph of z versus 1/m.

Figure 10. Moment about the centre of mass withknown masses in the hanger.

From figure 10:

m1 (L+ x)− mz− m2 (L− x)+Mx = 0

∴ (m1 + m2 +M)x = mz− (m1 − m2)L.

If m is varied while keeping the position ofthe hanger fixed, then a graph of x versus mz canbe plotted.

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Table 1. Position of the geometrical centre of the black box, q = 20.1 cm.

Mass in thehanger (mh) (g) m = mh + 1.0 (g) 1/m (g−1)

Position of thehanger h (cm) z = h− q (cm)

50.0 51.0 0.0196 28.9 8.8100.0 101.0 0.0099 24.7 4.6150.0 151.0 0.0066 23 2.9200.0 201.0 0.0050 22.4 2.3250.0 251.0 0.0040 21.9 1.8

Table 2. Varying the mass in the hanger and keeping the position of the hanger fixed = 35.0 cm (CG—centre ofgravity).

Mass in the hanger(mh) (g)

m = mh + 1.0(g) CG (cm)

z = 35.0− CG(cm)

x = CG− 20.1(cm) mz (g cm)

250.0 251.0 27.4 7.6 7.3 1907.6200.0 201.0 26.4 8.6 6.3 1728.6150.0 151.0 25.2 9.8 5.1 1479.8100.0 101.0 23.6 11.4 3.5 1151.4

50.0 51.0 21.3 13.7 1.2 698.7

Figure 11. Graph of x versus mz.

In that case,

x =

(1

(m1 + m2 +M)

)mz−

(m1 − m2)

(m1 + m2+M)L

Slope =1

(m1 + m2 +M).

(2)

Observations. See table 2 and figure 11.

Slope =1

(m1 + m2 +M)= 0.005 g−1

(m1 + m2 +M) = 200.0 g.

Figure 12. Mechanical black box suspended as abifilar pendulum.

Step 3: suspending the MBB as a bifilarpendulum and studying torsional oscillations ofthe box.

When two strings are symmetrically tied onthe two sides of the centre of mass of a bodyto allow it to have a torsional oscillation, thenthe suspension is referred to as bifilar suspension(figure 12). Here the MBB has been given atorsional oscillation about a vertical axis passingthrough its centre of mass.

(a) Two C-shaped plastic holders. Two C-shapedplastic holders (figure 13(a)) are used in step3 to hold the black box in a bifilar pendulumconfiguration, as shown in figure 13(b). Theholder can be clamped to the black box bytightening the screw with an Allen key.

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(a) (b)

Figure 13. (a) C-shaped plastic holder. (b) Plane side of the holders on the scale side of the black box.

Figure 14. Binder clip used to hold the thread.

(b) U-shaped channel with binder clips. TheU-shaped channel has holes drilled at regulardistances (1 cm apart). In step 3, threads canbe inserted through these holes. Binder clipscan be used to hold the thread in position, asshown in figure 14.

The time period of the torsional oscillationsof the bifilar pendulum is given by [3]

T = 2π

√Il

M′gd1d2(3)

where M′ = M + m1 + m2.I—total moment of inertia of the box about

an axis passing through its CG and perpendicularto its plane.

Observations. See table 3 and figure 15.

Slope = 36.072 cm s2

∴ I =T2d2M′gd1

4π2l

=(Slope)M′gd1

4π2l

Figure 15. Graph of T2 versus 1/d2.

Table 3. y = 2.0 cm, 2d1 = 12.0 cm, l = 33.0 cm.

2d2 (cm) 1/d2 (cm−1) T (s) T2 (s2)

22.0 0.0909 1.761 3.10120.0 0.1000 1.858 3.45218.0 0.1111 1.975 3.90115.0 0.1333 2.156 4.64813.0 0.1538 2.321 5.387

=36.072× 200.0× 980× 6.0

4π2 × 33.0= 32 594.5 g cm2.

The moment of inertia, I, comprises three parts

I = I1 + I2 + I3.

I1 = moment of inertia of the empty box =15 800 g cm2.

(Note: for the calculation of I1, the masses ofthe horizontal and vertical sections of the blackbox (figure 1) are separately provided.)

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I2 = moment of inertia of m2 with respect tothe centre of gravity, so I2 = m2(L+ y)2.

I3 = moment of inertia of m1 with respect tothe centre of gravity, so I3 = m1(L− y)2.

It needs to be mentioned in this connectionthat the diameters of unknown masses were muchless than L. Hence the individual moments ofinertia of the unknown masses are neglected.

