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Names of Participants _____________________________________ Date _________ Vector Addition of Forces INTRODUCTION : Newton's First Law states: "An object at rest will remain at rest and an object in motion will remain in motion, unless acted on by a net external force." The purpose of this lab is to show that, for equilibrium, the resultant of two or more forces acting at different angles must add to zero. THEORY : Two or more different forces often act simultaneously on an object at the same point. These forces may be replaced with a single force called the resultant (R). The magnitude and direction of this resultant force is determined by the magnitude of the forces acting on the object and the angles between them. The resultant of two forces can also be determined graphically as the diagonal of the force parallelogram of which the two forces are sides, or alternately using the "head-to-tail" method. According to Newton's First Law, for an object to remain stationary (in "equilibrium") the resultant force must be zero.  F = F1 + F2 + F3 + ... = 0 The above vector addition is calculated analytically by the method of components Rx = F1x + F2x + F3x + ... Ry = F1y + F2y + F3y + ... = F1cos1 + F2cos2 + ... = F1sin1 + F2sin2 + ... where Fx is the x-component of the force and Fy is the y-component of each force. In experimental situations, the resultant of the forces may not exactly equal zero and the remaining value is called the "residual". The equilibrant (E) is a term used to describe a single force that can establish an equilibrium with other forces. It is always equal in magnitude and opposite in direction to the resultant. APPARATUS : • Force Table with four pulleys • weight hangers • slotted masses • string • level • ruler • protractor PROCEDURE : In this experiment you will find the resultant of two or more vectors by three different methods (experimental, graphical, and analytical). You will also determine the components of each vector by each of the three different methods. For this experiment we will pretend that mass, measured in grams (g or g-N) represents force. This is not completely accurate as will be seen in Chapter 5. 1. Set up the force table with strings and weight hangers and perform the following cases of vector addition. 2. Use a bubble level and the adjusting screws on the legs of the force table to level the surface of the force table. 3. For each of the following experiments use the method of components to analytically add the two vectors to find the x and y components of the resultant. Draw a sketch of the vectors and use the "head-to-tail" method in order to add the vectors graphically. Vector addition 1 Arrange one force vector with magnitude F1 = 200 g-N at 30o. The equilibrant will be F2 = 200 g-N at 210 o on the force table. Don't forget to include the weight of the hangers. Find the x- and y-components of the equilibrant experimentally. Then determine the x- and y-components of the resultant. Vector addition 2 Arrange two force vectors with magnitudes F1 = 200g-N at 30o and F2 = 200 g-N at 330o. Experimentally determine the magnitude and direction of a third force (F3) that will keep the ring stationary. F3 is the equilibrant of F1+ F2. The resultant is at 180º from the eqilibrarant. The x-component of F1+ F2 is -F3 . The y-component of F1+ F2 is 0. Vector addition 3 Arrange two force vectors F1 = 200 g-N at 0o and F2 = 400 g-N at 90o. Experimentally determine the magnitude and direction of a third vector (F3) that will keep the ring stationary. F3 is the equilibrant of F1and F2. You might also consider F1and F2 to be the components of the equilibrant of F3 or the components of the resulant of F1+ F2. Vector addition 4 Arrange three force vectors F1 = 200 g-N at 0o, F2 = 300 g-N at 45o and F3 = 400 g-N at 225o. Experimentally determine the magnitude and direction of a fourth vector (F4) that will keep the ring stationary. After you determine F4 ( the equilibrant of F1+ F2+ F3), experimenatally determine the components of the resultant by removing F1, F2 and F3 and adding masses along the x- and y-axes. Vector addition 5 Arrange three different force vectors of different magnitudes at random angles so that the ring remains stationary. No two forces should form a right angle between them. (For example, you might hang 300g at 10° and 400g at 300°.) 4. Calculate the average of the absolute values of the ten residuals. (For example, Residual = |experimentally measured value of x-component - calculated value of x-component|)


2. Next use the graphical method to find the resultant force. Every group member should find create a tip-
to-tail graph to find the resultant. Using a ruler, protractor, and a full sheet of graph paper graphically
(using tip to tail method) find the resultant of the three vectors you have been assigned. Your graph should
be similar to figure 2. Be sure to include the scale relating length measured on the ruler to vector
magnitude. Record the magnitude and angle of the resultant as measured from each member's graph and
take the average of the different resultants and angles.

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