Physics Introductory Unit ~The Mathematical Background~

Math Skills There are several skills, some of which you have already learned, that you will need to use extensively in Physics. These include the following: –Algebra (manipulation of formulas) –Scientific Notation (very lg/sm numbers) –Significant Digits –Unit Conversions Math Rules!

Scientific Notation Scientific notation relies on exponential powers of ten (10 x ) to simplify extremely large and small numbers. In all cases, numbers written in scientific notation have a single digit in the ones place followed by the remaining digits placed to the right of the decimal point. This is called the coefficient. A multiplied power of ten is indicated afterwards. Standard Notation Scientific Notation Coefficient Power of Ten

Scientific Notation (Cont.) Large numbers correspond to positive powers of ten. Small numbers correspond to negative powers of ten. Figuring out the power on the ten relates to how many places you need to move the decimal point from its initial position. 4 Moves 2 Moves

Scientific Notation (Multiplication) At times, numbers in scientific notation will be multiplied as shown below. The trick is to combine the powers of ten with each other and the non-exponent terms with each other. Then simplify. Note: Remember that exponents add when like bases are multiplied.

Scientific Notation (Division) At times, numbers in scientific notation will be divided as shown below. As before, you need to combine terms. The exponent rule changes to subtraction when division is involved.

Scientific Notation (10 x ) Numbers that are simply powers of ten can be written in a shorter form without a coefficient. Consider the example dealing with 100,000. In simplified form it can be written as follows: The same holds true for small numbers.

Significant Digits Significant digits (sometimes called significant figures) are those digits that are considered important in a given number. In order to determine which digits are significant, one must look to the following rules. –All nonzero digits are significant. –Final zeros after the decimal point are significant. –Zeros between other significant digits are significant. –Zeros used solely for spacing are not significant.

Significant Digits (Special Cases) A bar can be placed over zeros that are not normally significant in order to make them significant. This usually occurs after some instances of rounding. Here a problem would specify to how many digits you must round. 1 Significant Digit3 Significant Digits1 Significant Digit2 Significant Digits

Significant Digits (Rounding) Instead of rounding to a place, you round a number to a specified number of significant digits. This is done by rounding up or rounding off the number that would constitute an extra place. Round the number to 3 significant digits. –How many significant digits does the number have? –Which digit must be rounded? –Round up or off? Round the number 6798 to three significant digits. Round Off!

Keeping correct significant digits while multiplying and dividing relies on the same process. –Count the number of significant digits in each of the numbers being multiplied or divided. –Calculate and round your answer to the number of significant digits found in the least significant input. –It is sometimes easier to write these problems horizontally. Significant Digits (Mult/Div) 2 3 Multiplying Dividing 41

Significant Digits (Add/Sub) Adding and subtracting rely on the same process when significant digits are being kept. –Align the addends (for addition) or the minuends and subtrahends (for subtraction) vertically. –Draw a vertical line down the least precise number (the one with least decimal places). –Add or subtract the values. –Round to the left of the vertical line. –Addition problems can have more than two numbers. Addition Subtraction

Units and Unit Conversion Anthony jumped in his car and drove 10 to the grocery store, where he bought 5. He returned within 30. WARNING: You will lose points for any answer that does not have proper units!!!

Units and Unit Conversion In this class we will use the MKS system. M  meter (m) … unit for length K  kilogram (kg) … unit for mass S  second (s) … unit for time All other units are derived units … they come from the 3 above. Standard Units

Unit Conversions We can multiply any number by 1 and not change its value.. How many m are there in 5783cm?

Practice Problem.

Practice Compound Problem.

Algebra Numerous times while studying Physics, you will be required to use algebra to solve equations. Isolating the variable involves the use of inverse order of operations to manipulate the variables. –Addition(+) and Subtraction(-) are inverse operations. –Multiplication(× or ·) and Division(÷) are inverse operations. –Squaring( 2 ) and square rooting(√) are inverse operations.

Find the value of x When solving for the value of an equation, you must use ORDER OF OPERATIONS Parenthesis (Grouping) Exponents / Powers Multiplication Division Addition Subtraction

When solving for a variable in an algebraic equation, you must use INVERSE ORDER OF OPERATIONS 1) Collect like terms 2) Addition / Subtraction 3) Move variable from denominator to the numerator a) Cross multiply b) Reciprocal c) Multiply both sides by the variable 4) Multiplication / Division 5) Exponents 6) Parenthesis (Grouping)

Algebra (Sample) Consider the formula shown. Solve the equation in terms of d. To do this, we must move t. What operation is t associated with? Division What is the inverse operation? Multiplication Perform the operation to solve for d. Some other problems may involve more than one step.

Other Algebra Samples Given the equation: Solve for t. Given the equation: Solve for v 2. Note: When you take the square root, a symbol must be included in front of the radical.

Unit Conversions

Conclusion Physics is a math-based science course. All four major skills will come into use during the course of the year, many as early as next section.