PHYSICS 1 REVIEW PACKET

Transcription

1 PHYSICS 1 REVIEW PACKET Powers of Ten Scientific Notation and Prefixes Exponents on the Calculator Conversions A Little Trig Accuracy and Precision of Measurement Significant Figures Motion in One Dimension Vectors

2 Standards of measurement: Physics I Review Packet mks cgs British Length meter (m) centimeter (cm) foot (ft) Mass Kilogram (Kg) gram (g) slug (sl) Time seconds (s) or (secs) seconds (s) or (secs) seconds (s) or (secs) Powers of 10: The exponent = the number of zero s after the 1 (in 10 5, the exponent = 5) 10 0 = 1* *Any number raised to the 0 power = 1 (5 0 = 1, = 1) 10 1 = =.1 10 = = = =.001 Prefixes: Very small femto- (f) 1 x Examples pico- (p) 1 x 10-1 nano- (n) 1 x nm = 10 nanometers = 10 x 10-9 m = m micro- (µ) 1 x 10-6 milli- (m) 1 x 10-3 centi- (c) 1 x cg = 7. centigrams = 7. x 10 - grams =.07 g deci- (d) 1 x 10-1 deka- (da) 1 x 10 1 hecto- (h) 1 x 10 kilo- (k) 1 x km = 5.1 kilometers = 5.1 x 10 3 meters = 5,100 m mega- (M) 1 x 10 6 giga- (G) 1 x 10 9 tera- (T) 1 x 10 1 Very Large peta- (P) 1 x Pb = 3 petabytes = 3,000,000,000,000,000 bytes Scientific Notation: A way to write very large or very small numbers using powers of 10 Examples: 1.45 x 10 6 = 1.45 x 1,000,000 = 1,450, x 10-4 = ,000 = Converting from Scientific Notation: Move decimal the number of places shown by the exponent Positive exponent: move decimal to the right Negative exponent: move decimal to the left 3.7 x 10 4 = = 3, x 10-3 = =.00461

3 Converting to Scientific Notation: Move the decimal until there is only number before it. The number of places moved is the exponent. moving decimal left, the exponent is positive moving decimal right, the exponent is negative = = x = = x 10-3 Exponents on the Calculator: Easiest Method: Most calculators have an EE button or something equivalent. The EE means x 10^ Example: To enter 3.7 x 10 5, type in 3.7 EE 5 To enter 1.6 x 10 -, type in 1.6 EE - Second option: When all else fails, just type it in as you see it. You will need to use the ^ key which means raised to the power of Example: To enter 3.7 x 10 5, type in 3.7*10^5 To enter 1.6 x 10 -, type in 1.6*10^- Conversions: Units can be converted from one to another when a conversion factor is known. Cancel units on the diagonal. Multiply across the top and divide by what s on the bottom. Examples: 1m 1. Convert 45 cm to m (1m = 100cm) 45 cm 100 cm = 4.5 m 1slug. Convert 35 kilograms to slugs (14.6 kg = 1 slug) 35kg 14. 6kg =.397 slugs =.4 slugs 3. Convert 5 days to seconds (1 day = 4hrs, 1 hr = 60 min, 1 min = 60 secs) 4hours 5 days 1day 60min 60secs = 43,000 seconds 1hour 1min 4. Convert 7 mph to Km/min (1mile = Km, 1 hour = 60 min) miles 7 hour 1.609Km 1mile 1hour 60min Km Km = = 0.7 min min

4 A Little Trig: Trigonometry involves right triangles only. Trig Functions: opposite sin = hypotenuse adjacent cos = hypotenuse opposite tan = adjacent csc = hypotenuse opposite sec = hypotenuse adjacent adjacent cot = opposite c a Opposite = side opposite the angle (side a for this triangle) Adjacent = side touching the angle (side b for this triangle) Hypotenuse = longest side (side c for this triangle) b Pythagorean Theorem: (hypotenuse) = (opposite side) + (adjacent side) for the above triangle: c = a + b Examples: 1. What are the lengths of side b and c in the given triangle? c sin35 = so c = 5sin35 = b 35 5 cos35 = 5 b so b = 5cos35 = 0.5 c. What is the length of side a, and the angle θ in the given triangle? a = = a = 146 a = 146 = 1.1 tan = 5 11 = tan = a 5

