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Mathematics Review Mathematics is a language. The Meaning of Numbers Algebra It is used to describe the world around us. Can you tell me what this means? N Trigonometry Geometry Choosing the Correct Equation(s) i 1 i I If you understand only “how to do” the math, then you will need to know the numbers to see any meaning behind this equation. However, if you understand the “meaning” of the math, then the equation itself tells you a great deal about how nature works. The equation says the following… 1 “The total torque acting on an object is the same as its moment of inertia multiplied by its angular acceleration.” Mathematics Review Mathematics is a language. N The Meaning of Numbers i 1 Algebra Trigonometry Geometry Choosing the Correct Equation(s) i I means that “The total torque acting on an object is the same as its moment of inertia multiplied by its angular acceleration.” You see i is any of N torques, I is moment of inertia and is angular acceleration. N means to sum all of what is behind it for every value of i from 1 to N . i1 2 But you still do not know the meaning of torque, moment of inertia and angular acceleration. Mathematics Review Mathematics is a language. The Meaning of Numbers Algebra Trigonometry Geometry Choosing the Correct Equation(s) N i 1 i I means that “The total torque acting on an object is the same as its moment of inertia multiplied by its angular acceleration.” But you still do not know the meaning of torque, moment of inertia and angular acceleration. Torque is a measure of how hard you are trying turn something. Moment of inertia tells us how hard it is to change how fast it turns. And angular acceleration measures how much it changes how fast it turns. 3 Mathematics Review Mathematics is a language. The Meaning of Numbers Algebra Trigonometry Geometry Choosing the Correct Equation(s) N i 1 i I means that “The total torque acting on an object is the same as its moment of inertia multiplied by its angular acceleration.” Torque is a measure of how hard you are trying turn something. Moment of inertia tells us how hard it is to change how fast it turns. And angular acceleration measures how much it changes how fast it turns. This same equation can describe a grinding wheel, the hands of a clock, the motion of a wrench, and an infinite number of other situations. 4 If I plug in the numbers as an example, I only learn ONE of the situations!! Mathematics Review The Meaning of Numbers Algebra Trigonometry You need to remember algebra. Here are some of the basics… If A B C then A B B C B Distributive property A B C A B A C ACB Geometry Choosing the Correct Equation(s) If A B C Commutative properties then A B / B C / B A B B A AC/B A B B A …and similarly for square roots, squares, subtraction and division except that subtraction and division are NOT commutative! 5 Mathematics Review Some simple examples… The Meaning of Numbers Algebra Trigonometry If F1x F2 x max then F1x F2 x max Distributive property m v1x v2 x mv1x mv2 x Geometry Choosing the Correct Equation(s) If maz FTOTALz then m FTOTALz / az Commutative properties m1 m2 m2 m1 max ax m …and similarly for square roots, squares, subtraction and division except that subtraction and division are NOT commutative! m1 m2 m2 m1 6 m / ax ax / m Mathematics Review Here are some other helpful concepts… The Meaning of Numbers Algebra Trigonometry If A C and B C then Ratios If AC D and BC E AB …even if YOU DON’T KNOW C! Geometry Simultaneous Equations Choosing the Correct Equation(s) A D then B E …even if YOU DON’T KNOW C! a1 A b1B c1 + a2 A b2 B c2 da1 a2 A db1 b2 B dc1 c2 7 usually we use this in such a way that one of the coefficients is zero Mathematics Review Here are some more simple examples… The Meaning of Numbers Algebra Trigonometry If F1x ma1 and F2 x ma1 Ratios If m1L T1 and m2 L K 2 then F1x F2 x …even if YOU DON’T KNOW m or a1! Geometry Simultaneous Equations Choosing the Correct Equation(s) m1 T1 then m2 K 2 …even if YOU DON’T KNOW L! 2 F 6Q 10 + 12 F 5Q 3 6 2 12 F 6 6 5 Q 6 10 3 8 usually we use this in such a way that one of the coefficients is zero 0 F 35Q 57 57 Q 35 Mathematics Review This is the unit circle… The Meaning of Numbers sine The second Also note that to The distance quadrant has the is the thecosine right on negative negative ifisthe cos-axiscosines the and positive sines. line is drawn cosine of theto and It axes are Algebra Trigonometry Geometry the left onIIthe angle. cos-axis. 1 2 We speak of Hereoften are some four otherquadrants. examples. 9 sin 3 2 cosine Choosing the Correct Equation(s) Note that the angle The thirdgoes quadrant always from has thenegative positivecosines cos-axis and sines. counterclockwise. 1 The first quadrant All lines drawn has positive cosines here have a and sines. length of 1 and an angle I equal to the angle we are working with. 150o o 330 1 2 3 2 o 45 30o 1 12 2 11 1 1 2 3 2 III IV cos Why is the sine The height negative along thehere? sinaxis the sine The is fourth of the angle. quadrant has positive cosines and negative sines. Mathematics Review Given a right triangle, the trigonometric functions for either non-right angle are given by the following… The Meaning of Numbers Algebra Trigonometry opposite (o) hypotenuse (h) Geometry Choosing the Correct Equation(s) θ adjacent (a) o sin h h csc o a cos h h sec a o tan a a cot o The value of the angle can also be determine by using any two of the sides. For example, o tan a 1 10 Mathematics Review Here is an example of how to use it… The Meaning of Numbers Algebra Trigonometry Geometry Choosing the Correct Equation(s) 3 5 csc 36.87 5 3 4 5 h=5 cos 36.87 sec 36.87 5 4 Note: This is NOT 3 4 tan 36.87 cot 36.87 drawn to 4 3 scale! o θ=36.87 sin 36.87 o=3 a=4 The value of the angle can also be determine by using any two of the sides. For example, 3 tan 36.87 4 1 11 Mathematics Review Here are some useful angle relations… a The Meaning of Numbers a Algebra b a b b a a b b b b a b 180 a Trigonometry a b 180 Geometry a b 180 b Choosing the Correct Equation(s) a a b c 180 a a 12 C c a b A c B A B C sin a sin b sin c a Mathematics Review For example… The Meaning of Numbers