ENGZ0002

Spring-Stiffness Experiment

Nabil Ibrahim
Anna Strzelecka
08/04/2018
Table of Contents
Introduction...........................................................................................................................................2
Objectives..............................................................................................................................................2
Apparatus..............................................................................................................................................3
Theory...................................................................................................................................................4
Procedure..............................................................................................................................................4
Results...................................................................................................................................................4
Graph.....................................................................................................................................................5
Least squared method...........................................................................................................................6
Discussion..............................................................................................................................................7
Direct stress and Direct strain...............................................................................................................8
Strain Gauges.........................................................................................................................................8
Conclusion.............................................................................................................................................9
References...........................................................................................................................................10

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Introduction.
Hooke’s law states that the strain of a solid object is directly
proportional to the force applied, up to the elastic limit [1].
Any object that quickly regains its shape after it’s been deformed by
force follows Hooke’s law. Due to this, Hooke’s law has a lot of real
life applications.
The lungs we use to breathe obey Hooke’s law, when we breathe,
the air we suck into our lungs forces them to get bigger and out of
shape, but once we breathe out again, they immediately return to
their original shape [1].
Other real life applications of Hooke’s law includes; springs in beds
and mattresses, diving boards at swimming pools, the suspension
used in cars, rubber bands, most polymers, pendulum clocks, hand
sheers, wind-up toys, watches and even the human skin, when
stretched returns to its normal shape [1].
Finally, Hooke’s law is an example of the first law of
thermodynamics, which states that any spring when compressed or
extended almost perfectly conserves the energy applied to it [2].

Objectives.
 To Investigate and understand Hooke’s law.
 Use the equation F=k x to measure the stiffness of a spring.
 Understand the definitions of stress, strain and young’s
modulus.

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Apparatus.
1. A helical spring
2. A weight hanger and masses
3. A meter ruler

Figure 1. Photo of apparatus

3
 k

Theory. N
x/
Force= K x Deflection m
m

Figure 2. Theory of
spring stiffness.

Procedure.
 Put up the spring on a hook and attach the weight hanger to
the bottom end of the spring.
 Increase the masses in intervals of 0.5kg till a mass of 3kg,
measure the distance the spring has moved from the reference
level and therefore the spring deflection.
 Plot a graph with the weight on the spring on the y-axis and the
deflection on the x-axis. Measure the gradient of the graph and
therefore determine the spring stiffness in N/m.

Results.
Mass/ kg Distance from Deflection/ mm
reference level/ mm
0 710 0
0.5 687 23
1 661 49
1.5 636 74
2 610 10
2.5 585 125
3 565 145
Table 1. Results of mass(kg) against Deflection

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 Load / N Deflection / m
0 0
0.5 x 9.81= 4.905 0.023
1 x 9.81= 9.81 0.049
1.5 x 9.81= 14.72 0.074
2 x 9.81= 19.62 0.10
2.5 x 9.81= 24.53 0.125
3 x 9.81= 29.43 0.145
Table 2. Table of Load (N) against Deflection (m)

Graph.
Spring Stiffness
35

30

25

20
Load N

15

10

0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Deflection (m)

20−0
Gradient= =200 N /m
0.1−0

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6
Least squared method.

Discussion.
Experiment:
 Hooke’s law states that the strain of a solid object is directly
proportional to the force applied.
Force = K x deflection
 Hooke’s law is obeyed by the spring used in the experiment,
this is because the graph obtained was linear.
 The spring stiffness is expressed by the gradient.
 The possible experimental errors of this experiment could be as
follows:
1. The ruler could have been moving during the experiment,
resulting in inaccuracies, this could be prevented by holding
the ruler using an adhesive.
2. The ruler might not have been read at eye level, this could
be prevented by having one person hold the ruler in place
and the other reading the result at eye level.
3. The spring could have been stretched past its elastic region,
this could be prevented by replacing the spring or making
sure the weights attached to it are not so heavy they
permanently deform it.
4. The tolerances in the weights could also affect results, this
won’t be a problem if the tolerances are taken into account.
5. Finally, if the distance was measured whilst the spring was
still oscillating, that could also cause inaccuracies, and this
could be prevent by waiting a few seconds for the spring to
stop oscillating.

