1 research question 2 introduction 2.1 rationale 2.2 … · center of the 50ccm mark should be its...

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1

1 Research Question

The effect of varying distances of the pivot from the geometric center of mass of a meter rule on the period of its oscillation.

2 Introduction

2.1 Rationale

Traditionally, simple pendulums, a point-massed spherical bob extended from a thin wire of a negligible mass, were “the

basis of virtually all accurate time-keeping devices” (Fitzpatrick, 2006). However, for these traditional clocks, I had observed

that the use of physical pendulum that is “a rigid body with a distributed mass” (Grieve, 2001) throughout, had not been widely

utilized. Just as the length of simple pendulum effects its period, I questioned whether a factor that may contribute to the period

of a physical pendulum is the distance it is pivoted away from its center of mass. Therefore, this brought rise to the research

question to be investigated for a common object, meter rule that is a cuboid, found in a school based laboratory: the effect of

varying distances of the pivot from the geometrical center of mass of a meter rule on the period of its oscillation.

2.2 Derivation for the Effect of Distance from the Center of Mass on the Period of a Physical Pendulum

The period, T, for a physical pendulum follows the relationship in [1] for small initial amplitudes. Here, 𝑀 is the mass of the

physical pendulum, 𝑔 is the acceleration due to gravity and 𝐼 is the moment of inertia of the physical pendulum pivoted a certain

distance, 𝐷, from its center of mass. The moment of inertia, 𝐼, could be viewed as an “object’s resistance to change in rotational

motion” (McFatridge, 2003) or as an equivalent to the inertial mass, 𝑚, in Newton’s Second Law for an object in linear

motion: 𝐹 = 𝑀𝑎.

𝑇 = 2𝜋√

𝐼

𝑀𝑔𝐷 (Nave, 2012)

I = ∫ 𝑟2 𝑑𝑀 (Kaldon, 2003)

[1]

[2]

This moment of inertia for a mass that is distributed continuously over a body is expressed in [2] as the integral or sum of

every infinitesimal point mass within the object, 𝑑𝑀, multiplied by the square of its separation, 𝑟, from its axis of rotation.

However, 𝐼 and 𝐷 are both constants dependent on one another; the independent variable 𝐷 could not be changed solely without

𝐼 varying. Considering the expression for 𝐼, if the rotational axis is changed such that its distance, separation from every point

mass is increased, 𝐼 should increase as well. Therefore, if the rotational axis of the meter rule were to be changed from its

geometric center of mass, where the range separation of each point mass is least from the rotational axis (see Figure 2.2.1), to

the end of the meter rule, where the range of separation of each point mass is greater (see Figure 2.2.2), 𝐼 should increase.

Figure 2.2.1: For a rotational axis at the geometric center of

the meter rule, the separation of each point mass should

least from the rotational axis should be least.

Figure 2.2.2: If the rotational axis were to be shifted to the

end of the meter rule there would be a greater separation,

𝑟2, of each point mass from the rotational axis.

This relationship in the change of 𝐼 for an object, if the axis of rotation parallel to one passing through the center of mass of

an object were to be shifted a distance 𝐷 is expressed as such in Figure 2.2.3. Here, 𝐼𝐶𝑂𝑀 that is a constant, is the moment of

inertia about the center of mass. Substituting this for 𝐼 in [1] yields the equation for the period such that if 𝐷 were to be changed,

no other variable would vary.


2

Figure 2.2.3: Graphical depiction of the theorem.

𝐼 = 𝐼𝐶𝑂𝑀 + 𝑀𝐷2

𝑇 = 2𝜋√𝐼𝐶𝑂𝑀 + 𝑀𝐷2

𝑚𝑔𝐷

𝑇2 =

4𝜋2

𝑀𝑔𝐷∙ (𝐼𝐶𝑂𝑀 + 𝑀𝐷2)

[3]

Assuming the meter rule that would be used for this investigation is a rectangular plate, 𝐼𝐶𝑂𝑀 , could be substituted for a

rectangular plate, [4], in terms of its length, 𝐴, and width, 𝐵.

𝐼𝐶𝑂𝑀 =

𝑀 ∙ (𝐴2 + 𝐵2)

12 (Richmond, 2015)

[4]

𝑇2 =

4𝜋2

𝑀𝑔𝐷∙ [

𝑀(𝐴2 + 𝐵2)

12+ 𝑀𝐷2]

𝑇2 =4𝜋2

𝑔∙ [

(𝐴2 + 𝐵2)

12𝐷+ 𝐷]

[5]

Equation [5] in this form suggests that the period squared, 𝑇2, for the physical pendulum is directly proportional to the sum

of the distance, 𝐷, it’s pivoted away from its center of mass and the moment of inertia of the rectangular plate at its center of

mass, divided by its mass. This is based on the assumptions that the moment of inertia of the meter rule is equivalent to that of

a plate and air resistance and frictional force have a negligible impact on the period.

