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Candidate Name: Aladdin Zaklouta

Candidate Number: 000790 – 029

November 2014

IB Physics Internal Assessment - SLFactors affecting the mass loaded on a cantilever

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Research Question:

How does the mass (50, 100, 150, 200, 250 grams) loaded on a cantilever affect the

period of oscillation of the cantilever?

Introduction:

A cantilever is a beam that is supported at only one end and carries a load at the other

end. Cantilevers are often used in architecture, as they allow for overhanging structures

that do not need to be supported from the outside. Airplane wings are also cantilevers,

as they are usually only supported from the body of the plane . Because only one end of

the beam is supported, a cantilever is typically subjected to a great deal of stress. The

top half of a cantilever experiences tensile stress, which elongates the fibers of the

beam. The bottom half is subjected to compressive stress, which compresses the fibers.

If it can't handle all of the stress, a cantilever may arc downward or even break. As such,

engineers and architects work to design beams and balance loads in such a way that

they will not suffer from such structural issues.

The aim of the experiment is to investigate the relationship between the mass loaded

on a cantilever and the time period of oscillation of a loaded cantilever. The masses used

that will test this relationship are 50g, 100g, 150g, 200g, and 250 grams. The period of

oscillation will be timed using a stop watch.

Hypothesis:

It is hypothesized that due to the increase of mass on the cantilever, the force will increase,

however the acceleration will decrease. This will result in different oscillations for different

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masses, and will further increase the time taken to complete 5 oscillations. The results obtained will

be used to construct a linear graph, due to the proportionality of mass and time.

M

T

Variables Description

Independent

Mass loaded on a cantilever The mass loaded on a cantilever will increase by 50grams, from 50grams to

250 grams.

Dependant

Time period for 5 oscillations The time period for 5 oscillations of the cantilever

will be dependent on the mass of loaded on the

cantilever.

Controlled

Cantilever The same cantilever will be used during the experiment to

minimise random errors. Therefore the cantilever will

not be substituted.Length of Cantilever The cantilever will be fixed on

the by a g-clamp. This means that the length of the cantilever will remain

constant throughout the experiment.

Number of oscillations The cantilever will be timed for 5 oscillations and this will remain the same throughout

the experiment.

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Apparatus

- 1x Meter ruler (±0.0005m)

- 1x Sticky tape

- 5x 50gram Mass

- 1x Stop watch (±0.01s)

- 1x Cantilever

- 1x Clamp

Method

1. Set up the experiment using the metre ruler, clamp and the retort stand. Use a high bench

to give a reasonable height for you to read measurements. Attach a ruler on a bench 5cm

away from the end using the clamp to the bench.

2. Measure your equilibrium position of the metre while no mass is attached to the ruler. This

will be the release point of all masses.

3. Attach the first set of mass (50 grams) onto the cantilever, raise the metre ruler to the

equilibrium position that was initially marked, and release the ruler. At this point you should

begin the stop watch and count for 5 oscillations. Ensure that the same person releasing the

loaded cantilever is holding the stopwatch and timing the 5 oscillations.

4. Repeat step 3 for 100 grams, 150 grams, 200 grams and 250 grams. After every trial make

sure that the cantilever isn’t swinging, and is idle before testing the time taken for 5

oscillations, this will minimise random errors.

5. Record results in a table, calculating the average time taken as well as uncertainty

Table 1: Table of results template of time period for 5 oscillations, and mass loaded on cantilever for 5 trials

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Time Period for 5 oscillations ±0.01 sMass loaded (g) ±0.001m

Trial 1(s) Trial 2 (s) Trial 3 (s) Trial 4 (s) Trial 5 (s)

50.000100.000150.000200.000250.000

Raw DataTable 2: Table of time period for 5 oscillations, and mass loaded on cantilever for 5 trials.

Time Period for 5 oscillations 0.01 (s)Mass loaded

(g) ±0.001Trial 1(s) Trial 2 (s) Trial 3 (s) Trial 4 (s) Trial 5 (s)

50.000 2.79 2.82 2.83 2.81 2.82100.000 3.10 3.19 3.09 3.19 3.19150.000 3.44 3.55 3.53 3.56 3.46200.000 3.75 3.82 3.78 3.81 3.75250.000 3.94 4.09 4.03 4.03 3.99

Table 3: Table of time period for 1 oscillation, and mass loaded on cantilever for 5 trials.

Time Period for 5 oscillations 0.01 (s)Mass loaded

(g) ±0.001Trial 1(s) Trial 2 (s) Trial 3 (s) Trial 4 (s) Trial 5 (s)

50.000 0.56 0.56 0.57 0.56 0.56100.000 0.62 0.64 0.62 0.64 0.64150.000 0.69 0.71 0.71 0.71 0.69200.000 0.75 0.76 0.76 0.76 0.75250.000 0.79 0.82 0.81 0.81 0.80

Mass loaded (g) ±0.001 Average time (s) Uncertainty of time (s)50.000 0.56 0.01

100.000 0.63 0.01150.000 0.69 0.01200.000 0.76 0.01250.000 0.80 0.02

Table 4: Table of average time period for 1 oscillation, mass loaded on cantilever and uncertainty of time.

