• Scalar – a quantity with magnitude only Speed: “55 miles per hour” • Temperature: “22 degrees Celsius” • Vector – a quantity with magnitude and direction • Velocity: “25 m/s Northwest” • Acceleration: “4.8 m/s2 at 90 degrees” • Component – one of the vectors given in the problem • Resultant – the “net” vector • Concurrent Vectors – vectors acting on the same point at the same time • Equilibrant – vectors that produce equilibrium; it is equal in magnitude and opposite in direction to the resultant • Examples: Let’s say you walk 10 m northeast and turn and walk 9m southeast. • Problem: Hurricane Frances is traveling at 8 mph West. A weather front approaches Frances at 20 mph Northeast. Find the resultant direction of Hurricane Frances. (“net”) • (OK – so it’s more complicated than that)… • What to do? • Go to: • http://www.sfu.ca/phys/100/lectures/Ol ManRiver.html • (Make sure the volume is turned UP!) • Watch the video. What does this video show? (Watch it more than once if necessary) • That the “river” was the paper being pulled at a constant rate? • That if the “boat” left perpendicularly to the “shore” it would end up “downstream”? • That the instructor must have calculated the proper angle needed based on the rate of “flow of the river” before he released the second “boat”? • This is vital information for pilots, ship navigators, athletes • A quarterback uses vectors to throw a ball to a receiver that is running to make a touchdown (but he probably doesn’t think of it as a vector) • Name some other examples in sports or other aspects of your life • Graphically – vectors are drawn using a ruler (measurements are done to scale) and a protractor (direction is noted as, for example, 20 degrees north of east) • Analytically – trig is used; right triangles use the Pythagorean Theorem (a2 + b2 = c2) and SOH CAH TOA. Triangles that are not right use LAW OF SINES or LAW OF COSINES • In fact, you will be taking one quiz in which you will have only a protractor and one quiz in which you will have only a calculator. • Resultant – is the “net” direction, or force, or acceleration, etc. • Use a protractor to draw a line going in the direction stated. The LENGTH of the line indicates the MAGNITUDE of the direction, force, etc. • The DIRECTION of the vector is in the stated direction and is carefully measured using the protractor • A plane’s engine pulls the plane 700m to the north. There is a strong wind pushing the plane 200 m to the west. • Let’s solve this problem graphically… • First, carefully draw an arrow pointing north that has a magnitude of 700 units. Then, draw an arrow pointing west that has a magnitude of 200 units. Notice that we have used a “tip-to- Now you have to tail method” to draw vectors. Two draw the Resultant, tails exist only where the resultant which shows the touches the “first” vector. “net” magnitude. • It is necessary to draw vectors using the Tip-To-Tail Method. That means that the tip of one vector can only touch the tail of a second vector in the final vector diagram. • See how this is done at this site, where the resultant is shown in RED: • http://www.walter- fendt.de/ph11e/resultant.htm Now that you know the magnitude of the resultant, you must report the direction of the resultant. (Remember – that’s what makes it a vector!) Put your protractor’s “origin” at the intersection between the original vector and the resultant. Find the angle and state the angle as, ie, “22 degrees N of E” You will probably note that these can’t officially be vectors since they’re not drawn in a tip-to-tail fashion! Let’s say you want to measure the angle between these two arrows… You could subtract 90.0 from 127.5 to get 37.5 degrees. You would record this as 37.5 degrees East of North. The ends of each arrow must rest in the “origin” of the protractor (the hole in the plastic) and one of the vectors must align Let’s move the with the arrows. How would marked black line. you measure this angle? You could subtract 0.0 from 78.5 to get 78.5 degrees. You could state that this is 78.5 degrees West of South. • Let’s take that first example of the 700 m displacement N and the 200 m displacement W. Since this makes a Right Triangle, we can use the Pythagorean Theorem to solve for the resultant. a2 + b2 = c2 Therefore, 7002 + 2002 = c2 And c = 728 m A vector diagram can be drawn for displacement in m, for velocity in m/s, for acceleration in m/s2 or for force in N but each vector diagram’s sides have consistent units. • Again, since it’s a right triangle, we can use Trig. • Since we know the opp and adj, and we know that tan q = opp / adj, calculate this, too. • For questions 5-8 on the “Vectors I” handout, 2 vectors act on a single point, like this… • …you must remember the tip-to-tail method so you will have to slide one vector to the end of another as shown on the site you saw earlier. Remember? Return to that site if needed. Here’s the link again. • http://www.walter- fendt.de/ph11e/resultant.htm You Must Pick One Vector to Slide Onto the End of the Other Vector • Let’s say you have one vector that is 7.8 m , one that is 2.3 m and you have to find the resultant. • If you know the sides (7.8 and Here are the sides (lower case) 2.3) and Angle C (110o) then you B must use the 7.8 m a Here are the angles (upper case) c2 = a2 + b2 – 2ab cos C c o C 110 Use this website to do a quick calculation: 2.3 m A http://hyperphysics.phy-astr.gsu.edu/hbase/lsin.html b Use the rest of the class time to work in small, quiet groups to draw and calculate the resultants.
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