Algebra Trigonometry Geometry Choosing the Correct Equation(s) 13 a If a 30 then b 150 b a b 180 Mathematics Review The Meaning of Numbers Algebra Trigonometry Geometry Choosing the Correct Equation(s) Here are some basic geometric and trigonometric formulae which we will use often in this and the next class… Circumference of a Circle Trigonometric Formulae sin 2 cos 2 1 Area of a Circle A r Surface Area of a Sphere A 4r 2 Volume of a Sphere 4 3 V r 3 Quadratic Formula A 2rL Ax 2 Bx C 0 V r 2 L B B 2 4 AC x 2A Surface Area of a Cylinder (not including end faces) Volume of a Cylinder 14 C 2r 2 sin A B sin A cos B cos A sin B cos A B cos A cos B sin A sin B Mathematics Review Finding out which equation or set of equations to use while solving a problem in physics is the most difficult part of the process. The Meaning of Numbers It is also the most crucial part! Algebra Trigonometry Geometry Choosing the Correct Equation(s) 15 Still, if you follow a few basic steps, the difficulty will be far less and you will need to spend much less time on PreAssignments, Homework and Exams. An example solved by a naïve student (Bailey D. Wonderdog’s nemesis, the neighbors cat, for instance) will help us see what the rules are and how to apply them. The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? Trigonometry Geometry If you look in your textbook, you will find the equation V IR Choosing the Correct Equation(s) At first this naively appears to be the simplest equation we can use for this problem. We might be tempted to guess that V is the velocity, I is the impulse , and R is the radius. 16 Let’s try this … Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? V IR Trigonometry Geometry Choosing the Correct Equation(s) When trying to plug in the numbers, we see our first challenge. There two different objects and each have different velocities. For this problem, the answers are… Which one do we choose? To answer this, we must ask ourselves two things. 17 1. 2. What physical quantity are we looking for? What object is that physical variable related to? 1. 2. Impulse The ball Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra Trigonometry Geometry Choosing the Correct Equation(s) The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? V IR Thus, we would use the quantities associated with the ball in this problem. Rule #1: We must know which object we are considering in a problem. Plugging in the numbers, we see that I 18 V 10 0.5. R 20 If we plug this into the homework software, it will tell us we are incorrect. What went wrong. Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra Trigonometry Geometry Choosing the Correct Equation(s) The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? V IR Well, first of all, the velocity is in m/s and the radius is in cm. So, we have to convert one of the units to make them the same. You will learn how to do this in the lecture called “Math for Physics”. If we do it properly in this case, we find that… m 10 V s 100 cm 1 I 50 s R 20 cm 1 m 19 Rule #2: Use the proper units. Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra Trigonometry Geometry Choosing the Correct Equation(s) The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? V IR Next we go back to the very first thing we learned in this lecture. The variables of physics are words in the language of math. If we look up the equation again and read carefully, we will find that it means… 20 The voltage across a resistive element in a circuit is the same as the current through it multiplied by its resistance. The variables are not even close to what we wanted to use!!!! Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? V IR Trigonometry Geometry Choosing the Correct Equation(s) Rule #3: Know the meaning of each variable. We now look up the word impulse in the index of our book or in the notes and find that the variable that represents it is J. We now find two equations that contain J on the website for the class. tf J Fdt t0 21 and But which one should we use? J mv Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? Rule #4: Use what is Trigonometry Geometry Choosing the Correct Equation(s) J mv known and unknown to sort out equations that are not useful. is the only one of the two equations for which we have all of the information to solve. It reads… The impulse on an object is the same as its mass multiplied by the change in its velocity. We know the mass and the change in the velocity of the ball. 22 The other equation needed force and time, neither of which is known. Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? Trigonometry Geometry Choosing the Correct Equation(s) J mv Now we just need to plug in the ball’s mass (40 g) and its change in velocity. It had 10 m/s to begin with and ended with 10 m/s as well. Thus, J mv 10 g 0 m / s 0 g m / s But the homework software stills says that we are incorrect! 23 Mathematics Review An example of choosing the correct equations. The Meaning of Numbers Algebra The velocity of a 40 g baseball is initially 10 m/s north. After it is hit by a bat that is moving at 5 m/s south, the ball is now moving 10 m/s south. The ball has a radius of 20 cm. What is the impulse that caused the ball to change its velocity? Trigonometry Geometry Choosing the Correct Equation(s) J mv Rule #5: Vectors! Gotta use vectors!!! The arrows on top of the variables J and v tell us that they are vectors. Which is the When we subtract the initial from the final velocity, we must also take correct into account their direction. (One is north, the other south). J mv 10 g 10 m / s 10 m / s 20 g m / s 24 answer !!!!!!!! Mathematics Review In summary.... The Meaning of Numbers Rule #1: We must know which object we are considering in a problem. Algebra Rule #2: Use the proper units. Trigonometry Rule #3: Know the meaning of each variable. Geometry Choosing the Correct Equation(s) Rule #4: Use what is known and unknown to sort out equations that are not useful. Rule #5: Vectors! Gotta use vectors!!! Follow these rules when solving problems and you will find that physics is not so bad. 25 This is what DR. Mike means when he says you must use concepts to solve problems in physics.