Simple Harmonic Motion

Aston University

Engineering and Applied Science – Physics Lab Report

01/11/2014

Determination by simple harmonic motion of the acceleration due to gravity

Introduction:
A system undertaking simple harmonic motion (SHM) can be restrained very accurately. The period of the SHM depends both on the mass of the system and the strength of the force tending to restore the system to its equilibrium state). Oscillations are a common part of life, for instance the vibrations of a musical instrument which helps make sounds or the foundations of a car suspension which are assisted by oscillations. The main aims of this experiment was to determine if the oscillation of a mass which hung vertically from a spring; this oscillating system was used to measure the acceleration of earth due to gravity and to determine the accuracy of experimental results precisely. (http://www.pgccphy.net/1020/phy1020.pdf

Theory: | | |
Acceleration due to gravity

The value of 9.8m/s/s acceleration is given to a free falling object, directing downwards towards Earth. Any object moving solely under the influence of gravity is known as acceleration of gravity and this vital quantity is denoted by Physicians as the symbol g. (http://www.physicsclassroom.com/class/1DKin/Lesson-5/Acceleration-of-Gravity)

Simple harmonic motion
This is everyplace where the acceleration is proportional and opposite to displacement to the continuous amplitude from the position of equilibrium.

Period
The time that it takes to complete one full cycle of oscillation which is measured quantitatively

Hooke’s law
‘Hooke’s law, law of elasticity discovered by the English scientist Robert Hooke in 1660, which states that, for relatively small deformations of an object, the displacement or size of the deformation is directly proportional to the deforming force or load. Under these conditions the object returns to its original shape and size upon removal of the load. Elastic behaviour of solids according to Hooke’s law can be explained by the fact that small displacements of their constituent molecules, atoms, or ions from normal positions is also proportional to the force that causes the displacement.’ (http://www.britannica.com/EBchecked/topic/271336/Hookes-law)

Springs constant
Newton’s third law states that for every force, there is an equal and opposite force. This refers to the spring which change shape when there is any kind of exertion of pressure, either stretching or compressing divided by the distance the spring stretches or compresses. The unit of spring constant can be worked out by using Hooke’s law. (http://www.education.com/science-fair/article/springs-pulling-harder/)

Equipment Required * One spring

* One mass hanger (approx. 10g)

* Nine 10g masses

* An electric timer

* A clamp stand

* One measuring ruler

Procedure

The clamp was set up on the table safely as this was needed for part 1 and part 2 of the lab, we then attached the spring to the clamp. The timer was set to 0 for part 2 of the lab.
We proceeded with the lab

PART 1 - Measurement of spring length
Once the experiment was set and safe. Part 1 of the lab consisted of a range of masses from 10g to 100g which was attached to the spring and the amount of stretch of the spring was measured using a measuring stick each time. It consisted if measuring the original spring length with no mass and we also did three trials for each mass added.
.Part 2 - Measuring the period of oscillation
Part 2 of the experiment consisted of the same 0g-100g of masses which was also attached 10g at time to the spring, and then, with a stopwatch we timed the time it took for the spring to make 10 complete periods of oscillations. The timer always started on 0 seconds. The experiment was also repeated thrice to ensure accuracy and reliability.

Theoritical Analysis:

Figure 1. section of relationship with mass and the pull of the string which leads to displacement of the spring form equilibrium level

Figure 1. shows the immediate position of mass hung vertically from a spring. The position where force of gravity is exactly balanced and an equilibrium position is reached is represented by X. The stretch on the spring was measured using a measuring ruler.
i.e. Fs = k. L

where Fs is the restoring force from the spring k is the spring constant L is the distance the spring has stretched
Fg = m . g

Where m is the mass g is the acceleration due to gravity

so, k . L = m . g or L = (g/k) . m

equation L = (g/k) . m tells us a straight line should pass through the origin with gradient g/k on a graph of a spring stretch versus mass.

Part 2 we measured the spring oscillations with an electric stopwatch. We started from the zero mass which hung vertically from a spring, then added a known mass starting from 10g, increasing the value of masses each time by 10g, until a final mass of 100g (1kg) was reached and counting the number of oscillations each time through observation, which depended on the spring constant and the mass.

i.e T = 2 . π / w

Where T is the time taken for one complete oscillation

So, T = 2 , π √ w

T = 2 . π . √ (m/k) or T² = (4 . π² / k) . m

T² = (4 . π² / k) . m tells us that a graph of period squared versus mass should be a straight line passing through the origin with the gradient of (4 . π² / k)

Fs; the restoring force is the cause that makes the mass oscillate. The mass will remain to travel up and down in a number of oscillations due to the conservation of momentum.

Table 1 shows the results for part one of the experiment, where different lengths of mass was added to the spring and the distance the spring stretched was measured with the measuring stick.

