Addition OF Vectors- week 3 report PDF

Preview

Summary

lab report...

Description

ADDITION OF VECTORS

Oluseye Dare

Gamoi Paisley

Objective: To demonstrate that the magnitude and direction of the resultant of several forces acting on an object may be determined by drawing a vector diagram and show that the particle is in equilibrium when the resultant force is zero. Introduction: Vectors can be described as quantities which have both magnitude and direction. Velocity, acceleration and force are some examples of vectors. When two or more vectors act upon an object from a point, the sum of the forces acting on the object will result in a single vector which will have the same effect on the object. This single vector is called the resultant vector. To completely cancel the effect of a resultant vector on an object, a force equal and opposite to the resultant force must be applied to the object, this force is known as the equilibrant. Vectors are normally represented graphically by drawing a straight line whose length is proportional to the magnitude of the vector. The direction of the vector can be indicated by an arrowhead at the end of the line. There are two methods which may be used to graphically represent vectors, they are the triangular method and the parallelogram method. When using the triangular method. The first vector is drawn from the point of origin in the direction of the vector, any other vectors are then drawn from the end point of the previous vector in its relevant direction. The resultant vector can then be determined by drawing a line from the origin to the tip of the last vector. The parallelogram method involves drawing both vectors from the point of origin in their respective direction. The parallelogram can then be completed by drawing the other two sides of the parallelogram parallel to the first two lines. The resultant vector would therefore be equal in magnitude and direction as the diagonal of the parallelogram which should be drawn from the point of origin. The resultant force of multiple vectors may also be calculated analytically by adding the x any y components of each vector to determine the x and y components of the resultant force. Pythagoras theorem may then be used to identify the magnitude of this resultant.

Lab Data: Table1: Analytical values obtained from using the analytical method for R 1 F1 x Y 94 34.2 Vector R1 (207g, 920)

F2 x -100

y 173

R1x F1x + F2X -6

R1y F1y + F2y 207

% Error Magnitude and Angle 0.5%

Table 2: Analytical values from using the analytical method for R2 R1 x Y -6 207 Vector R2 (164g, 1370)

F3 x -115

% Error Magnitude 3.5%, %Error of Angle 1.5%

y -96

R2x R1x +F3x -121

R2y R1y +F3y 111

Calculations: Analytical method for Resultant 1 

F1x = Cos 20 = F1x ÷ 100

F1y = Sin 20 = F1y ÷ 100

F1x = 100 × Cos 20

F1y = 100 × Sin 20

F1x = 95

F1y = 34



R1x = F1x + F2x = 94 + (-100) = -6



R1 = √(Rx2 + Ry2) = √-62 + 2072 = √42885 = 207g



R1 θ = tan-1 R1y ÷ R1x = tan-1 (207 ÷ 6) = 88o R1 θ from origin = (180 – 88) = 92o



Percentage error R1 Magnitude: |Experimental value – Analytical Value| ÷ Analytical Value × 100% |208 - 207| ÷ 207 × 100 1 ÷ 207 × 100 0.5%

Results and Discussion: In this experiment the magnitude and direction of two resultant forces was determined by both graphical and analytical methods. Using the analytical method, the resultant vector R1 which is the sum of the individual vectors F 1 and F2, was determined to have a magnitude of 207g at a 92 0 angle. The second resultant force R2, which was as a result of adding a third individual vector F3 to R1, was determined to have a magnitude of 170g at a 1390 angle. The analytical method uses the Pythagoras theorem to calculate the exact values of the resultant forces and their respective angles. We can therefore assume that the analytical method of vector addition will produce exact values of resultant forces. The graphical method of vector addition was also used to determine the resultant values of R1 (208g, 930) and R2 (170g, 1390). The graphical addition method uses lines proportional to the magnitude (1cm:20g) at angles from the origin of the vector. When compared to the analytical values, there was a minimal difference between the resultant values. Then graphical method produced errors less than 4%. These errors may have occurred due to random errors which may have occurred while the graph was being constructed such as shifting of the ruler during the measurement process or estimating values for lines which fell between lines on the ruler. A force table was used to test the values which were obtained,

by using the equilibrant, which is 180 0 and equal in magnitude to the resultant, along with the individual vectors to determine if the ring at the center of the table would remain in equilibrium when all forces were added to the table. The ring was observed to be in equilibrium when all forces were added to the table, this confirmed that both the analytical and graphical methods were sufficient in determining the resultant forces of two or more vectors.

Conclusion: The magnitude and direction of the resultant of several forces acting on an object was successfully determined using graphical addition methods. It was also proven, using a force table, that the resultant/net force acting on an object at equilibrium is zero.

Questions:

1. State how this experiment has demonstrated the vector addition of forces. In this experiment the force table was used to demonstrate the vector addition of forces using a force table. When two vectors of different magnitude and direction was placed on the force table, a single equilibrant force was used to bring the object in the center of the table back to equilibrium. This equilibrant force must have been equal to the sum of both independent vectors in order to oppose the force applied by each vector. This showed that two vectors acting upon an object will exert a force equal to both individual vectors. Graphical analysis of two vectors also showed that by adding one vector to another at the point which one ends produces a single vector from the origin equal to both vectors. 2. In procedure 3 could all four pulleys be placed on the same quadrant or in two adjacent quadrants and still be in equilibrium? Explain. All four pulleys could not be placed in the same quadrant to obtain equilibrium. To obtain equilibrium a force equal and opposite to the resultant force of the individual vectors had to be placed in the adjacent quadrant so that the resultant force would be canceled by that of the equilibrant. Since both forces are equal and opposite they object in the center of the table would remain in equilibrium. 3. State the condition for the equilibrium of a particle. For a particle to be in equilibrium, the total force acting upon that particle must be zero. Therefore when the equilibrant vector, which is equal and opposite to a resultant vector is applied to an object being acted on by two vectors. The sum of the total force will be zero and the object will be in equilibrium....