Vector Addition Lab Online

Vector Addition Lab Online Purpose The purpose of this lab is for the student to gain a better understand of the basic properties of vectors, and some simple vector mathematics . The student will see how vectors are related to the Right Triangle, and how vectors can be described in both the Rectangular Coordinate system, and the Polar Coordinate system. Also, the student will see how to convert a vector from one coordinate system to the other, and then back again. By adding different combinations of force vectors, and adding those vectors by different means, the student’s understanding of vectors, and vector addition, should be improved. Theory The Right Triangle, and Simple Trig Functions A triangle is a three sided, closed, geometric shape, whose three internal angles sum to 180o. A Right Triangle is a triangle that one of its internal Angles is 90o, which is called a right angle. If we pick one of the other two internal angles of a right triangle, and call it θ (theta), then we are able to label the three sides of the right triangle in reference to the angle θ. The three sides are: 1. Hypotenuse (hyp) opposite the right the longest of the 2. Adjacent (adj) – hypotenuse forms 3. Opposite (opp) – of the angle θ. – The side that is angle, and it is ALWAYS three sides. This along with the the angle θ. The side that is opposite Trig Functions relate the value of the angle θ to the value of the ratio of the lengths of two of the sides of a right triangle. There are three basic Trig Functions. 1. Sin(θ) – The sin function relates the value of θ to the value of the ratio of the opposite side and the hypotenuse. 𝐬𝐢𝐧(𝜽) = 𝒐𝒑𝒑 𝒉𝒚𝒑 2. Cos(θ) – The cos function relates the value of θ to the value of the ratio of the adjacent side and the hypotenuse. 𝐜𝐨𝐬(𝜽) = 𝒂𝒅𝒋 𝒉𝒚𝒑 3. Tan(θ) – The tan function relates the value of θ to the value of the ratio of the opposite side and the adjacent side. 1 𝐭𝐚𝐧(𝜽) = 𝒐𝒑𝒑 𝒂𝒅𝒋 There are also three basic Inverse Trig Functions. As the name suggest the Inverse Trig Functions do the exact opposite of what the Trig Functions do. As the Trig Functions relate the value of the angle θ to the value of a ratio of two sides of a right triangle, the Inverse Trig Functions relate the value of the ratio of two sides of a right triangle to the value of the angle θ. 1. sin−1 ( 𝑜𝑝𝑝 ℎ𝑦𝑝 ) – The inverse sin function relates the value of the ratio of the opposite side of the right triangle, and the hypotenuse. 𝐬𝐢𝐧−𝟏 ( 2. cos −1 ( 𝑎𝑑𝑗 ℎ𝑦𝑝 𝒐𝒑𝒑 )= 𝜽 𝒉𝒚𝒑 )- The inverse cos function relates the value of the ratio of the adjacent side of the right triangle, and the hypotenuse. 𝐜𝐨𝐬 −𝟏 ( 𝑜𝑝𝑝 𝒂𝒅𝒋 )= 𝜽 𝒉𝒚𝒑 3. tan−1 ( 𝑎𝑑𝑗 )- The inverse tangent function relates the value of the ratio of the opposite side of the right triangle, and its opposite side. 𝐭𝐚𝐧−𝟏 ( 𝒐𝒑𝒑 ) =𝜽 𝒂𝒅𝒋 The Pythagorean Theorem tell us that the hypotenuse is equal to the square root of the sum of the squares of the two other sides of the right triangle. Usually written as such: 𝒓 = √𝒙𝟐 + 𝒚𝟐 Coordinate Systems There are two basic coordinate systems we will be working with in this lab: the rectangular coordinate system, and the polar coordinate system. The rectangular coordinate system has the following properties. 1. It has two primary lines, called the Axes that intersect at right angles. The point these two lines intersect is called the Origin. • The horizontal axis is called the X-Axis. • The vertical axis is called the Y-Axis. • The Axes separate the rectangular coordinate system into four regions called quadrants. 2. A point (location) in the coordinate system is identified by a pair of numbers, called an ordered pair. • • • • The x-value is listed first, and the y-value is listed second. (x, y) The Origin has the value of (0, 0). The x-values are positive to the right of the Origin, and negative to the left of the Origin. The y-values are positive above the Origin, and negative below the Origin. Polar coordinate system has the following properties. 1. Centered on a point called the Origin. 