5 Measurement Accuracy vs Precision: Accuracy = How close your measured values are to the actual value Precision = How close your repeated measurements are to each other Significant Figures (Significant Digits): Refers to the reliably known digits of any measurement. The number of significant digits depends on the precision of the measurement tool. Example: Using the ruler shown, the arrow measures.9 cm. The ruler has a minimum interval of 0.1cm The last number () is an estimate cm.9 cm has 3 significant figures. All measurements made with this ruler will have significant digits after the decimal. (Thus if the arrow fell directly on the, the measurement would be.00 with 3 significant figures.) Rules of Significant Figures 1. All nonzero digits are significant: Example: 1.34 g has 4 significant figures. Zeroes between nonzero digits are significant: Example: 100 kg has 4 significant figures 3.07 ml has 3 significant figures. 3. Leading zeros are not significant. They are used to place the decimal point. Example: o C has 1 significant figure 4. Trailing zeroes that are also to the right of a decimal point are significant Example: ml has 3 significant figures 5. When a number ends in zeroes that are not to the right of a decimal point, the zeroes may or may not be significant. Example: 190 miles may be or 3 significant figures 6. When using scientific notation, all digits shown are significant. Example: has significant figures, but has 3 significant figures Rules for multiplying/dividing: The result should have the same number of significant digits as the measurement with the least amount of significant digits. Example: 3.5in x 15.15in = in = 355 in Since 3.5 has 3 significant figures, our answer can have no more than 3 significant figures as well. Rules for adding/subtracting: The result is rounded to the same number of decimal places as the measurement with the least amount of decimal places. Example: 3.5in in = 38.65in = 38.6in Since 3.5 has 1 place after the decimal point, the answer must also be rounded to 1 place after the decimal point. *Note: sometimes professors have their own rules for measurements so be careful!

6 Motion in One Dimension Distance: How far an object travels without regard to direction. The odometer on your car measures distance. Units: cm, m, miles, ft, etc Example: I traveled 15 miles Displacement ( x): Straight line distance between your starting point and your ending point. Units: cm, m, miles, ft, etc. Example: I traveled 15 miles North Speed: How fast an object travels without regard to direction. The speedometer on your car measures speed. Units: m/s, cm/s, ft/s, mph, etc. Example: John ran at 3 m/s Average Speed: average speed = total distance time Velocity (v): The speed of an object when moving along a straight line. Direction matters. Units: m/s, cm/s, ft/s, mph, etc. Example: John traveled north at 3 m/s Average Velocity (v ): average velocity = displacement time or v = x x = x t t t1 1 Instantaneous Velocity: The velocity of an object at a particular point in time. Example: A cars velocity at 3.s is 5 ft/s. Constant Velocity: Velocity doesn t change. Acceleration is 0. x = vt Acceleration (a): The rate at which velocity changes with time. Units: m/s, cm/s, or ft/s Average Acceleration ( a ): a = changein velocity = time v v v = t t t 00 Where: v 0 = Initial Velocity (velocity at t = 0) v = Final Velocity t t 0 = Elapsed time Constant Acceleration: When acceleration doesn t change over time use your Kinematic equations. Kinematic Equations: v = v 0 + at v = v 0 + a x x = v 0 t + 1 at v v x = 0 t Freefall: When air drag is neglected, any object traveling through the air without a jet pack or booster rockets falls with the same acceleration, gravity. On Earth, gravity = 3 ft/s = 9.8 m/s = 980 cm/s

