Mathematics Algebra Mathematics is a language The Meaning

Mathematics Algebra 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? Choosing the Correct Equation(s) 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 Algebra Mathematics is a language. The Meaning of Numbers Algebra Choosing the Correct Equation(s) means that “The total torque acting on an object is the same as its moment of inertia multiplied by its angular acceleration. ” You see is any of acceleration. torques, is moment of inertia and is angular means to sum all of what is behind it for every value of from 1 to 2 But you still do not know the meaning of torque, moment of inertia and angular acceleration. .

Mathematics Algebra Mathematics is a language. The Meaning of Numbers Algebra Choosing the Correct Equation(s) 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 Algebra Mathematics is a language. The Meaning of Numbers Algebra Choosing the Correct Equation(s) 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 Algebra The Meaning of Numbers Algebra You need to remember algebra. Here are some of the basics… If Distributive property then Choosing the Correct Equation(s) If Commutative properties then …and similarly for square roots, squares, subtraction and division except that subtraction and division are NOT commutative! 5

Mathematics Algebra Some simple examples… The Meaning of Numbers Algebra If Distributive property then Choosing the Correct Equation(s) If Commutative properties then …and similarly for square roots, squares, subtraction and division except that subtraction and division are NOT commutative! 6

Mathematics Algebra Here are some other helpful concepts… The Meaning of Numbers Algebra Choosing the Correct Equation(s) If and Ratios If and then …even if YOU DON’T KNOW C! Simultaneous Equations then …even if YOU DON’T KNOW C! + 7 usually we use this in such a way that one of the coefficients is zero

Mathematics Algebra Here are some more simple examples… The Meaning of Numbers Algebra Choosing the Correct Equation(s) If and Ratios If and then …even if YOU DON’T KNOW m or a 1! Simultaneous Equations then …even if YOU DON’T KNOW L! + 8 usually we use this in such a way that one of the coefficients is zero

Mathematics Algebra 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 Choosing the Correct Equation(s) Still, if you follow a few basic steps, the difficulty will be far less and you will need to spend much less time on Pre. Assignments, 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. 9 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 Algebra An example of choosing the correct equations. The Meaning of Numbers Algebra 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? If you look in your textbook, you will find the equation 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. 10 Let’s try this …

Mathematics Algebra 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? 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. Which one do we choose? To answer this, we must ask ourselves two things. 11 1. What physical quantity are we looking for? 2. What object is that physical variable related to? For this problem, the answers are… 1. Impulse 2. The ball

Mathematics Algebra 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? Choosing the Correct Equation(s) 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 12 . If we plug this into the homework software, it will tell us we are incorrect. What went wrong.

Mathematics Algebra 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? Choosing the Correct Equation(s) 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 doeit ar ect! properly in this case, we find that… w t 13 Rule #2: Use the proper units. of corr s rk e in o w e ar e om at w h e th h t s t y u Now a B ls s what l i t ? s

Mathematics Algebra 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? Choosing the Correct Equation(s) 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… 14 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 Algebra 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 #3: Know the meaning of each variable. Choosing the Correct Equation(s) 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. and 15 But which one should we use?

Mathematics Algebra An example of choosing the correct equations. The Meaning of Numbers Algebra 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? Rule #4: Use what is 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. 16 The other equation needed force and time, neither of which is known.

Mathematics Algebra 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? Choosing the Correct Equation(s) 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, Whic h But the homework software stills says that we are incorrect! 17 leads to a final princ iple

Mathematics Algebra An example of choosing the correct equations. The Meaning of Numbers Algebra 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? 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). answer !!!! 18

Mathematics Algebra 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. Choosing the Correct Equation(s) Rule #3: Know the meaning of each variable. 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. 19 This is what DR. Mike means when he says you must use concepts to solve problems in physics.