last physics 16july submitted

8/9/2019 Last Physics 16July Submitted

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Experimental

Investigation

July 16

2010Cantilevers By Arun


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2010 Melbourne High School Physics Department

THEORY

The external forces that a structure resists are called loads. Structures must also resist

their own weight and contents. Both the size of components and the materials that are

used are relevant to how well a structure can resist a load (Graeme Lofts et al l, 2010).

A typical structural cantilever is shown in diagram 1. A cantilever is made of a structured

section like a timber plank or concrete beam or steel. It has unsupported length L

(cantilever length) and a fixed end like that may be clamped into place and rigidly fixed. It

is free on the other end of the cantilever

.

Diagram 1: The turning force (Mortan, 2004).

The cantilever may have point vertical load F anywhere on the length at a distance of L

from the fixed end. The other vertical load is the self weight of the cantilever. The vertical

load on the cantilever causes clockwise torque at the fixed end (also called the pivot

point).

The turning effect of a force is called torque. It is produced by a force that acts at a

distance from a point of rotation. The variables that influence the magnitude of torqueare distance from the point of rotation or clamping, the direction of the force in relation

to that same point and the size of the force.

The turning force (torque) bends the section as shown in diagram 2.


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Diagram 2: diagram of bending (Mortan, 2004).

Bending is a term used to describe a torque force that causes a particular curving

deformation in a structure. When bending, a structure tends to curve. Some fibres of the

cross section move closer together and other fibres move further apart, causing

compression on one side and tension in the other. It is easier to break a structure by

bending than through sheer, so engineers take special care to ensure that structures are

strong enough to resist bending. When in tension, the molecules move further apart and

the material gets longer. When in compression they move closer together and the

material gets shorter.

The stress unit is usually used for comparisons of the performance of different sized loads

on different sized objects. However, the magnitude of the force and therefore torque can

be used when the size of the load, the size of the objects as well as the area the force is

applied are controlled in the experiment. (Graeme Lofts et all, 2010).

Deflection is a term that is used to describe the degree to w hich a structural element is

displaced under a load and the associated bending (curving of a section) it what causes

the free end of the cantilever to deflect downward (refer to diagram 3). The amount of

the deflection h depends on:

1. Applied load F.

2. Distance L of the load from pivot.

3. The width of the plank of wood b.

3. Cross section depth d, and width b and the section property called moment of inertia

I where I = (bd^3)/12


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4. Elastic modulus of the material of the cantilever E is higher in hard wood (Tasmanian

Oak) than soft wood (Pine Wood).

These can be linked by the equation

h=(FL^3)/(3EI)...................................... .....................equation 1

Under low loads the free end of the cantilever will return to the original point when the

load is removed.This means that the wood acted elastically.

Under a large load, torque, the free end of a cantilever does not return to the original

point and shows a permanent change due to the fibres in the section permanently

changing. This is known as the permanent vertical deformation and when this is not equal

to zero it is termed plastic deformation (G. Gorenc and R. Tinyou, 1984).

Diagram 3: Deflection and permanent vertical deformation.

Variables

*The distance of the weights from the point of clamping; L, measured in mm

*The mass of the weights added to the cantilever m, measured in kg [Weight (W) = Mass

(m) x acceleration due to gravity (g) = Force (F)]

*The deflection (dependant), h, measured in mm

*Type of wood (dependant)

*Permanent vertical deformation (dependant), p

*Torque (dependant),t, measured in kgm Throughout this report the use of the word

torque ignores the self-weight torque of the cantilever.


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Assumptions

In this experiment it was assumed that the weighing scale used to weigh the masses is

accurate. It is also assumed that the dimensions of any plank of either Tasmanian Oak or

Pine wood match the specifications for the planks tested and that the wood has a uniform

density. It is assumed that there are no pre-existing cracks exist in the wood and that eachwooden plank has comparable moisture content throughout the experiment.

In addition we have assumed that th e wood is not undergoing warping and that the

acceleration due to gravity is roughly ten meters per second per second where the

experiment in conducted. It is assumed that the humidity of the physics room is constant

throughout the investigation and that the experiment is taking place under Standard

Laboratory Conditions (SLC) which is defined as 1 atmosphere of pressure at 25*C.

The assumption that previous experimental work on the wooden beam will not affect

subsequent results was made. Since torque acts perpendicular to the surface of the

beam, it is assumed that despite bending, the force will continue to act perpendicular to

the woods surface.

Expectation

It is expected that the deflection of the plank of wood will increase as heavier masses are

attached to it at a constant distance from the point of clamping. If this distance is

decreased, the deflection will decrease. It is expected that this part of the experiment will

result in permanent vertical deformation in the wood and that the hard Tasmanian Oak

wood will break during the experiment. The permanent vertical deformation should

increase proportional to torque .

