Static equilibrium and the requirements of this principle - Lab Report Example

Static equilibrium (C2) Objectives The objectives for carrying out the static equilibrium experiment are To examine the static equilibrium of a system of forces of different magnitudes acting at different angles. 2. To compare the experimental results with those obtained by computing the resultant force. 3. To compare the experimental result and the calculated result with those obtained by graphical method. Abstract The main objectives of carrying out static equilibrium were: Firstly, to investigate the static equilibrium of a system of three forces in both horizontal and vertical planes. Secondly, to calculate the resultant force and compare its results with experimental results. And thirdly, to compare experimental results and calculated results with those obtained by graphical method. The first experiment was conducted on a horizontal plane and weights were set at different angles. Another weight set a different angle was used to balance these weights. A similar procedure was followed to carry out the second experiment but on a vertical plane. The final results got from the investigation and calculation of weights and angles were; a force of 2 N at an angle of 240° and a force of – 1.802 N at an angle of 238.923° in the first experiment. In the second experiment, a force of 3.5 N at an angle of 270° and a force of -3.348 N at an angle of 267.414° were obtained. Introduction The principle of moments states that when a force (F) is applied to an object that can turn around a pivot when acted on by forces, the turning effect of the body is equivalent to the moment (M) of the force. The moment equals the force multiplied by the perpendicular distance (d) from the pivot(Bird & ROSS, 2012). This means that the distance increases as the force decreases. The principle of moments has various practical uses in real life situations such as using a hammer to unscrew a nail, balancing objects around their pivots and among others. In regard to principle of moments, “the sum of the clockwise moments about any point must be equal the sum of the anticlockwise moments about that point” for a body to be in equilibrium. This illustrates that the body will attain static equilibrium, as long as the product of the force and the perpendicular distance on either side of the pivot is the same (Breithaupt, 2010). The illustration below strives to justify the concept of the principle of moment: M = F × d …….equation 1 Where F is the force of the load and is measured in Newton (N), d is the distance from the pivot and is measured in meter (m), and M is the moment given by the product of force and distance. It is measured in Nm. Illustration 1: The above diagram shows a balanced system with F1 and F2 pivoted at a point. The system balances because its clockwise moment and anticlockwise moment are equal. A body is said to be in static equilibrium when it is at a state of rest, that is, no motion. There are two conditions of equilibrium, when the vector sum of horizontal forces acting on a body is zero and also, when all vertical forces add up to give a zero sum (Eastern, 2014). That is, F = F1 + F2 + …+ Fn = 0, where n is the number of forces available. This means that the Fx = 0 (on x- axis) and Fy = 0 (on y-axis). An object is acted on by two or more forces which are in different directions to give a resultant force. This resultant force is the sum of the forces acting on an object and has the same effect of the individual forces. According to BBC (2011), to find the resultant force, the horizontal forces and vertical forces are added up to give a single horizontal force and a single vertical force respectively. Then the Pythagoras theorem is used to calculate the resultant force (FR). The angle of this resultant force is computed using the concept of trigonometry (BBC, 2011). Therefore, the condition for a system to reach equilibrium is that the resultant force of the object must be balanced using a force with the same quantity but acting in the opposite direction. The second condition of equilibrium is that the sum of the rotational forces on an object must be zero. This means that the sum of the clockwise forces must be equal to the sum of the anticlockwise forces so that they can cancel each other out. That is, τanticlockwise = τclockwise Illustration 2: Shows a resultant force that is due to the horizontal and vertical components of the force. Horizontal component, Fx Illustration 3: Materials and methods During the first experiment, Weights of 2 N and 1 N were attached to 3 load hangers which were attached to 3 copper wires. The wires were attached to a peg from one end and to the load hangers from the other end. These wires were passing through pulleys that were fitted on a horizontal plate. At the start, two weights were set at two different angles, 29° and 145° respectively. Then the weights were slowly made to balance by adding random weights on the third load hanger. The angle adjustment was done in random. Eventually, the obtained experimental weight and angle were brought into comparison with the weight and angle determined through calculation using the horizontal and vertical components. The results were too used in a graphical method to find out whether these weights and angles would make a perfect triangle or not. The results for the first experiment were: a. The result of the investigated force and angle = 2 N at 240° b. The result of the calculated force and angle = -1.802 N at 238.923° Table 