Mass(kg) | Spring length (m) | Spring stretch (m) | | | Trial 1 | Trial 2 | Trial 3 | average | | | 0 | 0.36 | 0.36 | 0.360 | 0.360 | 0 | | 0.01 | 0.36 | 0.36 | 0.360 | 0.360 | 0 | | 0.02 | 0.357 | 0.357 | 0.356 | 0.357 | 0.003 | | 0.03 | 0.354 | 0.355 | 0.354 | 0.354 | 0.006 | | 0.04 | 0.348 | 0.348 | 0.349 | 0.348 | 0.O12 | | 0.05 | 0.341 | 0.342 | 0.342 | 0.342 | 0.018 | | 0.06 | 0.337 | 0.338 | 0.337 | 0.337 | 0.023 | | 0.07 | 0.332 | 0.332 | 0.333 | 0.332 | 0.028 | | 0.08 | 0.330 | 0.330 | 0.330 | 0.330 | 0.030 | | 0.09 | 0.329 | 0.328 | 0.328 | 0.328 | 0.032 | 0.1 | 0.326 | 0.326 | 0.326 | 0.326 | 0.034 |

The relationship between spring stretch and mass

Graph 1 is plotted using Table 1

Table 1 shows the spring stretch for each individual mass when it is hung to the spring. In this experiment we have observed that as the mass increases the length of the spring increases. The original length of the spring is shown is 0.36 at 0kg. When the mass of 10g is added the spring the length remains constant. We can see an increase in spring length from 0.02 kg till 0.1 kg. The total average of all the spring lengths after their stretch is 0.343. The graph shows a positive correlation.
Table 2 shows the period of oscillation against mass. From the data we have collected the results shows that as mass increases, the period of oscillation increases. The lowest stretch was from 0 to 0.02kg of mass, which was a stretch of 0.003m. In table 2 the oscillation period was at its maximum rate when the weight of 0.02kg was added and the oscillation period was at its lowest when a weight of 0.1kg was added, oscillation at 0.454 seconds (Table 2). The average oscillation period of the total of the oscillations are 0.307 seconds. The graph again shows a positive correlation.
Table 1 and 2 include three trials to ensure accuracy and reliability so that the data can be reproduced, for example referring back to table 1 when we added on a mass of 0.05kg to the spring, at first, we measured the stretch at 0.0341 and then the second and third trials made it obvious that the actual length the spring stretched was 0.0342, meaning the first trial was likely to be an error. The average values vary between 0.362 and 0.360 in Table 1 and the average period of oscillation varies between 0.277 seconds and 0.454 seconds in table 2. The highest value that the mass was stretched was when the mass of 0.1kg when the spring was stretched from 0 to 0.034m (Table 1) The lowest stretch was from 0 to 0.02kg of mass, which was a stretch of 0.003m.
The graph was plotted to make it easier to observe and analyse results and even with the anomalies it showed the general pattern that the results indicated. The quantity the spring stretches intrigued against the weight put on to the spring in Table 1 gives relatively a straight line that goes through the origin, meaning that the extension of the spring directly depends on the mass force applied to it

Table 2. shows the results of the relationship between mass an the period of oscillation | Measurement Time (s) | | Mass (kg) | Trial 1 | Trial 2 | Trial 3 | average | Period (s) | Preriod2 (S2) | 0.00 | 0 | 0 | 0 | 0 | 0 | 0 | 0.01 | 0 | 0 | 0 | 0 | 0 | 0 | 0.02 | 2.73 | 2.79 | 2.76 | 2.77 | 0.277 | 0.077 | 0.03 | 3.10 | 3.16 | 3.24 | 3.17 | 0.317 | 0.100 | 0.04 | 3.45 | 3.40 | 3.30 | 3.38 | 0.338 | 0.114 | 0.05 | 3.50 | 3.42 | 3.39 | 3.44 | 0.344 | 0.118 | 0.06 | 3.90 | 3.88 | 3.78 | 3.85 | 0.385 | 0.148 | 0.07 | 4.05 | 3.91 | 3.88 | 3.95 | 0.395 | 0.118 | 0.08 | 4.30 | 4.10 | 4.20 | 4.20 | 0.420 | 0.148 | 0.09 | 4.43 | 4.56 | 4.47 | 4.49 | 0.449 | 0.202 | 0.1 | 4.52 | 4.48 | 4.62 | 4.54 | 0.454 | 0.206 |

Gradient = 4.π2 K
K = 4.π2 Gradient
K = 4.π2 2.2767
K = 17.34

The relationship between mass and oscillation period Graph 2. Is the plotted data from Table 2 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Calculating the value of gravity