Relevant Calculations

Torque (Force in Newtons * horizontal lever arm length in metres)

Weight: Mass (grams)* Acceleration due to gravity

Purpose

The purpose of this experimental investigation is to explore the structural properties of

wooden beams. This will involve observing elastic and plastic behaviours of two different

types of wood under different magnitudes of torque. The consequential affect of applying

a load to already bent wood, the deflection caused by a load and how these two

relationships vary due to the composition of the actual wood (hard and soft) will be

graphed and analysed to deduce the relationships between these three variables. In

addition, the breaking torque and corresponding permanent vertical deformation of both

wooden beams will be examined.


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This will be achieved by having a set of forces that will be applied to the free end of the

cantilever (plank of wood), and the lengths varied and deflections measured for each of

the lengths. This will be repeated for both types of wood, allowing for the differences in

resistance to bending between the two to be identified. This allows us to analyse the

dependant variable deflection, at the unsupported end (for each length and type of

wood). The permanent vertical deformation at each stage of this experiment due to the

effects off the bending will also be recorded. This allows us to account for the effects of

creep permanent vertical deformation, on the above relationship. The range of

weights that will be used will range from 1000g to 4000g.

The torque will be varied by maintaining a constant distance and varying the weight

applied using the mass range stated above. When the full set of masses have been

applied and the results recorded, the distance is varied in increments of 10cm and the

masses reapplied and the results recorded.

By unloading the beams, the relationship between the permanent vertical deformati onand torque was explored, and how this was affected by the wood co mposition of the

beam and the level of loading.

Materials

Ruler: Used to measure the deflection, the permanent vertical deformation and the

horizontal distance component of torque.

G-clamps: Used to keep the planks of wood in place.

Mixed masses/Weights: Used to provide force component of torque. Spare mixed masseswere also used to ensure the back of the plank does not lift up in the air thus increasing

leverage due to leverage and compromising the validity of the results.

Hooked mass holder: Used to hold the masses while they are attached onto the plank of

wood.

Hard wood plank: Used to examine its properties

Soft wood plank: Used to examine its properties


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Diagram 4: The experimental set up (not to scale) (Mortan, 2004).

Safety

Safety goggles were worn during the experiment.

Measuring instruments

Measure (ruler): A straightedge piece of wood that is calibrated with increments that

indicate length. The smallest division on the ruler used was 1mm. The mixed masses

were weighed using a laboratory scale.

METHOD

1. Safety glasses were worn2. The plank of wood is clamped with the G clamp using pieces of rough wood to

secure it, onto the table top as show in the diagram. The hooked mass holder was

attached to end of the plank.

3. To examine the soft pine wood, a mass was placed at the end of a plank of softpine wood using a hooked mass holder that allowed the masses to hang when the

wood was extended off the table. Whenever positioned on the table, the plank of

wood was secured into place with a G clamp and adjusted when needed. Sections

of wood that extended across the table where secured down using spare masses.

4. the plank length was increased (varied) in increments from 0cm to 100cm inincrements of 10cm. At first, only the hooked mass holder attached to the end of

the plank was examined to establish whether there was any observable bending

due to the mass of the hooked mass holder and plank.

5. The deflection of the plank was observed and recorded at each increment . Thedeflection was measured from the last measurement of permanent vertical


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deformation. However, the permanent vertical defor mation was always measured

from original height. Experimental method for experiment 1 deflection.

6. The wood was then extended by 10cm away from the edge of the table, anyexisting masses were removed from the hooked mass holder and one mass of

approximately 1kg placed on the mass holder.

7. Deflection was observed and a ruler was used to measure this. The results wererecorded. Experimental method for experiment 1 deflection

8. The mass was unloaded and permanent vertical deformation was observed. Aruler was used to measure this and the results were recorded. Experimental

method for experiment 2 permanent vertical deformation

9. The previous masses were reloaded an additional mass was attached to thehooked mass holder and steps 7 to 8 were repeated. This step was repeated thrice

(finally 4kg)

10.The above procedural steps (6-9) were repeated but an additional increment of10cm was made, cumulative of previous additions of 10cm.

11.The above step was repeated until the extension L was 100cm. When the woodsnapped during this process, the experiment proceeded to the step below rather

than proceeding till 100cm. Experimental method for experiment 3 the break

test

12.Steps 3-12 were repeated but for hard wood (Tasmanian Oak) rather t han softwood (Pine wood).

13.The results were recorded.

Variable control:

-Distance of mass from the point of clamping.

-The magnitude of the weight.

-Type of wood used.

The above variables were varied with intent during the experiment. Thus they were

controlled.

-The deflection (dependant variable) and the permanent vertical deformation

(dependant variable) were observed and recorded.

-The breaking torque of either type of wood (dependant variable) was observed

and recorded as applicable.

