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Engineering

Analysis,
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Beams are an essential part of engineering. A beam which is rigidly connected at one end to a fixed support and free to move at the other end is called a cantilever beam. Various infrastructures such as traffic lights, roofs, diving-boards, and bridges are constructed by employing the fundamental knowledge of cantilevers. As a child, I played Jenga- a game which employs the concept of cantilevers to evoke children’s interest. As the game progressed, more and more cantilevers were made, making the game full of suspense.

This curiosity rose further after realising the importance of cantilevers in mechanical engineering because I often watch videos of bridges, balconies and roofs collapsing just like the wooden blocks in Jenga. Hence, I decided to conduct an experiment to analyse the behaviour of a cantilever. I wanted to test a variable which is less commonly experimented. I chose the distance at which the force is exerted on the cantilever as my independent variable.

The aim of this experiment is to analyse the behaviour of a cantilever and then answer the following research question: How does the vertical depression of a cantilever respond to a change in the distance at which an external force is applied to the cantilever? Consider a cantilever, fixed at one end, M, and loaded at a distance(cm), d.

The end, N, is depressed to N’ and s or [N’-N] represents the vertical depression at the free end. Note that N may not be at the same height(cm) level as M.

There can be an initial vertical depression before any external force is exerted on the cantilever. The force exerted downwards by load- mg (Newtons), is equally and oppositely opposed by reaction force, R, acting upwards at the supported end M. If we exert the force at a distance farther from the suspension point M than d, the vertical depression(cm) is theoretically going to increase. However, as the force is exerted for longer periods of time and as the distance increases more and more, the upper layer of the filaments at the point of suspension are more likely to get elongated while the lower layer of filaments gets compressed. This may lead to a permanent deformation of the cantilever.

The relationship between distance d and vertical depression s is determined below. It is not a linear relationship, so I have used logarithms to linearize it: s ∝ dn s = kdn

(Equation 1) log(s) = log(kdn) log(s) = log(dn) + log(k) log(s) = nlog(d) + log(k)

(Equation 2)6 d = distance in centimetres s = vertical depression in centimetres n = constant k = constant I have decided to measure the vertical depression(cm) for the distances(cm): 10,20,30,40,50, 60,70,80 and construct a relationship between s and d, by calculating n and k for the cantilever in the experiment. I have chosen a cantilever with a low thickness (0.10cm) so that the vertical depressions in the experiment are large and observable.

This research is significant because the relationships between the distance(cm) and vertical depression(cm) of cantilevers can assist engineers to identify optimal lengths of materials for their industrial projects. Hence, it is a crucial part of mechanical engineering. Hypothesis There is a logarithmic linear relationship between the distance(cm) at which the force is exerted and the vertical depression(cm) on the cantilever, given that the force exerted on the cantilever remains constant. This is in the form: log(s) = nlog(d) + log(k), or, s=kdn, where s represents vertical depression (cm), d represents distance (cm), and k and n are constants.

Variables Independent Variable: Distance of slotted-mass from cantilever’s point of suspension (cm). The calibration on the beam is used to measure this variable. The slotted-mass is tied to a string of negligible mass and hung. Masking tape ensures that the string does not slide. Distance (cm) intervals are: 10cm, 20cm, 30cm, 40cm, 50cm, 60cm, 70cm, 80cm.

Dependent Variable: Vertical depression of the cantilever beam (cm). This is measured by placing a wooden ruler next to the end of the cantilever beam. Readings are taken at eye-level using a set-square once the metal beam stops vibrating.

Controlled Variables: Variable to be controlled Why and how the variable is to be controlled Mass of slotted-mass (g) The force exerted on the cantilever is directly proportional to the mass of the slotted-mass because Force=(mass)*(acceleration due to gravity), where force is in Newtons and acceleration due to gravity is in ms-2. Reduction in the mass will reduce the force, reducing the vertical depression and vice versa. Hence, this variable will be kept constant by using the same slotted-mass of 99.9 grams (or 0.0999 kilograms) throughout the experiment.

Any changes in the length, width, thickness and material will change the physical properties of the cantilever beam. This will affect the rate of change of vertical depression (∆s) during the experiment. Therefore, the easiest way to keep these properties controlled is by using the same beam throughout the experiment. Length of metal beam suspended from the table If there is an increase in the length of beam suspended, the suspended mass of the beam will increase, increasing the initial force (mg) on the beam, causing a higher vertical depression. This change in vertical depression can be mistaken as a change due to change in distance, and this reduces the accuracy and precision of the results. Therefore, a G-clamp will be firmly attached, so that the length of beam suspended remains constant throughout the experiment

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