3 Hypothesis

Given that the assumptions aforementioned are valid, graphing the quantities that yield a direct proportionality on the Y

and X axis respectively for distances above the center of mass, a straight-line graph through the origin should be drawn within

each of its errors. Alternatively, since T2 should be the same for similar distances pivoted below the center of mass, a reflection

of the previous graph should also be obtained, producing an absolute value function (see Figure 3.1).

Furthermore, the absolute value of both gradients for distances above and below the meter rule’s center of mass should be

equivalent to each other and 4𝜋2 ∙ 𝑔−1.

4 Experimental Design

4.1 Variables

Figure 3.1: Depiction of the hypothesized result.


3

Independent Variable: The independent variable would be the varying distances the meter ruler would be

pivoted away from its geometric center. The pivot points were made by drilling holes

along a drawn line that goes through the center of the ruler (see Figure 4.1.1), at distances

46cm, 42cm, 38cm, 44cm, 30cm and 26cm both above and below from the center of the

ruler’s 50cm mark that should be the geometric position of its center of mass (see Figure

4.1.1), assuming the meter rule’s mass is uniformly distributed (Hall, 2015). These

distances were measured, verified again using another meter ruler, extended from the

center of the 50cm mark to the center of the drilled hole. Furthermore, the drill bit used

to drill the holes was used as the pivot to ensure the distance from the center of mass the

pendulum oscillates is not increased slightly as a consequence of a smaller, loose fit.

Figure 4.1.1: Pivot points are to be made along the drawn line along the center of the ruler. The center of the 50ccm mark should be its geometric center of mass.

Figure 4.1.2: Meter rule pivoted at an arbitrary distance such that the line through the center of the ruler rests at the 90° mark of the protractor.

Dependent Variable: The dependent variable would be the period of one oscillation by the meter ruler that is

pivoted a distance away from its center. This would be obtained by placing a photogate

below the meter ruler and measuring the time it takes for it to complete oscillation or

when the photogate switches state from being blocked to unblocked twice till its first

blocked again (Vernier.com, n.d.). To ensure the reliability of the results, 10 such repeats

would be taken.

Controlled Variables:

Initial Angle of the Swing The initial angle of the swing would be controlled ensuring the center of the meter ruler

is displaced and released from the same small angle of 5 degrees each time. This would

be ensured by attaching a protractor to the point of suspension of the meter rule on the

retort stand, and aligning it such that the drawn line that goes through the center of the

pivoted meter ruler, rests at the 90-degree mark on the protractor (see Figure 4.1.2). The

line drawn through the center of the rule would then be displaced 5° which would be

measured using the protractor attached.

Position of Center of

Mass on the Meter Rule

Assuming the mass of the meter rule is uniformly distributed, its geometric center of mass

should be conserved and not shifted by drilling holes symmetrically along the drawn line

that goes through the center of the meter ruler. Therefore, holes were drilled at distances

46cm, 42cm, 38cm, 44cm, 30cm and 26cm both above and below the center of the ruler’s

50cm marks to ensure symmetry.

Length and Width of the

Ruler

The same meter ruler was used throughout the experiment whose length and width was

verified by measuring it utilizing another meter rule and Vernier caliper respectively, at

5 random positions along it. Their average width and length were taken to ensure the

validity of the assumption regarding the uniform length and width of the meter rule. This

should ensure the moment of inertia of the meter rule, that is assumed to be equivalent

to a rectangular plate, remains constant.

4.2 Apparatus


4

1 Protractor ±0.5° 1 15cm Ruler ±5×10−4𝑚

1 Vernier Photogate ±1×10−6𝑠 1 Drill Bit

2 Meter Rulers ±5×10−4𝑚 1 Drill

1 Vernier Caliper ±2.5×10−5𝑚 1 Retort Stand

4.3 Procedure

1. First, Using the Vernier caliper and other meter rule, take 5 measurements of the meter rule’s depth, at random

positions along the ruler, and length respectively.

2. Using a 15cm ruler, measure the width of a meter ruler and mark its center at both ends. Then using another meter

ruler, draw a line along the center of the meter ruler that joins both marks (see Figure 4.1.1).