2. All points (locations) are identified by two numbers, called an ordered pair. • The first number is the point’s straight line distance from the Origin, and normally denoted by ‘r’. • The second number is the point’s orientation (direction), normally denoted by ‘θ’. Unless otherwise stated, the orientation is taken to be measured from what would be the x-axis in the rectangular coordinate system. • Collectively this ordered pair is written as such (r, θ). • A counterclockwise rotation results in a positive value for the orientation. While a clockwise rotation results in a negative value for the orientation. Converting between coordinate systems Sometimes we have a point that is described by one coordinate system, but for various mathematical reasons we need it to be described by another coordinate system. So we must convert between coordinate systems. We can, and will, use the properties of the Right Triangle to convert between our polar and rectangular coordinate systems. Converting from polar coordinate system to a rectangular system. 1. Let us assume we have a point in the polar coordinate system described by the order pair (r, where θ is measured from what would be the xaxis in the rectangular coordinate system. 2. If we extend a line straight down from that point to the x-axis it forms a right triangle with the length r playing the role of the hypotenuse in the triangle. θ), 3. In this simple example, the x-leg of our right triangle is the adjacent side, and the y-leg is the opposite side. 4. Applying the trig functions to this we get the following: cos(𝜃) = 𝑥 𝑟 and sin(𝜃) = 𝑦 𝑟 If we multiply either by r we get: 𝑟cos(𝜃) = 𝑥 and 𝑟 sin(𝜃) = 𝑦 • Resulting in the conversation from polar to rectangular coordinates. (𝒓, 𝜽) = (𝒓 𝐜𝐨𝐬(𝜽), 𝒓 𝐬𝐢𝐧(𝜽)) = (𝒙, 𝒚) • Please note that the conversion method described assumes that θ is measured from the x-axis. If θ is measured from the y-axis you get a slightly different conversation relationship. (Try it, and see what you get) Converting from rectangular coordinates to polar coordinates. 1. Let us assume we have a point in the rectangular coordinate system described by (x, y) 2. Extend a line directly y). Call this line r, and x-axis. Finally, a line point (x, y) to the xtriangle. from the origin to the point (x, it will form an angle θ with the that runs vertically from the axis will complete a right 3. If we apply the properties of the right following: triangle to this, we get the 𝑟 = √𝑥 2 + 𝑦 2 • 𝑦 tan−1 ( 𝑥 ) = 𝜃 Resulting in coordinates. converting to polar 𝒚 (𝒙, 𝒚) = (√𝒙𝟐 + 𝒚𝟐 , 𝐭𝐚𝐧−𝟏 ( )) = (𝒓, 𝜽) 𝒙 • Please note getting the orientation θ can be a bit tricky due to the possible signs of the rectangular coordinates, and the rules of simple arithmetic. If the point is in the First or 𝑦 Fourth quadrants then tan−1 ( 𝑥 ) will yield the correct value for 𝜃. However, if the point 𝑦 is in either the second or third quadrants tan−1 ( 𝑥 ) will NOT yield the correct answer, and you will need to add 180o to get the correct value for 𝜃. 4 Vectors and Right Triangles A vector is a mathematical ‘object’ that has both a magnitude (size), and a direction, and it requires at least two numbers to describe a vector. There are two basic forms that those two bits of information about a vector are written in; Pure Form, or Component Form. 1. The Pure Form is called that because the two pieces of information about the vector, magnitude, and directions are completely separated from each other. So one number gives you purely information about the magnitude, and the other number gives you purely information about the direction. 󰇍 = (𝑉, 𝜃 ) • Let’s assume we have a vector 󰇍𝑉 , then in pure form it would be written as 𝑉 • Where the first number V is the magnitude, while the second number θ is the direction of the vector. 2. The Component Form still describes the vector by using two numbers, but the information about magnitude, and direction are sort of ‘mixed up’ between both of the numbers. • Let’s assume the same vector 𝑉󰇍 as before is written in component form instead, then • that would be 󰇍󰇍󰇍𝑉 = (𝑉𝑥 , 𝑉𝑦 ). The first number 𝑉𝑥 is displacement in the x-direction, and the second number 𝑉𝑦 is the displacement in the y-direction. 