1: Showing the given weights and angles and the experimentally found equilibrium force and angle. Forces Magnitudes (N) Angles (°) Force 1 2 29 Force 2 1 145 Force 3 (FEquilibrium) 2 240 Table 2: The given weights and angles and the calculated equilibrium force and angle. Forces Magnitudes (N) Angles (°) Force 1 2 29 Force 2 1 145 Force 3 (FEquilibrium) 1.80 N 238.9 Calculating the components of F1: Fx1 = 2 × cos (29) = 1.7492 N Fy1 = 2 × sin (29) = 0.9696 N Calculating the components of F2: Fx2 = 1 × cos (145) = − 0.8191 N Fy2 = 1 × sin (145) = 0.5736 N Calculating the Resultant Force (Equilibrium Force, FE): FE = √ (Fx1 + Fx2) 2 + (Fy1 + Fy2)2 FE = √ (1.5432)2 + (0.9301)2 = 1.8018 N Calculating the Resultant angle (ΘR) and Equilibrium angle (ΘE): ΘR = tan-1 [ (Fy1 + Fy2) / (Fx1 + Fx2) ] ΘR = tan-1 (1.5432 / 0.9301) = 58.9225° ΘE = ΘR + 180 = 58.9225 + 180 = 238.9225° In the second experiment, similar procedure was followed but using a vertical board. A peg was positioned at the center of an A3 paper which was fitted on the board. This peg was adjusted to balance by adding random weights on every weight hanger. Then lines on the paper sheet corresponding to the angles were drawn and the angles between the lines on the paper were measured. The results for this experiment were as follow: 1. The result of the investigated force and angle = 3.5 N at 270. 2. The result of the calculated force and angle = -3.348 N at 267.414°. Table 1: The forces and angles that caused the peg to be centered in the middle of the taped paper. Forces Magnitudes (N) Angles (°) Force 1 3 127.5 Force 2 2.2 26 Force 3 (FEquilibrium) 3.5 270 Table 2: The calculated equilibrium force and angle that caused the peg to be centered in the middle of the taped paper. Forces Magnitudes (N) Angles (°) Force 1 3 127.5 Force 2 2.2 26 Force 3 (FEquilibrium) 3.3 267.4 Calculating the components of F1:. Fx1 = 3 × cos (127.5) = −1.8263 N Fy1 = 3 × sin (127.5) = 2.3801 N Calculating the components of F2: Fx2 = 2.2 × cos (26) = 1.9773 N Fy2 = 2.2 × sin (26) = 0.9644 N Calculating the Resultant Force (Equilibrium Force, FE): FE = √ (Fx1 + Fx2) 2 + (Fy1 + Fy2)2 FE = √ (0.1510)2 + (3.3445)2 = 3.3479 N Calculating the Resultant angle (ΘR) and Equilibrium angle (ΘE): ΘR = tan-1 [ (Fy1 + Fy2) / (Fx1 + Fx2) ] ΘR = tan-1 (3.3445/ 0.1511) = 87.4138° ΘE = ΘR + 180 = 58.9225 + 180 = 267.4138° Discussion In the former experiment, a weight of 2 N was used to the other weights at an angle of 240°. It acts as balancing weight for this matter. Using, the known weights and angles the weight (FE) and its angle were computed. These weights and angles used were 2 N at 29° and 1 N at 145° and the end result of the calculated FE and its angle were 1.802 N at 238.923°. There was a small gap portraying on drawing the triangle because it did not perfectly close as expected especially if forces balance each other. This gap would have been eliminated if the calculated force was used instead of the experimentally observed force. Having followed the same procedure in the later experiment, the same procedure was done but on a vertical plane instead of a horizontal one. The result of the investigated FE and its angle were -3.5 N at 270° and the result of the calculated FE and its angle were -3.348 N at 267.414°. There was also a small gap on drawing the triangle. This was as a result of some experimental errors that led to slight variance between the calculated and investigated results. Some of these errors were the friction that was due to the pulleys that were being used in both experiment and parallax. Seeing the precise angles on the protractor were difficulty, in the first experiment, this is because the protractor was fixed to the plate. Another problem encountered in this experiment is that pulleys moved whenever a weight was added causing the angle to change, and thus affecting the result. It was difficult to draw lines on the paper since the plate was vertical and the wires holding the weights were slightly far from the plate. Conclusion In conclusion, the final results of the investigated and calculated weights and angles of the first experiment were 2 N at 240° and -1.802 N at 238.923° respectively and the second experiment -3.5 N at 270° and -3.348 N at 267.414° respectively. This lab report has in details depicted how to reach static equilibrium and the requirements of this principle. The report has showed that the difference between the expected and measured results is negligible. Therefore, results obtained are not 100% accurate due to errors and difficulties encountered during the conduction of the experiment. The report too showed how the resultant force of random forces acting on a body at different angles can be calculated using the components of each of the forces (Sundararajan V., 201). Reference List BBC, 2011. Force, mass and Acceleration. [Online] Available at: http://www.bbc.co.uk/schools/gcsebitesize/science/add_aqa_pre_2011/forces/forcemassrev1.shtml [Accessed o6 march 2014]. Bird, J. & ROSS, C., 2012. Mechanical Engineering Principles. 2nd ed. New york: Routledge. Breithaupt, J., 2010. physics. 3rd ed. London: palgrave Macmillan.Eastern, l. u., 2014. Static Equilibrium. [Online] Available at: http://www.ux1.eiu.edu/~cfadd/1150/08Statics/ToC.html [Accessed 06 march 2014]. Eastern Illinois University (2014) Static Equilibrium. Available at: http://www.ux1.eiu.edu/~cfadd/1150/08Statics/ToC.html (Accessed: 06 march 2014). Sundararajan V., M., 2010. Principles of biomedical engineering. Boston: Artech house. Read More