3. At distances, 46cm, 42cm, 38cm, 44cm, 30cm, 26cm and 0cm both above and below the ruler’s 50cm mark, draw a line

across the ruler’s width.

4. Then, at each intersection of the drawn lines, excluding the one at the meter rule’s 50cm mark, drill holes utilizing a drill

bit.

5. Measure and verify that the distances from the central, undrilled intersection point to the center of the drilled holes are

46cm, 42cm, 38cm, 44cm, 30cm and 26cm above and below it respectively.

6. Clamp the drill bit to the retort stand and place either hole that is 46cm away from the meter rule’s center through the

clamped drill bit.

7. Then, attach the protractor on the drill bit, and align it such that such that the drawn line that goes through the center

of the pivoted meter ruler, rests at the 90-degree mark on the protractor (see Figure 4.1.2).

8. Place, a photogate that is connected to the data logger below the pivoted meter ruler. Ensure that the meter ruler is low

enough such that is the state of the photogate is blocked and the data logger is in a mode, ‘Pendulum’.

9. Displace the meter ruler by an angle of 5 degrees with reference to the drawn line along its center. Then, start data

collection on the data logger and let the meter rule to oscillate freely.

10. Record the period for the first 10 oscillations of the meter rule. A single oscillation would have occurred when the

photogate switches state from being blocked to unblocked twice till it’s first blocked again.

11. Repeat steps 5 to 10 for the 11 other drilled holes.

4.4 Safety Considerations

The use of the drill invites several concerns regarding safety. Therefore, whilst drilling holes into the meter rule, the use of

safety goggles and thick gloves was considered. This ensured any wooden material does not fall into the eye and that an injury

as a result wooden splinters or hot temperatures was avoided. Prior to drilling, the meter rule was secured firmly with clamps

to mitigate any accident as a result of its slipping and long clothing such as ties that may get entangled with the drill was taken

off. Furthermore, the drill was unplugged whilst not in use and the drilling was conducted a distance away from surrounding

people.

5 Raw Data

5.1 Quantitative Observations

5.1.1 Raw Data Table to Determine the Average Values of A and B

Measured Lengths of A (Width) and B (Length)

Trial A (m) B (m)

±0.000025 ±0.0005

1 0.02670 0.999

2 0.02675 0.999

3 0.02700 1.000

4 0.02685 0.998

5 0.02685 1.000

5.1.2 Raw Data Table Displaying the Period for a Distance Away from the Ruler’s Center of

Mass


5

Period (T) of the Meter Ruler Pivoted a Distance Away From it Centre of Mass (D)

D (m) T (s)

±0.0005 ±0.000001

Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7 Trial 8 Trial 9 Trial 10

0.460 1.612229 1.612449 1.613166 1.613845 1.613410 1.611128 1.613252 1.613709 1.613597 1.613744

0.420 1.577239 1.577044 1.576322 1.575424 1.574071 1.573291 1.571644 1.577319 1.575850 1.577541

0.380 1.559229 1.560052 1.560163 1.561087 1.560912 1.561957 1.559912 1.560891 1.561084 1.560797

0.340 1.534268 1.534636 1.532641 1.531826 1.530509 1.528610 1.526366 1.524344 1.521616 1.517736

0.300 1.521629 1.522739 1.517375 1.513243 1.506666 1.515665 1.516432 1.511321 1.507372 1.499169

0.260 1.526027 1.528336 1.528365 1.523310 1.523426 1.521938 1.521199 1.522275 1.517409 1.524041

-0.260 1.497864 1.513420 1.508436 1.499114 1.498607 1.491676 1.508255 1.503622 1.514136 1.501430

-0.300 1.526418 1.523375 1.523676 1.524068 1.520430 1.526338 1.528319 1.524111 1.529267 1.520089

-0.340 1.531071 1.527468 1.526313 1.536455 1.536623 1.534605 1.533377 1.531921 1.530601 1.527972

-0.380 1.551624 1.550589 1.549361 1.551254 1.554931 1.554133 1.550423 1.549107 1.550504 1.553119

-0.420 1.567861 1.567163 1.569725 1.569244 1.567421 1.568365 1.564389 1.566840 1.565252 1.562468

-0.460 1.607977 1.607191 1.605903 1.603554 1.599546 1.600723 1.604047 1.608097 1.605447 1.605460

5.2 Qualitative Observations

1. For distances 34cm, 30cm and 26cm both above and below the meter rule’s center of mass, the initial amplitude of 5°

decreased significantly, within 5 oscillations such that the photogate continually remained in a blocked state, preventing

a period measurement to be taken. Therefore, for these distances the first 3 oscillations were taken and repeated such

that 10 periods were obtained.