3. If you look at the same vector drawn out together in both forms a few things should pop out at you. • First that both forms of the vector start and stop at the same locations. That is because they are the same vector, just being described in two different ways. • Second, together they make a RIGHT TRIANGLE. The pure form is just the vector written in polar coordinates, and the component form is the vector written in rectangular coordinates. We can use the properties of the right triangle to convert back and forth between the two forms as needed. • The Pure form vector plays the role of the hypotenuse of the right triangle, and the component form plays the role of the two other legs of the right triangle. Adding Vectors The summation of a set of vectors is called the resultant vector. We will be examining three different methods of adding vectors to obtain the resultant vector: graphical, mathematical and experimental. Regardless of the method used when adding a set of vectors, one should obtain the same resultant. 5 Graphically – In this method vectors are denoted by arrows, also called rays. A ray has a front end and a back end, the tip, and the tail respectively. The length of the ray represents the magnitude of the vector, the orientation of the ray, from tail to tip, is the direction of the vector. Adding vectors graphically is a rather straightforward method. 1. First you pick a scale to be used for the magnitude of the vectors you are going to add. So that 1 unit of length represents so many units of whatever magnitude the vectors have. • As an example 1.0 cm = 5.0 Newtons. 2. Then take one of the vectors you are adding together, and draw it with its tail at the origin of a coordinate system. The length of the vector is to be determined by the scale you chose, and its orientation is the orientation that is given for it. 3. Then you draw the next vector with its tail at the tip of the first vector. Again, its length is to be determined by the scale you chose, and its orientation is the orientation that is given for it. 4. Repeat step 3 till there are no more vectors to add. 5. Then draw a line from the origin to the tip of the last vector you added. This line is called the ‘resultant’, and it is the vector summation of all the vectors you added. Its length gives you the magnitude of the resultant, and its orientation, from tail to tip, gives you the direction of the resultant. Analytical – A simple method for adding vectors mathematically is, always make sure the vectors are in their component form before you add them. The reason for this is that while adding vectors in their pure form can get rather complicated, vector components add like simple scalars. Meaning vector components obey the rules of simple arithmetic. Let us examine this by adding two vectors 𝐴, and 𝐵󰇍 together to get the resultant vector 𝑅󰇍 𝐴 + 𝐵󰇍 = 𝑅󰇍 6 If we break down our vectors from their pure form to component form then we can simply add up all the x-components to get the x-component of the resultant, and add up all the ycomponents to get the y-components of the resultant. 𝐴𝑥 + 𝐵𝑥 = 𝑅𝑥 𝐴𝑦 + 𝐵𝑦 = 𝑅𝑦 Once you have the resultant, then you simply use triangle to convert the √𝑅𝑥2 + 𝑅𝑦2 = 𝑅 tan−1 ( =𝜃 components of the the properties of the right resultant to the pure form. 𝑅𝑦 ) 𝑅𝑥 Of course, this method can be extended to the summation of any number of vectors. There is another vector that balanced out the original vectors called the Equilibrant, and it makes the entire system of vectors sum to zero. The Equilibrant has the exact same magnitude of the Resultant of the original forces, but the exact opposite direction. 󰇍 = (𝑅, 𝜃) = −𝐸 󰇍 = (𝑅, 𝜃 ± 180𝑜 ) 𝑅 7 Procedure Part 1 1. In the White Box near the bottom right of your screen that contains the vectors 𝑎, 𝑏󰇍, 𝑎𝑛𝑑 𝑐, Click and Drag vector 𝑎 onto the coordinate system so that its tail is at the origin. a. Click and Drag the head of vector 𝑎 around the coordinate system till in the Grey Box at the top of your screen it reads magnitude |𝑎| = 18.0 and its angle 𝜃 = 33.7. Record these values for 𝑎, and its components in Table 1. 2. Click and Drag the vector 𝑏󰇍 from the White Box near the bottom right of your screen into the coordinate system with its tail on vector 𝑎′𝑠 head. a. Click and Drag the head of vector 𝑏󰇍 around the coordinate system till in the Grey Box at the top of your screen its reads magnitude |𝑏󰇍| = 25.1 and its angle 𝜃 = 175.4 . Record these values for 𝑏󰇍, and its components in Table 1. 3. Click and Drag the vector 𝑐 from the White Box near the bottom right of your screen into the coordinate system with its tail on origin. a. Click and Drag the head of vector 𝑐 to vector 𝑏󰇍 ′𝑠 head. b. 𝑐 is our Resultant vector. Record its magnitude, angle, and components in Table 1. 