-The torque (dependant) is dependent on the mass and the horizontal distance

from the point of clamping. However, it was indirectly controlled by the control of each of

its composing values.


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Collected Data

The allowable bending stress of Hard Tasmanian Oak wood cannot be calculated from

the available data since the cross -sectional area cannot be defined (note: the cross

sectional area for this stress is not the cross sectional area for each plank - which is

known). The breaking applied mass and length and thus torque was recorded in lieu of

this.

Recalling that the plank length was increased (varied) in increments from 0cm to

100cm in increments of 10cm. At first, only the hooked mass holder attached to the

end of the plank was examined to establish whether there was any observable

bending due to the mass of the hooked mass holder and plank . From the

experimental method The weight of the hooked ma ss holder was found to be

insignificant and would not compromise the integrity of the results.

While the masses used were (g): 1008

They have beensimplified in the

graphs to 1, 2, 3

and 4 kg

2009

3006

3998

UNITS used mm -

deflection

mm

permanent

vertical

deformation

g mass of

weights

Nm -

torque

cm

distance

from point

of clampingSI units

M MKg Nm M

Justification

for not

using SI

units

Convenience

Convenience

Convention Accuracy

Convenience


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Diagram 7


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Diagram 8


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Diagram 5


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Diagram 6


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SOFT WOOD

Length (cm) Mass (g)

Deflection

(mm)

Permanent

vertical

deformation

(mm) torque (Nm)

10 1008 0.8 0.1008

10 2009 1 0.2009

10 3006 1.5 0.3006

10 3998 2 0 0.3998

20 1008 3.5 0.2016

20 2009 8 0.4018

20 3006 13 0.6012

20 3998 17 0 0.7996

30 1008 11 0.3024

30 2011 23 0.6033

30 3006 38 0.9018

30 3998 52 2 1.1994

40 1008 20 0.4032

40 2012 52 0.8048

40 3006 80 1.2024

40 3998 102 11 1.5992

50 1008 52 0.504

50 2013 103 1.0065

50 3006 165 1.503

50 3998 232 41 1.999

60 1008 92 0.6048

60 2014 196 1.2084

60 3006 279 1.8036

60 4000 333 421 2.4

80 1008 203 0.8064

80 2014 318 1.6112

80 3006 380 2.4048

80 4000 440 620 3.2


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HARD WOOD

Length (cm) Mass (g)

Deflection

(mm)

Permanent

vertical

deformation

(mm) torque(Nm)

10 1008 1 0.1008

10 2009 2 0.2009

10 3006 2.5 0.3006

10 3998 3 1 0.3998

20 1008 3 0.2016

20 2009 7 0.4018

20 3006 10 0.6012

20 3998 13 2 0.7996

30 1008 6 0.3024

30 2011 14 0.6033

30 3006 23 0.9018

30 3998 32 3 1.1994

40 1008 15 0.4032

40 2012 32 0.8048

40 3006 49 1.2024

40 3998 66 5 1.5992

50 1008 29 0.504

50 2013 61 1.0065

50 3006 92 1.503

50 3998 126 10 1.999

60 1008 47 0.6048

60 2014 96 1.2084

60 3006 132 1.803660 4000 213 18 2.4

70 1008 73 0.7056

70 2014 145 1.4098

70 3006 260 2.1042

70 4000 276 29 2.8


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80 1008 106 0.8064

80 2014 208 1.6112

80 3006 294 37 2.4048

100 1008 135 1.008

100 2014 265 2.014

100 3006 367 3.006

100 4000 460 4

At an 80cm section of the beam off the edge of the table, with a weight mass of 4kg and

thus a torque of 3.2Nm - the soft wood plank snapped.

*Graphs of both the deflection and the permanent vertical deformation at each increment

of length are available in the appendix.

Analysis

The hard wood plank did not break throughout the course of the experiment. This

indicates that the hard wood plank is stronger than the soft wood. The soft would

snapped at 3.2Nm but the hard wood was safe (see diagram 6).

Highlighted in yellow on the table above are two data points pertaining to the soft wood

beam where the permanent vertical deformation was recorded as zero. This means that

when the load was removed from the end of the cantilever, it returned to its original,

unbent position. This is an example of elastic deformation. Every other data point in the

grouping column for permanent vertical deformation is an example of plastic

deformation. By comparison, the hard wood did not expe rience any elastic deformation

at all. Both of these observations are represented in the graphs of permanent vertical

deformation for hard and soft wood.

As indicated in both the graphs for hard and soft wood, a positive line of best fit indicates

positive relationship between the deflection and the torque. The almost linear positive

relationship that exists between torque and permanent vertical deformation is

compatible with this result. This linear relationship is further demonstrated by the graphs

of permanent vertical deformation that are available in the appendix.