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Direct stress and Direct strain.
Definitions.
 The limit of proportionality is the point in which Hooke’s law is
no longer correct when stretching the material.
 The elastic region is a point during the stretching of a material
when the material goes back to its original shape when the
force is removed [3].
 The plastic region is when the material is stretched past the
elastic region and becomes fully deformed, it doesn’t return to
its original shape [3].
 Young’s modulus = stress/strain. This is the equation of young’s
modulus, it’s a numerical value which tells us how easily a
material can stretch and deform [4].
 Ultimate tensile stress is the maximum stress a material can
take while being stretched or pulled [3].

Strain Gauges.
 A strain gauge is a sensor whose resistance varies with applied
force. It’s simply a length of wire formed into the shape of a
continuous grid, which is attached to a non-conductive surface
[5]. It converts force, pressure, tension, weight and so on, into
a change in electrical resistance. They are used for the
measurement of mechanical quantities and in this case used to
measure strain. The strain is positive if the material is
stretched, and negative if it’s compressed [6].
 Strain gauges could be used in smart bridge technology to
detect structural problems early. They are installed on
structural components in a bridge or building to measure stress
and compare them to analytical models and stress calculations
[7].

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 Strain gauges are bonded directly to structural load bearing
components on an aircraft to measure and detect stress for
wing deflection. Strain gauges are used to numerous locations
throughout the aircraft structure. The load equations are
derived after ground calibration testing based on strain gauge
measurements [7].

Another application of strain gauge technology is on railroads,

it measures stress and strain on rails through strain gauges
attached to each rail. They measure axial tension or
compression without affecting the rails. When a decline in the
structure occurs, the strain gauges can generate a warning so
maintenance can be done early to prevent accidents and rail
traffic [7].

Conclusion.
The aim of the experiment was to investigate Hooke’s law and prove
the theory that F=kx using a helical spring, weights and a ruler. The
experiment was set up as shown in figure 1. The mass of the weights
which were put on the spring was increased in intervals of 0.5kg, the
distance the spring stretched in relation to the reference level was
measured and from that the deflection of the spring was measured.
The load of the weights were calculated by multiplying the mass of
the weights by gravity (9.81), the deflection was then converted into
meters.
A graph of load (N) and deflection (m) was plotted, with load on the
Y-axis and deflection on the x. The gradient of the graph, which stood
for spring stiffness, was measured to be 200Nm. Hooke’s law states
that the strain of an object is directly proportional to the force
applied, up to the elastic limit, the graph obtained was linear which
meant the spring did obey Hooke’s law.

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There were a few possible experimental errors which includes; the
movement of the ruler, readings not done at eye level,
measurements taken while spring is still oscillating. Etc.

References.

[1]2018. [Online]. Available:

http://www.answers.com/Q/Hooke's_law_application_in_daily_life#slide=2. [Accessed:
13- Mar- 2018].

[2]"What is Hooke's Law?", Phys.org, 2018. [Online]. Available:

https://phys.org/news/2015-02-law.html. [Accessed: 13- Mar- 2018].

[3]C. Engineering, C. Construction, P. Concrete-II, C. Estimation, E. Materials, S.

Engineering, M. 1, M. 2, S. 1, S. 2, S. Levelling, S. 2, E. Engineering, E. Engineering, E.
psychology, E. planning, E. Design, W. Engineering, F. 1, F. 2, H. Engineering, I.
Engineering, E. Hydrology, P. Foundation, E. Geology, T. structures, D. Structures, S.
Structures, R. methodology, T. Engineering, C. Us and M. 1, "Elastic Region | Plastic
Region | Yield Point | Plastic Strain | Strain Hardening | Necking", Civil Engineering
Terms, 2018. [Online]. Available: http://www.civilengineeringterms.com/mechanics-of-
solids-1/elastic-region-plastic-region-yield-point-plastic-strain-strain-hardening-necking/.
[Accessed: 13- Mar- 2018].

[4]"Young Modulus", Physicsnet.co.uk, 2018. [Online]. Available:

http://physicsnet.co.uk/a-level-physics-as-a2/materials/young-modulus/. [Accessed: 13-
Mar- 2018].

[5]E. Hearn, Mechanics of materials. Oxford: Pergamon Press, 1982. Page 171, paragraph
five, line 1-2.

[6]O. Engineering, "What is a strain gauge? | Omega Engineering", Omega.co.uk, 2018.

[Online]. Available: https://www.omega.co.uk/prodinfo/straingauges.html. [Accessed:
13- Mar- 2018].

[7]S. Systems, "Strain Gauge Technology in Field Testing", Sensing-systems.com, 2018.

[Online]. Available: http://www.sensing-systems.com/blog/strain-gauge-technology-in-
field-testing. [Accessed: 13- Mar- 2018].

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