##### Jacob Ward (jjw3g12 ID:25579363) *“I am aware of the requirements of good academic practice and the potential penalties for any breaches”*

### Introduction

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

Hooke’s Law equation is written as:

F = -kx

whereis the force applied inFNewtons(N).

whereis the rate at which the material is displaced, orkspring constant, inNewtons per metre(N/m).

whereis the displacement of the material inxmetres(m).

Hooke’s Law applies, so long as the material is within it’s **elastic limit**. When an amount of force has been applied, so as to extend the material beyond it’s elastic limit, the material is in it’s **plastic range**, where applying further force causes permanent displacement of the material.

In this experiment three materials will be used to determine their behaviour according to Hooke’s Law, two within their elastic limit and one in it’s plastic range.

### Apparatus and Method

The Apparatus required to carry out the experiment are:

- Clamps
- Frame / Ring Stand
- Material to be tested (Spring)
- Known Mass
- Metre Rule

As shown in Fig 1.

In order to carry out the experiment, the apparatus is first set up as shown in Fig 1, followed by these steps for each material being tested:

- The top of the material to be tested is attached to the clamp at the top and is hanging parallel to the metre rule.
- A known mass is attached to the bottom of the material, causing the material to be displaced.
- The material’s displacement is measured using the metre rule and noted down.

### Data and Analysis

Shown below in Fig 2 are the results collected after performing the experiment and analysing the data. Where *y1* is material 1, *y2* is material 2 and *z* is material 3.

###### Fig 2. Table showing results and analysis of data

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 |

As shown in Fig 3, OpenOffice was used to analyse and calculate the data.

###### Fig 3. Functions 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).

Shown in Fig 4 and Fig 5 are graphs of the results plotted with trend lines.

###### Fig 4. Graph with trend lines plotting materials *y1* and *y2* against *x*

###### Fig 5. Graph with a trend line plotting *z* against *x*

From looking at Fig 4 it is possible to estimate the meeting point of *y1* and *y2* as *2.30, 5.00*

By resolving the simultaneous equations, the precise meeting point of *y1* and *y2* is calculated as:

*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*

### Conclusions

From the results obtained by this experiment it is possible to confirm that Hooke’s Law holds true.

A linear relationship between the force applied, *x*, and the displacement, *y*, is shown for materials *1* and *2*, shown in the results as *y1* and *y2*, respectively, showing that they are both within their elastic limit.

Material *2*, having a steeper trend line than material *1*, shows that less force is required to displace the material and that it can be considered to be more elastic.

The exponential trend line for material *3*, represented by *z*, shows that as force is applied the material is permanently displaced and that it is within it’s plastic range.

Despite the results of this experiment showing what Hooke’s Law predicts, there are a number of possible areas of error which could lead to inaccurate results, such as:

- The accuracy of the metre rule being used to take measurements.
- Parallax error from the person performing the experiment reading the measurement from an angle.
- The mass being used to exert force on the material not being accurately measured.
- Rounding numbers down when resolving the simultaneous equations would lead to slightly inaccurate results.

If conducted again, it would be advantageous to minimise these potential sources of error in the experiment to achieve more accurate results.

### 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].

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].

Ward, J., “Hooke’s Law Experiment” (2012), *Jacob Ward*, [Online].

Available at: http://www.jacobward.co.uk/computer-applications-assignment-1/ [Accessed 9 November 2013].

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