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Table 1 shows the spring stretch for each individual mass when it is hung to the spring. In this experiment we have observed that as the mass increases the length of the spring increases. The original length of the spring is shown is 0.36 at 0kg. When the mass of 10g is added the spring the length remains constant. We can see an increase in spring length from 0.02 kg till 0.1 kg. The total average of all the spring lengths after their stretch is 0.343. The graph shows a positive correlation.
Table 2 shows the period of oscillation against mass. From the data we have collected the results shows that as mass increases, the period of oscillation increases. The lowest stretch was from 0 to 0.02kg of mass, which was a stretch of 0.003m. In table 2 the oscillation period was at its maximum rate when the weight of 0.02kg was added and the oscillation period was at its lowest when a weight of 0.1kg was added, oscillation at 0.454 seconds (Table 2). The average oscillation period of the total of the oscillations are 0.307 seconds. The graph again shows a positive correlation.
Table 1 and 2 include three trials to ensure accuracy and reliability so that the data can be reproduced, for example referring back to table 1 when we added on a mass of 0.05kg to the spring, at first, we measured the stretch at 0.0341 and then the second and third trials made it obvious that the actual length the spring stretched was 0.0342, meaning the first trial was likely to be an error. The average values vary between 0.362 and 0.360 in Table 1 and the average period of oscillation varies between 0.277 seconds and 0.454 seconds in table 2. The highest value that the mass was stretched was when the mass of 0.1kg when the spring was stretched from 0 to 0.034m (Table 1) The lowest stretch was from 0 to 0.02kg of mass, which was a stretch of 0.003m.
The graph was plotted to make it easier to observe and analyse results and even with the anomalies it showed the general pattern that the results indicated. The quantity the spring stretches intrigued against the weight put on to the spring in Table 1 gives relatively a straight line that goes through the origin, meaning that the extension of the spring directly depends on the mass force applied to it

Analysis Under Hooke’s law the spring has returned to its original shape and proves that the gravitational force exerted on the mass is the same as the force exerted by the spring, the displacement of the size of the spring was directly proportional to the force in this case as the stretch increased with the increase of masses. A true relationship between the period, mass, and the spring constant was shown by our results which match the theory that the period of oscillation and the amount a spring stretches is dependent on the mass. The dots have been connected correctly as I have a positive correlation which is dependent to the value of acceleration. These results can be put into application in real if engineers who are trying to build a trampoline try to apply the theory into building a safe trampoline so when a person with a normal weight pushes up and down on it they are able to oscillate pulling from a position of equilibrium, allowing the springs to acquire potential which assists the upward bounce without anything going wrong as all anomalous results can be identified through such lab tests and then avoided in real life. The little differences between experimental and theoretical theory and anomalous results maybe due to certain things that have not been taken into account, such as stopping the stop watch couple of seconds later, which is a human error or because the spring length was not quantified beforehand and the spring could have possibly be compressed which had an effect on the size, and also the surface the experiment was carried out on could have influenced measurements. Taking different settings into account would give precision with our results as not all SHM systems in real life are applied to the settings that were used in the Lab, it is important to be able to generalise the information gained. Low cost materials were used throughout the experiment, hence why it was easier to analyse the connections between the variables that shape the SMH of a spring mass system.

Percentage error
Gradient = (4.π2/g)
Gradient = 6.2866 % Error = Original value – Result g value X 100 Original value
= 9.8 - 6.28 X 100 9.8
= 35.92%

Precaution

* Ensure equipment works properly e.g. stopwatch.

* Avoid human error by being very careful repeating experiments many times and working out averages to deducting anomalous results and ensuring accuracy

* The spring should return to the original shape after the mass is taken off, hence why the limit of mass attached should not exceed the limit.

* Using the identical spring throughout with the same mass, length, coils etc. to make the experiment fair. CONCLUSION
The hypothesis which states that ‘the length of spring is directly proportional to the applied force’ is correct as it caused a larger change in length of spring as a greater force was applied as the masses increased. The oscillations are characterised by the principle variables. The safeguarding of the identical spring throughout the experiment meant that the spring constant was always maintained for any value of force and spring extension. To improve this experiment similar studies in different setting could be carried out.
The lab has aided me to understand that the elasticity force means that if the spring is stretched within its limit, it will always return to its original shape. This can be applied in real life which has been reviewed earlier in lab; the trampoline apparatus which is sufficient for any person with the weight that is the maximum limit of the SMH system of a trampoline will be able to jump on without the trampoline losing shape. Other real life examples are the clock pendulum and a swing. The restoring forces in the swing are sufficient enough to make the swing move forward and travel upwards when it is being pushed; the opposite direction gives out a restoring force which pushes the swing back down. Again, this force is not great enough to overpower equilibrium, and travel in the other side. The swing will swing in a constant period and amplitude as long as the process is repeated, oscillations will carry on.
As discussed earlier experimental errors and my anomalies could be due to not measuring correctly or pulling the spring at different strengths each time, this errors need to be corrected in real life application.

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D.G. Simpson, Ph.D (2013); Waves, Acoustics, Electromagnetism, Optics, and Modern Physics; Introductory Physics; 11; (2); P.8

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The Editors of Encyclopædia Britannica (2014) – Britain Enclopaedia; Accessed 28 Nov 2014

Erin Bjornsson (2010); Hooke's Law; Calculating Spring Constants; 6; (4); P.4

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