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Vector Addition

Vector Addition. Simple addition (with signs!) and Pythagorean addition for vectors at right angles. Warmup. Santa has decided to supercharge his sleigh. He buys several rocket-powered cyborg reindeers and gets rid of the non- cyborg reindeers. A cyborg reindeer produces 200N of force.

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Vector Addition

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  1. Vector Addition Simple addition (with signs!) andPythagorean addition for vectors at right angles

  2. Warmup • Santa has decided to supercharge his sleigh. He buys several rocket-powered cyborg reindeers and gets rid of the non-cyborg reindeers. A cyborg reindeer produces 200N of force. • First, Santa hooks up two cyborg reindeers next to each other, so that they both pull the sleigh straight ahead together. Solve for the resultant. • Next, Santa hooks up the same two cyborg reindeer so that they are 90 degrees apart, together they’re pulling the sleigh in a V. Solve for the resultant. • Which is the best way for Santa to arrange his new cyborg reindeer? Why?

  3. Resultants and Nets and Force, Oh My! • Another way we will ask you for the resultant is to ask for the “net force”. • This is just a fancy way of saying “the answer once we’ve added up all the vectors.” • Later on in the year, when we start figuring out exactly what force does to the motion of objects, the net force will be very important. • For now, the only math we will have to do is to simply find the net force/resultant.

  4. Vector Addition and Straight Lines • By now, we have learned the two rules for vector addition. They’re very simple. 1) Put signs on the vectors based on the direction they’re going. 2) Add ‘em. • Sometimes, we will ask you to figure out what a missing force is if we give you the resultant and one other force. • We can use algebra to easily solve these problems. We will use the equation: Resultant = Force 1 + Force 2

  5. Let’s Try it Out • Superman is flying due east with a net force of 175N. • Friction is acting against the force he’s producing. What direction will we draw the friction vector in? • The friction vector has a magnitude of 55N. What sign do we put on it? • Draw the FBD for this situation. • Finally, figure out how much force is Superman using to fly. Net force = Superman’s force + force of friction. (Hint: remember to use proper signs, and our rules for rearranging algebra problems.)

  6. Storytime! • In whiteboard pairs, draw a FBD of two (or more) force vectors acting on the same object in a straight line . Make up numbers for each vector (make sure to use correct signs!!!). Then solve for the resultant. • Then, come up with a short ‘story’ to explain what might give us a FBD like you drew. For example “While Christmas shopping, Bob and Betty both grabbed the same toy , and each is pulling on the it exactly as hard as the other. Because the vectors sum up to zero, the forces are balanced and the toy is in equilibrium.” • Be prepared to share your story with me when I come to check on your group’s progress.

  7. Vectors at Right Angles • When working with vectors at right angles, we have to rearrange them “head to tail”. This will tell us the direction of the resultant if we’re not sure. • On whiteboards, answer the following four questions. • 1) If two vectors are going north and west, what direction does the resultant go? • 2) North and east? • 3) South and east? • 4) South and west?

  8. And Now, the Math • When we have any right triangle, we solve for the longest side (the hypotenuse) with Pythagorean. • The resultant will ALWAYS be the hypotenuse. • To find the resultant, we square both vectors, then we add them, then we take the square root of our answer. Remember order of operations: square both first, then add the two squared answers, and then square root that final number. • The “Smell Test”: your resultant will ALWAYS be bigger than either one of your two forces, but will ALWAYS be smaller than both added together. If this isn’t the case, you’ve gotten the wrong answer.

  9. Testing It Out – Work the Following Four Problems on you Whiteboards • A 55N force due north and a 72N force due east are acting on an object. What is the magnitude and direction of the resultant? • A 12N force due north and an 82N force due west are acting on object. What is the magnitude and direction of the resultant? • A 91N force due south and a 27N force due east are acting on an object. What is the magnitude and direction of the resultant? • A 77N force due south and a 73N force due west are acting on an object. What is the magnitude and direction of the resultant? • DO NOT ERASE YOUR ANSWERS YET. You may erase your work (but NOT the answers) once I have checked your whiteboard.

  10. Followup • A mad scientist has decided that all objects, everywhere, must be in equilibrium, always. • For each of the four examples you just solved, find the magnitude and direction of the vector required to balance our resultants. • What’s the pattern we see here? (How does the magnitude of the resultant compare with the magnitude of the balancing force? How does the direction of the resultant compare with the direction of the balancing force?)

  11. Exit Ticket • Your Exit Ticket will be the same every day for the rest of the year. • Write, in complete sentences, one thing you learned or one thing you did not understand. You MUST be specific.

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