A particularly meaningful result was singled out in diagram no.5. It was apparent that

identical values of torque could yield significantly differe nt values of deflection. This is

actually because torque is not actually directly proportional to deflection. Themathematical relationship between torque and deflection (refer to equation 1 from the

theory section) means that the different in the hypothetical points of data 2kg x 20cm

which has the same torque as a hypothetical 4kg at 10 result in very different deflections.

Deflection = (FL^3)/(3EI) which can be rewritten as: h=(TL^2)/(3EI) - L represents the

length of the beam extended off the table length . This component of the equation is

consistent with the empirical observation of the discrepancy in deflection for consistent


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values of torque. L increases at a greater rate (^2) compared to the load. Therefore,

deflection is proportional to the square of the length. Therefore, the length has a far more

significant positive bearing on the overall deflection than the weight component of

torque. Again this is consistent with the corresponding data values from the data table for

that value of torque. This has been conveniently highlighted in the collected data in red.

I represent the moment of inertia. This is the same across the experiment, for a constant

cross sectional area of the beam. Since both beams have the same cross sectional area

the moment of inertia is insignificant in explaining the above results.

The soft wood graph is steeper when imposed on the same axis as the hard wood graph.

This can verify by referring to the light blue highlights within the above table tables. The

highlighted data compares equivalent data points on each of the graph to illustrate the

conclusion that the soft wood is more plastic than the hard wood.

E is the modulus of elasticity, which is unique for different materials. The Pinewood has a

lower and the Tasmanian Oak has higher modules of elasticity. From the comparative

gradients (The deflection graph pine wood is steeper) of either wood, it may be noticed

that hard wood deflects less than soft wood for the same torque. This can also be

accounted for by the deflection equation. A lower modulu s of elasticity corresponds with

greater steepness in a graph of deflection. Since the modulus of elasticity is inversely

proportional to the deflection, the results produced are theoretically sound. (G. Gorenc

and R. Tinyou, 1984)

Percentage error/Absolute error

Absoluteerror of

Weight:

+/-0.0005kg (derived from the

smallest increment on the ruler)

Absolute

error of

Distance:

+/-0.0005m (derived from 1g

the smallest increment of the

ruler)

For simplicity, the calculation of the uncertainty of the torque was placed at the

'maximum' uncertainty, since each single point will have its own uncertainty due to

the fact that it is calculated from the uncertainty of weight (A) and the uncertainty of

distance (B) with the equation (DeltaA/A)+(DeltaB/B) which is dependent on the

individual value of each point of data. Instead, the following simplification utilised the

maximum possible error of the vertical axis for this experiment for every value oftorque for uniformity : +/- 0.0055Nm.

The uncertainty for the deflection and the permanent vertical deformation (the

vertical error bars was measured normally and is represented in the error bars on the

graphs)

Absolute error of deflection/permanent vertical deformation: +/-0.5mm


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2010 M lb Hi h S h l Ph i D t t

*note*parallax error has not been accounted for here since this error has

presumably been eliminated due to the use of a sheet of white paper placed

behind measuring instruments to aid the naked eye in judging measurements.

Conclusion

The structural properties of the wooden beams were mostly successfully examined. The

exceptions are the experiment into the breaking stress of the Tasmanian Oak wood,

which was not found during the practical component of the investigation. As a result, this

experiment was not a success. However, the elastic and plastic behaviours of two

different types of woods were identified, measured and recorded. The breaking torque

for soft wood was identified and the mechanism of permanent vertical deformation was

successful investigated. The deflection caused by the load and how these two

relationships vary due to the composition of the actual wood (hard and soft ) was also

successfully graphed and analysed. This was then compared to a mathematical

relationship that explained the results. Every experiments results were consistent withthe associated theory. A possible error consideration /possible reason for discrepancy

from theory could have been that the uncertainties of the calculated torque was

determined loosely and broadly by using the rules for calculating uncertainty and

absolute error (See the error and uncertainty in the results section for details) rather

than precisely for each data point . This experiment could be improved by using a

compass to measure the angle formed between the normal and the bent wood with every

change in mass to ascertain the precise perpendicular component of torque around the

point of clamping by inducting the associated data into a suitable trigonometric equation.

Acknowledgements

Mr.Haydn For passing on his theoretical knowledge and directing the practical

investigation.

Scott L and Mitchell C - for their contribution and support during the experimental

investigation.

References

Graeme Lofts et all (2010). Jacaranda physics. 2 : VCE physics units 3 & 4 . Milton, Victoria:

Jacaranda.

G.Gorenc and R.Tinyou (1984) Steel Designers Handbook. Kensington, New South Whales:

New South Wales University Press LTD.

Wayne Mortan. (2004). S.H.M of a Cantilever. Available:

http://www.practicalphysics.org/go/textonly/Experiment_430.html. Last accessed 28

May 2010.

last physics 16july submitted

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