7 To Solve Motion Problems: 1. Draw a picture. Write down the value of all the variables you know. 3. Write down what variable you are looking for. 4. Choose a direction to be positive (+) and make sure your values have the proper sign. 5. Find the equation or equations that best suit your needs. Sometimes you will need to use two equations to get one answer. Motion in one Dimension Examples: 1. It took me hours to run 4 laps around a ¼ mile track. If I finish at the same point at which I started, a) What is my distance traveled? Distance = 4(¼ mile) = 1 mile b) What is my displacement? 0 c) What is my average speed? 1 mile Avg speed = =.5mph hrs d) What is my average velocity? 0 v = = 0 hrs. If I travel at a constant ft/s for 1 minute, how far have I traveled? x = s ft (60s) = 13 ft 3. A penny is dropped from the top of a building. a) How fast is it falling in 1second, seconds, 3 seconds, in ft/s? Solution: The penny falls with an acceleration of gravity = 3 ft/s. This means the penny increases it s velocity by 3 ft/s every second. So, v 1 = 3 ft/s, v = 64 ft/s, v 3 = 96 ft/s. b) If the penny lands in 3 seconds, how tall is the building? Solution: We know acceleration = gravity = 3 ft/s down, time = 3s, v = 96 ft/s and since the ball is dropped, we know v 0 = 0. Take down to be +. x = v 0 t + 1 at = 1 (3ft/s )(3) = 144 ft ( I could have also used v = v 0 + a x) 4. A ball is thrown up with an initial velocity of 50 m/s from the top of a 10 m cliff. a) How long does it take to reach the peak of it s trajectory? Solution: Take up to be +. v 0 = 50 m/s, a = g = -9.8 m/s, v = 0 v = v 0 + at 0 = 50m/s (9.8m/s )t t = 5.1s b) How fast will the ball be traveling just prior to impact if it lands at the base of the cliff? Solution: Take up to be +, v 0 = 50 m/s, a = -9.8 m/s, x = -10 m (- since below your start) v = v 0 + a x v = (50) + (-9.8m/s )(-10m) v = 69.7 m/s. c) How long will the ball be in the air? Solution: Take up to be +. v 0 = 50m/s, v= m/s, a = -9.8 m/s. v = v 0 + at -69.7m/s = 50m/s (9.8m/s )t t = 1.s

8 Vectors Scalar: Has magnitude only, no direction. Examples: Time = 3sec, Temperature = 5 F, speed = 60mph, distance = 80 feet Vector: Has both magnitude and direction. Often drawn as an arrow. In many physics books, vectors are either in bold, or have an arrow over their symbol. Example: Velocity = ran 30 meters east, Displacement = walked 50 miles west Resultant: The result of adding two or more vectors together. Components of a Vector: All vectors can be broken up into vertical and horizontal components using trigonometry. Definitions: Vector vertical component or y-component horizontal component or x-component Examples: 1. What are the components of a vector 56 miles at 40 North of East? 56 miles y-comp = 56sin40 = 36 miles 40 x-comp = 56cos40 = 43 miles. What are the components of a vector 100 miles at 50 South of West? y-comp = 100sin50 = 77 miles x-comp = 100cos50 = 64 miles miles ** Do not assume that y is always sin and x is always cos. It depends on the placement of the angle as shown in the earlier trig exercise.

9 Vector Addition: Vectors can be added either graphically or algebraically. Given the following vectors: 15 Adding Vectors Graphically: To add vectors graphically, you must measure and find the angle of the resultant vector. All vectors must be drawn to scale at the proper angle. The vectors must be placed tip to tail. The order in which you add the vectors does not matter. The resultant vector connects your starting point to your ending point. Example: Re-arrange the given vectors by placing them tip to tail as shown. All angles must remain the same and the length of each vector must be drawn to scale. 4 cm 5 cm 4 cm cm 8 cm If you draw this out, you should measure a Resultant of about 1.5cm at an angle of 5 ( measured with a protractor) Resultant 5 cm (units were added here for convenience) 8 cm 6 cm Adding Vectors Algebraically: To add vectors algebraically, you need to add the x and y components of each vector. Then use Pythagorean theorem to determine the length of the resultant, and inverse tangent to determine the angle of the resultant. Example: Tabulate the x and y components of each vector. Sum up each column. The totals are the components of the Resultant vector. Use those components to determine the magnitude and direction of the Resultant vector. x-component y-component 6 cos(0 ) = 6 6 sin(0 ) = 0 8 cos(30 ) = sin(30 ) = 4-5 sin(15 ) = cos(15 ) = cos(15 ) = sin(15 ) = cm 9.87 cm Resultant x-comp = 7.78cm *Not drawn to scale y-comp = 9.87cm Resultant = = tan = = 1.57cm Resultant = (xcomp) = tan -1 ycomp xcomp (ycomp)