Graph 1: Graph of Curve fit of Mass loaded on cantilever versus Average time period for 1 oscillation.

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As seen in the graph above, the results gave a curved graph. In order to linearise the curve, the average time and uncertainty of time must be square rooted. The manipulated results are shown below, which will produce the linear graph.

Processed Data

Table 5: Table of Mass loaded on cantilever, average time squared and uncertainty squared.

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Mass loaded (g) ±0.001 Square root of Average time (s)1/2

Square root of Uncertainty of time (s)1/2

50.000 0.75 0.01100.000 0.79 0.01150.000 0.83 0.01200.000 0.87 0.01250.000 0.89 0.01

Graph 2: Graph of Linear fit of Mass loaded on cantilever versus Square root of Time period for 1 oscillation

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Graph 3: Graph of Minimum and Maximum fit of Mass loaded on cantilever versus Square root of Time period for 1 oscillation

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Conclusion and Evaluation

It is hypothesised that the mass applied on the cantilever is directly proportional to the time it takes

to oscillate. Graph 1 shows the results obtained as a curve fit of time period for 1 oscillation, which

disregards the hypothesis. However graph 2 shows that a linear fit is obtained after processing the

data. This was obtained through square rooting time as well as time uncertainty, which gave a linear

graph. The equation y = mx + c gives the equation of the line for graph 3.

Y = mx + c

Linear Fit = m = 0.0007200 s1/2 g-1

Gradient of Maximum Line = 0.0007900 s1/2 g-1

Gradient of Minimum Line = 0.0006000 s1/2 g-1

Uncertainty in slope = 0.0007200 – 0.0007900 = 0.00007 s1/2 g-1

0.0007200 – 0.0006000 = 0.00012 s1/2 g-1

Therefore, Highest deviation = 0.0007200 ± 0.00012

0.0007200 ± 0.03 s1/2 g-1

Y = mx + c

√ t = 0.0007200 ± 0.03M + c

The equation of the line derived through the calculation of the uncertainty of the minimum and

maximum line of the linear fit graph. This was then substituted into the equation y = mx + c to give

the equation of the linear fit line, where y is substituted as √ t and the slope being 0.0007200 ± 0.03

multiplied by the substitution of x as Mass.

Graph 1 shows the curve fit of mass loaded on a cantilever versus time period for 1 oscillation. The

hypothesis suggested that graph 1 would have a linear fit, where mass loaded on cantilever verse

time to be directly proportional. This was not met due to the fact of the graph having a curve fit, and

more deeply, not met due to errors that were instigated during the experiment.

During the process of loading the mass on the cantilever, and having to lift the metre ruler back to

the marked equilibrium position, some error was involved, called parallax error. This error was

caused due to the fault in our eyes, where the perfect angle in respect to the equilibrium position

was not obtained, hence the release point varied between trial to trial, ultimately affecting the time

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period of the oscillation. The readings obtained were in error of ±0.5cm of the original equilibrium

position. This error could have been minimised by adding a platform to level the metre ruler with,

rather than levelling it with the mark drawn.

Another source of error was human reaction time error; this played a much larger role in the scale of

errors. As the time taken was the main variable of the practical, perfecting the time was crucial.

When releasing the loaded cantilever the same person was needed to record the time taken for it to

complete 5 oscillations. However, the reaction of releasing the loaded cantilever and the initiation of

the stop watch was not accurate; therefore this played a role in the accumulation of the results into

a curved graph rather than a linear. Due to the speed of oscillations fir the first few masses (50

grams – 150 grams) one wasn’t able to keep up with the 5 oscillations, therefore the exact point of

oscillation was not apprehended. This could have been improved by increasing the trials, or

beginning the experiment with a mass of 150 grams, in order to read the time period for 5

oscillations. One should take into account the reaction time and therefore time before releasing the

loaded cantilever, and stopping before the exact point of 5 oscillations, to account for the human

reaction time error.

In conclusion, the hypothesis was not met, where the results obtained will construct a linear graph,

as mass loaded on a cantilever and time period is proportional. This was due to the errors which

deviated the results into a curved graph. These errors were the human reaction time error and the

parallax error. The errors could be avoided in further practicals by taking into account the human

reaction time error and both; timing and stopping before you have released the metre ruler with the

loaded cantilever attached. As for the parallax error, it could be avoided by preparing a suitable

arrangement in the method process, where the release point of the metre ruler was accurate to the

equilibrium position.

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