2. Due to a loss of equipment, during experimentation the meter rule had to be pivoted on a smoother, but slightly larger

thin metal rod than the drill bit used to drill the holes for distances, 46cm, 42cm, 38cm, 34cm, 30cm and 26cm below

the meter rule’s center of mass.

3. For certain trials, the meter rule rocked sideways after which in some occasions did not oscillate perpendicular to the

pivot and the placed photogate.

6 Processed Data

6.1 Data Processing

6.1.1 Avg. Value of 𝑨 (𝑨𝑨𝒗𝒈.)

𝐴𝐴𝑣𝑔. =1

5∙ ∑ 𝐴𝑛

5

𝑛=1

∆𝐴𝐴𝑣𝑔. =𝐴𝑀𝑎𝑥. − 𝐴𝑀𝑖𝑛.

2+ 2.5×10−5𝑚

Sample Calculations:

𝐴𝐴𝑣𝑔. =0.02670 + 0.02675 + 0.02700 + 0.02685 + 0.02685

5𝑚

= 0.02683𝑚

∆𝐴𝐴𝑣𝑔. =0.02700 − 0.02670

2𝑚

= 1.500×10−4𝑚

Here, the precision of the Vernier Caliper, 2.5×10−5𝑚, was not added as its insignificant in comparison to 𝐴𝐴𝑣𝑔. by a degree of

10−3.

6.1.2 Avg. Value of 𝑩 (𝑩𝑨𝒗𝒈.)

𝐵𝐴𝑣𝑔. =1

5∙ ∑ 𝐵𝑛

5

𝑛=1

∆𝐵𝐴𝑣𝑔. =𝐵𝑀𝑎𝑥. − 𝐵𝑀𝑖𝑛.

2

Sample Calculations:

𝐵𝐴𝑣𝑔. =0.999 + 0.999 + 1.00 + 0.998 + 1.00

5𝑚


6

= 0.999𝑚

∆𝐵𝐴𝑣𝑔. =1.00 − 0.998

2𝑚

= 1.00×10−3𝑚

Again, the precision of the meter rule, 5×10−4𝑚, was not added as its insignificant by a degree of 10−3in comparison to the

value of 𝐵𝐴𝑣𝑔.

6.1.3 Determining the Avg. Period Squared (𝑻𝑨𝒗𝒈.𝟐)

𝑇𝐴𝑣𝑔.2 = [

1

10∙ ∑ 𝑇𝑛

10

n=1

]

2

∆𝑇𝐴𝑣𝑔.2 = 2 ∙ [

𝑇𝑀𝑎𝑥. − 𝑇𝑀𝑖𝑛.

2 ∙ 𝑇𝐴𝑣𝑔.

] ∙ 𝑇𝐴𝑣𝑔.2

Sample Calculations of TAvg.2 for a Distance of 0.460m Away from the Center of Mass:

𝑇𝐴𝑣𝑔.2

= [1.612229s + 1.612449s + 1.613166s + 1.613845s + 1.613410s + 1.611128s + 1.613252s + 1.613709s + 1.613597s + 1.613744s

10]

2

= 2.601940s2

∆𝑇𝐴𝑣𝑔.2 = 2 ∙ [

1.613854s − 1.611128s

2 ∙ 1.613053s] ∙ 2.601940s2

= 4.397183×10−3s2

6.1.4 Determining [𝑨𝟐+𝑩𝟐

𝟏𝟐𝑫+ 𝑫]

[𝐴2 + 𝐵2

12𝐷+ 𝐷] =

𝐴𝐴𝑣𝑔.2 + 𝐵𝐴𝑣𝑔.

2

12𝐷+ 𝐷

∆ [𝐴2 + 𝐵2

12𝐷+ 𝐷] = [2 ∙

∆𝐴𝐴𝑣𝑔.

𝐴𝐴𝑣𝑔.

+ 2 ∙∆𝐵𝐴𝑣𝑔.

𝐵𝐴𝑣𝑔.

+0.0005𝑚 ∙ 12

12 ∙ 𝐷] ∙ [

𝐴𝐴𝑣𝑔.2 + 𝐵𝐴𝑣𝑔.

