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Mathematics Algebra 1 Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Trig and Geometry Mathematics is a language. It is used to describe the world around us. Can you tell me what this means? 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… “The total torque acting on an object is the same as its moment of inertia multiplied by its angular acceleration.”

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Mathematics Algebra 2 Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Trig and Geometry Mathematics is a language. 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 torques, is moment of inertia and is angular acceleration. means to sum all of what is behind it for every value of from 1 to. But you still do not know the meaning of torque, moment of inertia and angular acceleration.

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Mathematics Algebra 3 Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Trig and Geometry Mathematics is a language. 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.

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Mathematics Algebra 4 Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Trig and Geometry Mathematics is a language. 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. If I plug in the numbers as an example, I only learn ONE of the situations!!

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

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

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

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

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9 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry cos sin 45 o 30 o 150 o 330 o III IIIIV This is the unit circle… It axes are sine and cosine All lines drawn here have a length of 1 and an angle equal to the angle we are working with. The height along the sin- axis is the sine of the angle. The distance to the right on the cos-axis is the cosine of the angle. Here are some other examples. Note that the angle always goes from the positive cos-axis counterclockwise. Also note that the cosine is negative if the line is drawn to the left on the cos-axis. Why is the sine negative here? We often speak of four quadrants. The first quadrant has positive cosines and sines. The second quadrant has negative cosines and positive sines. The third quadrant has negative cosines and sines. The fourth quadrant has positive cosines and negative sines.

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10 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry Given a right triangle, the trigonometric functions for either non-right angle are given by the following… θ hypotenuse ( h ) opposite ( o ) adjacent ( a ) The value of the angle can also be determine by using any two of the sides. For example,

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11 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry Here is an example of how to use it… θ =36.87 o h =5 o =3 a =4 The value of the angle can also be determine by using any two of the sides. For example, Note: This is NOT drawn to scale!

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12 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry Here are some useful angle relations… a a a b a a a a b b b b b b a b c A C a c B b a a

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13 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry For example… a b

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14 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry Here are some basic geometric and trigonometric formulae which we will use often in this and the next class… Circumference of a Circle Area of a Circle Surface Area of a Sphere Volume of a Sphere Surface Area of a Cylinder (not including end faces) Volume of a Cylinder Trigonometric Formulae Quadratic Formula

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15 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry Finding out which equation or set of equations to use while solving a problem in physics is the most difficult part of the process. It is also the most crucial part! 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?

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16 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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. Let’s try this …

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17 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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? 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. 1.What physical quantity are we looking for? 2.What object is that physical variable related to? 1.Impulse 2.The ball For this problem, the answers are…

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18 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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? 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. If we plug this into the homework software, it will tell us we are incorrect. What went wrong.

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19 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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? 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… Rule #2: Use the proper units. But the homework software stills says that we are incorrect! Now what?

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20 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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? 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… 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!!!!

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21 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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? 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 But which one should we use? Rule #3: Know the meaning of each variable.

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22 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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? 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. The other equation needed force and time, neither of which is known. Rule #4: Use what is known and unknown to sort out equations that are not useful.

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23 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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? 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, But the homework software stills says that we are incorrect! Which leads to a final principle

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24 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry An example of choosing the correct equations. 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? The arrows on top of the variables J and v tell us that they are vectors. When we subtract the initial from the final velocity, we must also take into account their direction. (One is north, the other south). Rule #5: Vectors! Gotta use vectors!!! Which is the correct answer !!!!!!!!

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25 Algebra Trigonometry Geometry The Meaning of Numbers Choosing the Correct Equation(s) Mathematics Algebra Trig and Geometry In summary.... Rule #1: We must know which object we are considering in a problem. Rule #2: Use the proper units. 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. This is what DR. Mike means when he says you must use concepts to solve problems in physics.

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