## Computer Applications Assignment 1

##### Jacob Ward *ID: (25579363)*

*“I am aware of the requirements of good academic practice and the potential penalties for any breaches”*

### Hooke’s Law

Hooke’s Law is a scientific law which concerns itself with the elasticity of materials. It states that when a force is applied to a spring, the displacement of that spring will be directly proportional to the amount of force applied.

As an equation, it can be written as:*F = -kx*

where ** F** is the force applied to the spring (

**N**).

where ** x** is the displacement of the spring (

**m**).

where ** k** is the spring constant, the rate at which the spring is displaced (

**N/m**).

Hooke’s law applies, as long as the material is within it’s elastic limit.

Once a sufficient amount of force has been applied, so as to extend the material beyond it’s elastic limit, the material enters it’s plastic region. With the material in it’s plastic region, the force applied causes permanent displacement of the material.

### Introduction

An experiment is to be undertaken to determine the behaviour of three materials, in relation to Hooke’s law. Two of the materials within their elastic regions and one in it’s plastic region.

The results will then be analyzed, interpretting them to determine what is happening physically and the differences between the materials, and presented with a conclusion.

### Experimental Method

The steps to perform the experiment are as follows:- 1. Set up the apparatus as shown in Fig 1.
- 2. The material (spring) to be tested is fixed at one end of the clamp, hanging parallel with the ruler (meter stick).
- 3. A known force is applied to the material, causing it to be displaced.
- 4. It’s displaced length is measured and noted.
- 5. Steps 3 to 4 are repeated a number of times (in this case nine times), each time applying a greater force.

The experiment is repeated for all three materials.

### Results

The results of the experiment are shown below in Fig 3.

Where *y1* is material 1, *y2* is material 2 and *z* is material 3.

The results for *y2* and *z* were calculated in OpenOffice using the following equations and formulae (Fig 2.)

###### Fig 2. Equations and Formulae Used to Calculate *y2* and *z*

Value | Math Function | OpenOffice Formula |
---|---|---|

y2 |
f(x) = (a + 0.5) x + c |
=(1.5583333333+0.5)*(A2:A10)+0.2 |

z |
f(x) = x^{3} + b |
=(A2:A10^3)+1.375 |

where *c = 0.2* (as given on data sheet).

where *a = 1.5583333333* (as calculated using OpenOffice).

where *b = 1.375* (as calculated using OpenOffice).

###### Fig 3. Table of Results

x (Force applied in Newtons N) |
y1 (Deformation in mm) |
y2 (Deformation in mm) |
z (Deformation in mm) |
---|---|---|---|

1.00 | 3.00 | 2.26 | 2.38 |

2.00 | 4.50 | 4.32 | 9.38 |

3.00 | 6.00 | 6.37 | 28.38 |

4.00 | 7.50 | 8.43 | 65.38 |

5.00 | 9.00 | 10.49 | 126.38 |

6.00 | 10.50 | 12.55 | 217.38 |

7.00 | 13.00 | 14.61 | 344.38 |

8.00 | 14.00 | 16.67 | 513.38 |

9.00 | 15.00 | 18.72 | 730.38 |

The results of the experiment plotted on graphs with trend lines are shown in Fig 4. and Fig 5.

###### Fig 4. Graph Showing *y1* and *y2* Plotted Against *x*

###### Fig 5. Graph Showing *z* Plotted Against *x*

Looking at Fig 4., the meeting point of *y1* and *y2* is estimated as *2.30, 5.00*

The actual meeting point is calculated by resolving the simultaneous equations:

Manually solving simultaneous equations to get

*x, y*values

*y1 = 1.5583x + 1.375
y2 = 2.0583x + 0.2*

2.0583x + 0.2 = 1.5583x + 1.375

2.0583x – 1.5583x = 1.375 – 0.2

0.5x = 1.175

∴ **x = 2.35**

y = (a + 0.5)x + c

y = (1.5583 + 0.5) * 2.35 + 0.2

*∴ y = 5.037
*

**Actual meeting point:** *2.35, 5.037*

### Conclusion

The results obtained from the experiment confirm that Hooke’s Law is true.

Materials *1* and *2*, whose displacement is shown in the results as *y1* and *y2*, respectively, are in their elastic region. As such, a linear relationship between the force applied (*x*) and their displacement (*y*) is shown in the graph Fig 4.

This indicates that as force is applied, the material is displaced directly proportionally.

Material *3*, shown in the results as *x*, on graph Fig 5., is indicated to be in it’s plastic region. The exponential trend line shows that as force is applied, the material is permanently displaced.

The steeper trend line in the graph for material *2*, shows that less force is required to result in a greater displacement of the material than for material *1*, indicating that it is more elastic.

A comparison of the graphs shows that for material *3* a far greater amount of force is required than for materials *1* or *2* to achieve any displacement.

Though the data appears to represent the physical results of the experiment, there are a number of possible errors:

- The ruler used to make the measurements may not have been accurate.
- User error when reading the measurements from the ruler, such as a parallax error by reading the measurement from an angle or choosing the closest marker to represent the measurement.
- Inaccurate forces being exerted over the material, such as the mass not being exactly as stated.
- The overall calibration of the apparatus being used to conduct the experiment may have been inaccurate.
- When resolving the simultaneous equations, the values
*1.5583333333*and*2.0583333333*were rounded to four decimal places this would have produced a slight error in the final results of the calculation.

If the experiment was to be conducted again, these possible sources of error would have to be minimised as far as possible, either through the use of more precise and accurate apparatus or more vigilant monitoring of the data being collected and used.

#### References

Bird, J., Ross, C., 2012. *Mechanical Engineering Principals*, 2nd edition. London and New York: Routledge.

Burton, P., “Forces & Elasticity” (2010), *Physics Net*, [Online].

Available at: http://physicsnet.co.uk/gcse-physics/forces-elasticity-hookes-law-spring-constant-elastic-potential-energy/ [Accessed 13 November 2012].

Croft, A., Davison, R., 2008. *Mathematics for Engineers*, 3rd edition. London: Pearson.

OpenStax College. “Hooke’s Law: Stress and Strain Revisited” (2012), Connexions, [Online].

Available at: http://cnx.org/content/m42240/1.5/ [Accessed 13 November 2012].

License: Creative Commons Attribution – 3.0 Unported (CC BY 3.0): http://creativecommons.org/licenses/by/3.0/

Stanbrough, JL., “How Does A Spring Scale Work?” (2002), *Mr Stanbrough’s Classes*, [Online].

Available at: http://www.batesville.k12.in.us/physics/phynet/mechanics/newton3/Labs/SpringScale.html [Accessed 13 November 2012].

It was such a brilliant experiment conducted, straight forward results and very easy to understand.