2

12𝐷] + 5×10−4𝑚

Sample Calculations for a Distance of 0.460m Away from the Center of Mass:

[𝐴2 + 𝐵2

12𝐷+ 𝐷] =

(0.02683𝑚)2 + (0.999𝑚)2

12 ∙ 0.460𝑚+ 0.460𝑚

= 0.641𝑚

∆ [𝐴2 + 𝐵2

12𝐷+ 𝐷] = [2 ∙

1.500×10−4𝑚

0.02683𝑚+ 2 ∙

1.00×10−3𝑚

0.999𝑚+

0.0005𝑚 ∙ 12

12 ∙ 0.460𝑚] ∙ [

(0.02683𝑚)2 + (0.999𝑚)2

12 ∙ 0.460𝑚] + 5×10−4𝑚

= 0.003𝑚

6.2 Processed Quantitative Observations

[A2+B2

12D+ D] Against TAvg.

2

[A2+B2

12D+ D] (m) TAvg.

2 (s2)

0.641 ±0.003 2.601940 ±4.397183×10−3

0.618 ±0.003 2.482435 ±9.291163×10−3

0.599 ±0.004 2.435499 ±4.257340×10−3

0.585 ±0.004 2.335564 ±2.582751×10−2

0.578 ±0.005 2.289657 ±3.566521×10−2

0.580 ±0.005 2.321456 ±1.669292×10−2

-0.580 ±0.005 2.260981 ±2.516775×10−2

-0.578 ±0.005 2.324433 ±1.399286×10−2

-0.585 ±0.004 2.345923 ±1.579121×10−2

-0.599 ±0.004 2.407166 ±9.035962×10−3

-0.618 ±0.003 2.455090 ±1.113708×10−2

-0.641 ±0.003 2.575365 ±1.372260×10−2


7

7 Data Analysis

7.1 Graphical Presentation and Justification of Anomaly

The processed data table was graphed; however, for clarity 𝑇𝐴𝑣𝑔.2 for distances above and below the meter rule’s center of

mass were graphed onto separate graphs respectively. This enables a clear trend to be seen within 𝑇𝐴𝑣𝑔.2 that varies little, by a

maximum of 0.314384, that is less than 1.

y = -3.99xR² = 0.9225

Max: y = -5.22x - 0.735

Min: y = -3.55x + 0.267

2.20

2.25

2.30

2.35

2.40

2.45

2.50

2.55

2.60

2.65

-0.650 -0.640 -0.630 -0.620 -0.610 -0.600 -0.590 -0.580 -0.570

T^

2 (

s^2

)

[((A^2+B^2)/12D)+D (m)

Graph Displaying the Linearized Relationship Between the Centre of Mass and the Period of a Physical Pendulum

Max: y = 5.17x - 0.690

Min: y = 3.57x + 0.295

y = 4.03xR² = 0.9797

2.20

2.25

2.30

2.35

2.40

2.45

2.50

2.55

2.60

2.65

0.570 0.580 0.590 0.600 0.610 0.620 0.630 0.640 0.650

T^

2 (

s^2

)

[((A^2+B^2)/12D)+D (m)


Figure 7.1.1: Period of the physical pendulum for distances above the center of mass of the meter rule.

Figure 7.1.2: Period of the physical pendulum for distances below the center of mass of the meter ruler. Here, a line of best fit through the origin cannot be drawn through the error bars of the anomaly.


8

For distance above the center of mass of the meter rule, a line of best fit through the origin could be drawn within the 𝑋 and

𝑌 error bars of the collected data, suggesting a direct proportionality with a regression of 0.96. However, this was not possible

for distances below the meter rule’s center of mass; a line of best fit through the origin cannot be drawn through the error bars

of 𝑇𝐴𝑣𝑔.2 for 26cm below the center of mass of the meter rule that deviates from the trend. Furthermore, its 𝑇𝐴𝑣𝑔.

2 value is lower

than for a distance of 26cm by 6.6485×10−2𝑠2. Although apparently small, this difference is significant given the number of

trials obtained for it, 10, and the precision of the period measurement, ±1×10−6𝑠. Furthermore, its high 𝑌 error bars that is the

second highest from the collected data, ±2.516775×10−2𝑠2, is typical of an anomalous result.

With the omitted anomaly, a line of best fit can then be drawn through the origin and within all the error bars that is

consistent with the hypothesized direct proportionality. A higher regression of 0.98 than previously 0.92 was obtained which

could be further corroborating evidence for the identified result being an anomaly, qualifying its omission.

Hence, the anomalous result was omitted. Since the absolute values of both gradients (𝐺1 for distance above and 𝐺2 for

distances below with the omitted anomaly) should be the same, their absolute average was obtained, 𝐺𝐴𝑣𝑔., with its error

propagated with the maximum and minimum possible gradients formed. This average gradient would be used as a reference

point for the percentage error of the experiment; the deviation from the experimental gradient from the hypothesized gradient.

