GENERAL PHYSICS I

EXPERIMENT NO. 5

FORCE TABLE — ADDITION AND RESOLUTION OF VECTORS INTRODUCTION Forces add together as vectors. For example, if two or more forces act at a point, a single force may act as the equivalent of the combination of forces. The resultant R of the sum of two force vectors A and B is a single force which produces the same effect as the two forces, when these pass through a common point (see figure). The equilibrant E is a force equal and opposite to the resultant. A vector may be broken up into components. To find the components of a vector in a particular coordinate system, one must find the vectors aligned with each of the perpendicular axes of the coordinate system which give the original vector when added together.

METHODS FOR ADDING VECTORS PARALLELOGRAM METHOD & POLYGON METHOD To add two vectors, A + B by the parallelogram method a parallelogram is drawn of which two adjacent sides have magnitude and direction of A and B respectively. The diagonal of the parallelogram then has the same magnitude and direction as the resultant, R. Thus A + B = R. An equivalent method which may be used to add more than two vectors is to draw the vectors to be added “head to tail” (head of A to tail of B, head of B to tail of C, etc.). An arrow connecting the tail of the initial vector to the head of the final vector describes the vector sum of the vectors added. This is sometimes called the polygon method. COMPONENT METHOD To add two dimensional vectors by the component method, all vectors are regarded as beginning at the origin. The vectors are then resolved into x and y components. These components are then added algebraically, and the sum of the x and y components represent the components of the sum of the vectors. ANALYTICAL METHOD USING TRIGONOMETRIC IDENTITIES The law of cosines and the law of sines can be used to find the magnitude and direction of the resultant when two vectors are added together by the head to tail method. Consider a triangle whose sides a and b represent two vectors added together, and side c represents the resultant vector. Then let angle α be opposite side a, angle β opposite side b and angle γ opposite side c. The law of cosines is then c2 = a2 + b2 − 2ab cos γ, and the law of sines is sin α sin β sin γ = = . a b c These formulas can be proved from the basic trigonometric definitions. Note that in the special case of γ = 90◦ , the law of cosines becomes the Pythagorean theorem, and the law of sines follows from the definition of the sine function. 1

GENERAL PHYSICS I EQUIPMENT Force table, Four pulleys, Four plastic hangers (5 grams each), Weight set (weights of 5, 10, 20 ,50 and 100 grams), String, Bull’s eye level, Meter stick or other measuring device, Protractor. PROCEDURE

EXPERIMENT NO. 5

0. When adding weight make sure you consider the weight of the hanger! Since the hanger weighs 5 grams, if you want 50 grams you need add 45 to the hanger, etc. 1. Check to see that your force table is level, using the bull’s-eye level provided. (If not, you may adjust the legs slightly by unscrewing them.) 2. Attach three pulleys to the force table. (Do not tighten the screws too tightly. This can damage the pulleys.) Tie three strings on the ring and hang each over a pulley. These should be tied loosely enough to be able to slide around. 3. Attach a weight hanger on each string by winding the string around the top of the weight hanger a couple of times. (It should hold without a knot.) Be sure the string is not so long that the hangers touch the table. 4. Given the two vectors, F1 = 0.075 kg × g Newtons at 120◦ and F2 = 0.050 kg × g Newtons at 45◦ , (g = 9.8m/s2 is the acceleration due to gravity) find their vector sum or resultant by the following procedures. (a) Graphical Method. Use the parallelogram method of vector addition. (Be sure to use a scale large enough to minimize errors.) (b) Analytical Method using Trigonometric Identites. Use the law of cosines to compute the magnitude of the resultant force, and the law of sines to calculate the angle of orientation. (c) Experimental Method. Use the force table to determine the resultant force, by finding the equilibrant force using trial and error until the force balances the other two. Start with the ring held in place by the plastic bolt in the center of the force table. Once the forces are in equilibrium, the bolt can be removed and the ring should return to the center when it is moved around. 5. Repeat step 4 for F1 = 0.100 kg × g Newtons at 20◦ and F2 = 0.075 kg × g Newtons at 150◦. 6. Repeat step 4 for F1 = 0.200 kg × g Newtons at 0◦ and F2 = 0.150 kg × g Newtons at 90◦ . Note that F1 and F2 are the x and y components of a vector F = |F1 |ˆ + |F2 |ˆ. ı  ◦ 7. Given a force vector of 0.150 kg × g Newtons at 70 , resolve the vector into its components graphically, analytically (using trig identities), and experimentally. 8. Add the three vectors F1 = 0.050 kg × g Newtons at 30◦ , F2 = 0.075 kg × g Newtons at 80◦ , and F3 = 0.125 kg × g Newtons at 225◦ to find F = F1 + F2 + F3 , using all three methods: (a) Graphically — Using the “polygon” method. (b) Analytically — Using the component method. (c) Experimentally — Using the force table. (For this, you will have to add another string, pulley and weight hanger to the table.) QUESTIONS — For the writeup. See instructions. 1. Discuss the possible sources of error for each method. How would you rank the three methods as far as accuracy is concerned?

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