I = I1 + I2 + I3

I = I1 + m2(L+ y)2 + m1(L− y)2

32 594.5 = 15 800+ m2(L2+ 2Ly+ y2)

+ m1(L2− 2Ly+ y2)

(m1 + m2)L2− 2y[(m1 − m2)L]

+ (m1 + m2)y2− 16 794.5 = 0

(M′ −M)L2− 2y[(m1 − m2)L]

+ (M′ −M)y2− 16 794.5 = 0

(200.0− 115.0)L2− 2× 2.0× [449.28]

+ (200.0− 115.0)× 2.02− 16 794.5 = 0

(85.0)L2− 1797.12+ 340.0− 16 794.5 = 0

L2= 214.72 cm2 L = 14.653 cm

(m1 − m2)L = 449.28 g cm

(m1 − m2) = 30.66 g, (m1 + m2) = 85.0 g

m1 = 57.83 g and m2 = 27.17 g

m1 = 58± 2 g, m2 = 27± 2 g

and L = 15± 2 cm.

Actual values:

m1 = 58.0 g, m2 = 29.0 g

and L = 15.0 cm.

ConclusionThe mechanical black box (MBB) problem isinteresting because it can be used in severalvariations. We are in a position to determinethree unknowns by framing three independentequations. So the problem of the MBB maybe framed in several distinct ways by changingunknown quantities while always keeping theirnumber to three. However, in the process we needto provide some additional information dependingon the problem. Here are a few examples ofdifferent problems with MBBs where the numberof unknowns may be different from that discussedhere.

(a) m1 (one unknown mass), L (location of theunknown mass with respect to the midpoint

of the empty box), M (mass of the emptybox) (as shown in figure 2).

(b) m1 (one unknown mass), m2 (secondunknown mass), L (equal distance from themidpoint of the box), M (mass of the emptybox is supplied) (as shown in figure 3).

(c) m1 (one unknown mass), m2 (secondunknown mass), M (mass of the empty box).(Two unknown masses are kept on the twosides of the midpoint of the box and thedistance(s) L1 and L2 are supplied.)

(d) m1 (mass of each equal unknown mass), L1(distance of one mass from the midpoint), L2(distance of other equal mass from themidpoint), M (mass of the empty box issupplied) (as shown in figure 4).

One can actually handle the MBB with onlytwo unknowns (say a mass and its locationinside the box) just by taking moments about themidpoint of the box and about any other suitablepoint. So, by using equations (1) and (2) withthe value of the mass of the empty box supplied,unknowns can be determined. That would actuallybe the simplest type of MBB, where the bifilaroscillation is not necessary, meaning that theset-up could act as an introductory MBB in amiddle school class where the students have justlearnt about the moments of forces and turningeffects, but have not yet been introduced tooscillations.

Another important aspect of this exercise liesin the fact that the students can actually fabricatethe MBB with easily available inexpensivematerials and can perform the experiment bythemselves. In fact, it is easy to make multiplesets and give them to a number of students whocan then perform the experiment simultaneouslyin a laboratory situation.

AcknowledgmentsThe authors would like to acknowledge theexcellent academic input provided by ProfessorD A Desai through some useful discussions.Thanks are also due to the supporting staff of thePhysics Olympiad cell at Homi Bhabha Centre forScience Education for preparing and fabricatingthe experiment.

Received 24 December 2012, in final form 12 February 2013doi:10.1088/0031-9120/48/4/477

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References[1] Experimental Question 2004 International

Physics Olympiad 2004, Pohang, SouthKorea http://ipho.phy.ntnu.edu.tw/problems-and-solutions.html#2004

[2] Experimental Question 2011 InternationalPhysics Olympiad 2011, Bangkok,Thailand http://ipho.phy.ntnu.edu.tw/problems-and-solutions 5.html#2011

[3] Worsnop B L and Flint H T 1927 AdvancedPractical Physics for Students (London:Methuen) p 61

Bhupati Chakrabarti is an associateprofessor in the Department of Physicsin City College, Kolkata, India. Apartfrom teaching he writes popular sciencearticles and conducts workshopsinvolving lectures and hands-on physicsfor teachers and students. He is theGeneral Secretary of the IndianAssociation of Physics Teachers. Hisparticular interest is in developingphysics experiments suitable for highschool students. He is involved inPhysics-Olympiad-related activities inIndia.

Shirish Pathare is a research student atthe Homi Bhabha Centre for ScienceEducation (TIFR), India. His researchtopic is students’ alternative conceptionsin heat and thermodynamics. He is incharge of the experimental developmentprogramme for Physics Olympiadactivity in India.

Saurabhee Huli is a project assistant atthe Homi Bhabha Centre for ScienceEducation (TIFR). She is involved in theexperimental training of PhysicsOlympiad students and also in differentteacher training programmes of theinstitute.

Madhura Nachane is a project assistantat the Homi Bhabha Centre for ScienceEducation (TIFR). She is involved in theexperimental training of PhysicsOlympiad students and also in differentteacher training programmes of theinstitute.

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