This would enable a conclusion to be made as to whether the hypothesis should be accepted.

7.2 Experimental Error Calculations

7.2.1 Determining 𝑮𝑨𝒗𝒈.

𝐺𝐴𝑣𝑔. =𝐺1 + |𝐺2|

2

∆𝐺𝐴𝑣𝑔. = [𝐺1.𝑀𝑎𝑥 − 𝐺1.𝑀𝑖𝑛

2+ |

𝐺2.𝑀𝑎𝑥 − 𝐺2.𝑀𝑖𝑛

2|] ∙

1

2

Calculations:

𝐺𝐴𝑣𝑔. =4.03𝑚𝑠−2 + |−4.01𝑚𝑠−2|

2

= 4.02𝑚𝑠−2

∆𝐺𝐴𝑣𝑔. = [5.17𝑚𝑠−2 − 3.57𝑚𝑠−2

2+ |

(−3.01𝑚𝑠−2) − (−5.29𝑚𝑠−2)

2|] ∙

1

2

y = -4.01xR² = 0.9828

Max: y = -5.29x - 0.735

Min: y = -3.01x + 0.267

2.20

2.25

2.30

2.35

2.40

2.45

2.50

2.55

2.60

2.65

-0.650 -0.640 -0.630 -0.620 -0.610 -0.600 -0.590 -0.580 -0.570

T^

2 (

s^2

)

[((A^2+B^2)/12D)+D (m)


Figure 7.1.3: Period of the pendulum for distances below the center of mass of the meter ruler with the omitted anomaly. Here, a line of best fit through the origin can be drawn for all data.


9

= 0.970𝑚𝑠−2

7.2.2 Determining the Random Error of 𝑮𝑨𝒗𝒈. (𝜹𝑹𝒂𝒏𝒅𝒐𝒎)

𝛿𝑅𝑎𝑛𝑑𝑜𝑚 =∆𝐺𝐴𝑣𝑔.

𝐺𝐴𝑣𝑔.

×100

Calculations:

𝛿𝑅𝑎𝑛𝑑𝑜𝑚 =0.970𝑚𝑠−2

4.02𝑚𝑠−2×100

= 24%

7.2.3 Determining the Percentage Error of 𝑮𝑨𝒗𝒈. (𝜹%)

𝛿% = |1

𝐺𝐴𝑣𝑔.

∙ [4𝜋2

𝑔− 𝐺𝐴𝑣𝑔.] ×100|

Calculations:

𝛿% = |1

4.02𝑚𝑠−2∙ [

4𝜋2

𝑔− 4.02𝑚𝑠−2] ×100|

= 0.64%

8 Conclusion

The average absolute value of the experimentally obtained gradient was 4.02ms−2 ± 0.970ms−2, having a random error of

24%. The percentage error an indicator of the accuracy of this experimentally obtained gradient, 0.64%, being within its random

error of 24% and less than 1% away from the hypothesized gradient, allows the hypothesis to be accepted. Moreover, the

hypothesis is further validated by the direct proportionality obtained with high regressions 0.96 and 0.98 that is greater than

0.95 when the identified anomaly was omitted. However, the random error of 24% being greater than the percentage error of

0.64%, suggests that random errors were the main contributors to the error of the experiment.

9 Evaluation

9.1 Strengths of the Investigation

The direct proportionality obtained for both graphs, distances above and below the meter rule’s center of mass, is reflective

of the negligible presence of systematic errors in the experiment, after the justified anomalies were omitted. This validated the

assumption regarding the position of the meter rule’s center of mass being at its geometric center and the methodology for not

causing a significant deviation in its position. If the center of mass had deviated, the absolute value graph (see Figure 3.1) or the

position of its origin would have shifted along the 𝑋- axis, resulting in a positive and negative 𝑌-intercept for the line of best fit

for either graph. Since, this was not the case as a direct proportionality had been obtained, it could be asserted that this

assumption was valid and the methodology conserves its position.

Although random errors, 24%,were the main contributors to the error of the experiment, it being just less than 25% for a

school based experiment is low. This extent is further evident through the 𝑌 and 𝑋 error bars, not exceeding ± 3.566521×10−2s2

and ± 0.005m. The low deviation, random error, an indicator of the reproducibility or precision of the measurements made is

attributed to the use of a programmed photogate in measuring the period of the meter rule. This mitigates any source of human

errors if the period were to be measured in the traditional way where one would time 20 oscillations to obtain the average

period of 1 oscillation. Alternatively, this could be attributed to the number of trials taken, 10, and an increase in this should not

discernably reduce the random error of the experiment.