4. Take a screen shot, save it, and turn it in with your lab worksheet. Procedure Part 2 1. Click on the Eraser near the bottom right of the coordinate system to clear vectors from it. 2. In the White Box near the bottom right of your screen that contains the vectors 𝑎, 𝑏󰇍, 𝑎𝑛𝑑 𝑐, Click and Drag vector 𝑎 onto the coordinate system so that its tail is at the origin. a. Click and Drag the head of vector 𝑎 around the coordinate system till in the Grey Box at the top of your screen its reads magnitude |𝑎 | = 11.2 and its angle 𝜃 = 153.4. Record these values for 𝑎 , and its components in Table 2. 3. Click and Drag the vector 𝑏󰇍 from the White Box near the bottom right of your screen into the coordinate system with its tail on vector 𝑎′𝑠 head. b. Click and Drag the head of vector 𝑏󰇍 around the coordinate system till in the Grey Box at the top of your screen its reads magnitude |𝑏󰇍| = 31.6 and its angle 󰇍 , and its components in Table 2. 𝜃 = −18.4 . Record these values for 𝑏 4. Click and Drag the vector 𝑐 from the White Box near the bottom right of your screen into the coordinate system with its tail on the origin. c. Click and Drag the head of vector 𝑐 so it is on vector 𝑏󰇍′𝑠 head. d. 𝑐 is our Resultant vector. Record its magnitude, angle, and components in Table 2. 5. Take a screen shot, save it, and turn it in with your lab worksheet. Analysis of Vector Addition Lab Name Stefany Pascua Course/Section PHY-1951-005 Physics for Scientist and Engineers 1 Laboratory Instructor Stephen Flowers Tables 1 󰇍 | 𝜽 𝑽𝒙 |𝑽 18 33.7 15 𝑎 󰇍 25.1 175.4 -25 𝑏 𝑐 15.6 129.8 -10 󰇍 15.6 -50.2 10 𝐸 Complete Table 1 with the values of the Equilibrant vector. (15 points) Vector 𝑽𝒚 10 2 12 -12 1. Using trig and the properties of vectors show that for each vector the given polar coordinates are equal to the component coordinates. Show work for credit. (10 points) Given (𝒓, 𝜽) = (𝒓 𝐜𝐨𝐬(𝜽), 𝒓 𝐬𝐢𝐧(𝜽)) = (𝒙, 𝒚) Vector A: (15, 10) Vx = rcosθ = 18cos (33.7°) = 14.97 = 15 Vy = rsinθ = 18sin (33.7°) = 9.98 = 10 Vector B: (-25, 2) Vx = rcosθ = 25.1cos (175.4°) = -25.02 = -25 Vy = rsinθ = 25.1sin (175.4°) = 2.01 = 2 Vector C: (-10, 12) Vx = rcosθ = 15.6cos (129.8°) = -9.99 = -10 Vy = rsinθ = 15.6sin (129.8°) = 11.99 = 12 Vector E: (10, -12) Vx = rcosθ = 15.6cos (-50.2°) = 9.99 = 10 Vy = rsinθ = 15.6sin (-50.2°) = -11.99 = -12 2. Show that 𝑎 + 𝑏󰇍 = 𝑐. Show work. (10 points) Ax + B x = C x 15 + (-25) = - 10 Ay + B y = C y 10 + 2 = 12 Tables 2 󰇍 | 𝜽 |𝑽 11.2 153.4 𝑎 󰇍𝑏 31.6 -18.4 𝑐 20.6 -14 󰇍 20.6 166 𝐸 Complete Table 1 with the values of the Equilibrant vector (15) Vector 𝑽𝒙 -10 30 20 -20 𝑽𝒚 5 -10 -5 5 3. Using trig and the properties of vectors, show that for each vector the given polar coordinates are equal to the component coordinates. Show work for credit. (10 points) Vector A: (-10, 5) Vx = rcosθ = 11.2cos (153.4°) = -10.01 = -10 Vy = rsinθ = 11.2sin (153.4°) = 5.01 = 5 Vector B: (30, -10) Vx = rcosθ = 31.6cos (-18.4°) = 29.98 = 30 Vy = rsinθ = 31.6sin (-18.4°) = -9.97 = -10 Vector C: (20, -5) Vx = rcosθ = 20.6cos (-14°) = 19.99 = 20 Vy = rsinθ = 20.6sin (-14°) = -4.98 = -5 Vector E: (-20, 5) Vx = rcosθ = 20.6cos (166 °) = -19.99 = -20 Vy = rsinθ = 20.6sin (166°) = 4.98 = 5 4. Show that 𝑎 + 𝑏󰇍 = 𝑐. Show work. (10 points) Ax + B x = C x -10+ 30 = 20 Ay + B y = C y 5+ (-10) = -5 5. Two vectors are being added, one at an angle of 20.0 , and the other at 80.0 . The only thing you know about the magnitudes is they are both positive. Will the equilibrant vector be in the (a) first quadrant, (b) second quadrant, (c) third quadrant, (d) fourth quadrant or (e) you cannot tell which quadrant from the available information? Justify your answer. (10 points) The equilibrium vector will be in the first quadrant due to the parallelogram law of vector addition since both vectors have positive magnitudes, the equilibrium vector must be between them. 6. Imagine two force vectors, one of magnitude 2.00 N, and the other of magnitude 3.00 N. The directions of both vectors are unknown. Which best describes the limitations on the magnitude of R, the resultant vector? Justify your answer. (10 points) (a) R  5N (b) 2N  R  3N (c) R  3N (d) 1N  R  5N (e) R  2N Rmax = F2 + F1 3N + 2N = 5 N Rmin = F2 - F1 = 1 3N – 2N = 1 (d) 1N ≤ R ≤ 5N best describes the limitations on the magnitude of R, the resultant vector. 7. Screen shots attached (5 points each, for a total of 10 points)