Physics 521 Math Review

SCIENTIFIC NOTATION Scientific Notation is based on exponential notation (where decimal places are expressed as a power of 10). The numerical part of the measurement is expressed as a number between 1 and 10 multiplied by a whole-number power of 10. M * 10 n, 1≤ M < 10, where n is an integer (+ or - #). Standard Notation 2000 → 2 * 10 3 Standard Notation 180 g → 1.8 * 10 2 g or 1.8 * kg

SIGNIFICANT FIGURES Significant Figures - The number of digits is rough but useful indication of a measurements precision. Each digit obtained as a result of measurement is a significant figure. The last digit of each measured quantity is always estimated. The zeros in a number warrant special attention. A zero that is the result of a measurement is significant, but zeros that serve only to mark a decimal point are not significant.

Example: A)65 ml(2 sig figs) B)173.4 (4 sig figs) C)13.2 g (3 sig figs) D)5 ml (1 sig figs)

Rules for Significant Figures 1.Non-zero digits are always significant. Ex.A) L(4 sig figs)B) kg(4 sig figs) 2.A zero between other SF is significant. Ex.A) 1.05 (3 sig figs) B) 2001 m (4 sig figs)

3. Final zeros to the right of the decimal point are significant Ex.A) 6.30 g(3 sig figs)B) ml(4 sig figs) 4. Initial zeros are not significant and serve only to show place of decimal. Ex.A) km(2 sig figs)B) (3 sig figs)

5.Final zeros in numbers with no decimal point may or may not be significant. A) 20 marbles*count, exact, infinite* B) 2000 m*1* C) 20 lbs*1*precise from D) 2.0 x 10¹ lbs*2* precise from E) 2.00 x 10¹ lbs*3*precise from F) 1 km = 1000 m *definition, infinite* Example:

200 lbs*1* significant figures... better written 2 * 10 2 lbs 200 lbs*2* significant figures... better written 2.0 * 10 2 lbs 200 lbs*3* significant figures... better written 2.00 * 10 2 lbs Interpretations of Significant Figures

COUNTS, CONSTANTS, DEFINITIONS All have an infinite number of significant figures.(∞) COUNT - Ex. 10 marbles, 3 people.... Exact CONSTANTS - Ex. Consider a + 2b = c. The number 2 is a constant. DEFINITIONS - Ex. 1 km = 1000 m, 12 = 1 dozen

SIGNIFICANT FIGURE CALCULATIONS The result of any mathematical calculation involving measurements cannot be more precise than the least precise measurement. (Assume all the numbers below are from a measuring)

Addition and Subtraction When adding or subtracting measured quantities, the answer should be expressed to the same number of decimal places as the least precise quantity used in the calculation. ( If needed use a LINE OF SIGNIFICANCE to aid in solving these.) A) = = Example: B) = = 1.70 C) = 7230 = 7230 D) = 107 E) = 1000 = * 10 3 F) = 1049 = 1.0 * 10 3

Multiplication, Division, and Square Root When multiplying, dividing, or finding the square root of measured quantities, the answer should have the same number of significant digits as the least precise quantity used in the calculation. Example: A) * 0.020= = 0.11 B)2500 * 2= 5000 C)26.3 * 35= = 920

ROUNDING When completing calculations, do not round any of the intermediate answers on your way to finding a solution to a problem. The only rounding that should occur is in the final answer that is being reported. Example: Now, at the very end round to 1 sig fig Answer is “1”

INVERSELY AND DIRECTLY PROPORTIONAL When considering what effect changing one or more variables has on another variable in mathematical relationships inversely and directly proportional relationships are used. C ↑ A↑, or C↓ A↓, if C doubles then A doubles D ↑ C↓, or D↓ C↑, if D doubles then C becomes half C ∝ A, C ∝ 1/D

Directly Proportional Quantities Quantities that are directly proportional to one another occupy the same position on opposite sides of the proportion sign (either both located in the numerator position or both in the denominator position). Quantities that are directly proportional to one another increase or decrease by the same factor. AX = ZP Z & X are directly proportional to each other. ( Z α X )

Inversely Proportional Quantities Quantities that are inversely proportional to one another change by the reciprocal of one another (or 1/x of one another). In a proportion, quantities that are inversely proportional to one another occupy opposite positions on opposite sides of the proportion sign. Z & M are inversely proportional to each other. ( Z α 1/M)

Example: Given the following formula, what would happen to v if r is doubled and T is tripled? Answer: v would change by a factor of,

Example: Given the following formula, what would happen to m c if T is changed by a factor of 2 and G by a factor of ½? Answer: v would change by a factor of,

Unit Analysis Often we need to change the units in which a physical quantity is expressed. For example we may need to change seconds and minutes, hours, days or even years, to do this we use conversion factors. Example:and When a quantity is multiplied by conversion factor it does not change the amount of quantity just the units the quantity is measured in. When a conversion factor is evaluated its value is equal to 1. Any number multiplied by 1 remains unchanged.

Example: Second are cancelled which leaves units of minutes Answer: (3 min)

Example: How many centimetres are in 1 km? 1*10 5 cm

Example: How many seconds are in one day? sec

Example: Convert 2.4 km/hr to m/s 0.666… m/s

The metric system

Re-arranging formulas Solve for q: Given the following formula,

Solve for T:

Given the following formula, Solve for v f :