𝑇2 ∙ 𝐷 =

4𝜋2

𝑔∙

𝐼𝐶𝑂𝑀

𝑀+

4𝜋2

𝑔∙ 𝐷2

[6]

Despite the drilled holes on the meter rule, it was assumed that its moment of inertia about its center of mass is still

equivalent to that of a metal plate. Therefore, to test this assumption TAvg.2 ∙ 𝐷 was graphed against 𝐷 for both above and below

the meter rule’s center of mass. The 𝑌-intercept of both graphs should yield a value proportional to 𝐼𝐶𝑂𝑀 of the meter rule,

divided by its mass (see Figure 9.1.1 and 9.9.2). This linearized equation to be graphed, [6], was obtained by rearranging, [3]. If

the experimental 𝐼𝐶𝑂𝑀 of the meter rule, divided by its mass corresponds closely to 𝐼𝐶𝑂𝑀 of a rectangular plate, [4], divided by


10

its mass the assumption could be argued as valid and accurate. For distances, above the meter rule’s center of mass the

experimental value of 𝐼𝐶𝑂𝑀 divided by its mass was 4.5% away from the assumed value that was calculated using 𝐴𝐴𝑣𝑔. and 𝐵𝐴𝑣𝑔.

(see Table 9.1.3). Alternatively, for distance below the discrepancy was 0.89% (see Table 9.1.3). Both these percentage deviations

being less than 5% away from the experimental value, renders the assumption to be valid for this investigation.

Percentage Variation (𝜹𝑰𝑪𝑶𝑴𝑨) in Experimental 𝑰𝑪𝑶𝑴

𝒎

Values from Assumed Values for Distances Above the

Meter Rule’s Center of Mass:

Percentage Variation (𝜹𝑰𝑪𝑶𝑴𝑩) in Experimental 𝑰𝑪𝑶𝑴

𝒎

Values from Assumed Values for Distances Below the

Meter Rule’s Center of Mass:

Figure 9.1.1: The maximum and minimum gradients were not drawn since only the Y intercept for the line of best fit is necessary to evaluate the assumption.

Figure 9.1.2: The anomaly for a distance 26cm below the meter rule’s center of mass was omitted. Here, the error bars are too small to visualize the anomalous result.

y = 4.13x + 0.319R² = 0.9993

0.55

0.65

0.75

0.85

0.95

1.05

1.15

1.25

0.040 0.090 0.140 0.190 0.240

(T^

2)D

(m

s^2

)

D^2 (m^2)

(T^2)D Against D^2 For Distances Above the Meter Rule's Center of Mass

y = -3.97x - 0.338R² = 0.9993

-1.3

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

0.040 0.090 0.140 0.190 0.240

(T^

2)D

(m

s^2

)

D^2 (m^2)

(T^2)D Against D^2 For Distances Below the Meter Rule's Center of Mass


11

𝛿𝐼𝐶𝑂𝑀𝐴 = |𝑔

4𝜋2∙ 𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 −

(𝐴2 + 𝐵2)

12| ∙ [

12

𝐴2 + 𝐵2] ×100

Calculation:

𝛿𝐼𝐶𝑂𝑀𝐴 = |𝑔

4𝜋2∙ 0.319𝑚2 −

1.500×10−42+ 0.9992

12∙ 𝑚2|

∙ [12

1.500×10−42+ 0.9992

𝑚−2] ×100

= 4.5%

𝛿𝐼𝐶𝑂𝑀𝐵 = |𝑔

4𝜋2∙ 𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 −

(𝐴2 + 𝐵2)

12| ∙ [

12

𝐴2 + 𝐵2] ×100

Calculation:

𝛿𝐼𝐶𝑂𝑀𝐵 = |𝑔

4𝜋2∙ 0.319𝑚2 −

1.500×10−42+ 0.9992

12∙ 𝑚2|

∙ [12

1.500×10−42+ 0.9992

𝑚−2] ×100

= 0.89%

Table 9.1.3: Calculations displaying the difference between the experimental value of 𝐼𝐶𝑂𝑀divided by the meter rule’s mass and the assumed value.

9.2 Limitations and Improvements

The deviating result obtained for distances below the center of mass of the ruler, was justified as an anomaly on the basis of

a limited range of the independent variable, 6. Therefore, to justify and omit this result on a strong basis so as to reliably reflect

the random or percentage error of the experiment on which a strong, conclusion could be made, there should have been a greater

range for the distances on the meter rule where the period measurement should be taken. A possible range is 46cm, 43cm, 40cm,

37cm, 34cm, 31cm, 28cm and 25cm both above and below the center of mass.

Figure 9.1.1: If the approach of the meter rule were perpendicular to

the photogate, the angular it would have to travel to block it would

be least.

Figure 9.2.2: If the angle of approach were to vary, the angular

distance the meter rule would have to travel to block the photogate

would be greater and vary. Thus, period measurements would vary.

The meter rule was observed to wobbled sideways after being released which may have contributed to the random error of

the experiment. This sideways wobbling motion varies the angle at which the meter rule approaches the photogate. Hence, the

photogate would reach the final blocked stage that determines the end of one oscillation later, by varying amounts, as the meter

rule would have to travel a greater distance to block the photogate if it were approaching it angled (see Figure 9.1.2) than

perpendicular (see Figure 9.1.1). The discrepancy between these period measurements when perpendicular or angled during

its wobbling motion should contribute to the random error and is significant given the high precision of the photogate,

±1×10−6𝑠. Therefore, to mitigate this behavior the meter rule could have been pivoted between two large, secured washers

that are sufficiently close to each other. This ensures oscillations are perpendicular to the pivot.

It is still unclear as to how the low anomaly, for a distance of 26cm below the meter rule’s center of mass, may have been

brought about. Typically, the factor explored above could systematically increase period measurements rather than reduce

period measurements to bring about the anomaly. Furthermore, its random error, ±2.516775×10−2𝑠2, not being the highest

and looking back at the raw data, suggests that there wasn’t any data that may have significantly reduced the average. Since how

the anomaly may have been brought about is inconclusive, it would be best to repeat the measurements taken for this distance

of 26cm below the meter rule’s center of mass.

The amplitude was qualitatively observed to have deteriorated significantly that is likely to be the consequence of frictional

force opposing its motion, invalidating this assumption. This could be attributed to the slightly larger metal rod used as the pivot

after a loss of equipment whose tight fit may have accentuated the role of frictional force. Though the period is independent of

the amplitude (Burley et al., 1997), this significant reduction of the meter rule’s amplitude to within a few oscillations disallowed

its period to be measured due to the continual blocking of the photogate. Therefore, to mitigate this behavior and the role of

friction, the drilled holes and the pivot could have been greased.


12

10 Extensions

The investigation established effect of varying distances of the pivot from the center of mass of rectangular plate on the

period of its oscillation. Upon further research, it was found that there exists a minimum period for a certain distance the

rectangular plate is pivoted from its center of mass. Therefore, factors such as the rectangular plate’s length, width and depth

on the effect of this minimum period could be investigated given more time. This could possibly aid in determining or shortlisting

the length of pendulum required for a certain measurement of time. Alternatively, the investigation could have possible

extended by comparing the periods obtained for different regularly shaped objects or on different axis of rotations.

11 Bibliography

Burley, I., Carrington, M., Kobes, R. and Kunstatter, G. (1997). The Simple Pendulum. [online] Theory.uwinnipeg.ca.

Available at: http://theory.uwinnipeg.ca/physics/shm/node5.html [Accessed 8 Apr. 2017].

Fitzpatrick, R. (2006). The Simple Pendulum. [online] Farside.ph.utexas.edu. Available at:

http://farside.ph.utexas.edu/teaching/301/lectures/node140.html [Accessed 9 Apr. 2017].

Grieve, D. (2001) Compound Pendulum. [online] Available at:

http://www.tech.plym.ac.uk/sme/mech226/compoundpend.htm [Accessed: 26 February 2017].

Hall, N. (2015). Center of Gravity. [online] Grc.nasa.gov. Available at: https://www.grc.nasa.gov/WWW/k-

12/airplane/cg.html [Accessed 12 Apr. 2017].

Kaldon, P. (2003). Moment of Inertia (Rotational Mass). [online] Homepages.wmich.edu. Available at:

http://homepages.wmich.edu/~kaldon/classes/ph205-11-moment-of-inertia.GIF [Accessed 11 Apr. 2017].

McFatridge, J. (2003). Rotational Motion. [online] Www2.hawaii.edu. Available at:

https://www2.hawaii.edu/~jmcfatri/labs/rotmotion.html [Accessed 11 Apr. 2017].

Nave, R. (2012). Physical Pendulum. [online] Hyperphysics.phy-astr.gsu.edu. Available at: http://hyperphysics.phy-

astr.gsu.edu/hbase/pendp.html [Accessed 11 Apr. 2017].

Vernier.com. (n.d.). Photogate User Manual. [online] Available at: https://www.vernier.com/manuals/vpg-btd/ [Accessed

12 Apr. 2017].

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