P PHYSICS SUMME SSIGNMENT: Calculators allowed! 1 The Metric System Everything in physics is measured in the metric system. The only time that you will see English units is when you convert them to metric units. The metric system is also called SI. In the SI system fundamental quantities are measured in meters, kilometers, and seconds. The equations in physics depend on unit agreement. So you must convert in most problems to arrive at the correct answer. Name of prefix Numerical value bbreviation Pico- 10-12 p Nano- 10-9 n Micro- 10-6 µ Milli- 10-3 m Centi- 10-2 c Kilo- 10 3 k Mega- 10 6 M Giga- 10 9 G This is the same thing as stoichiometry in chemistry. Show all work. Use another sheet of paper if necessary. a. 4008 g = kg b. 1.2 km = m c. 823 nm = m d. 298 K = o C e. 0.77 m = cm f. 8.8x10-8 m = mm g. 1.2 atm = Pa h. 25.0 m = m i. 2.65 mm = m j. 8.23 m = km k. 5.4 L = m 3 l. 40.0 cm = m m. 6.23x10-7 m = nm n. 1.5x10 11 m = km Significant Figures When using a measuring device, you MUST estimate between the smallest marks on the instrument. For example, if a ruler is marked off in increments of whole millimeters, you estimate the length of an object to the closet tenth of a millimeter. In laboratory work, you are only as precise as the tool you use. Precision is also important in labs and when solving problems. In Physics, the same thing is true you can not round numbers at will. You must obey the rules for significant figures. Example Non-zero numbers are significant. 123 245 g 6 Zeros between non-zero numbers are significant 12 027 m 5 Zeros at the end of a number to the right of the decimal are significant Zeros in front of non-zero numbers are not significant Zeros at the end of the number to the left of the decimal are not significant unless they were measured. Use scientific notation for clarity. 23.00 kg 4 0.0502 3 1000 m 1.00x10 3 m # of Significant Figures Unknown, could be 1 to 4 3, zeros would not be shown unless they were measured
P PHYSICS SUMME SSIGNMENT 2 When adding or subtracting numbers, the precision of the answer can be no greater than the precision of the least precise value. 9 7. 3 (Least precise value, your answer will be rounded to the same place as this value) 4. 3 2 + 0. 1 4 7 1 0 1. 7 6 7 rounded to the nearest 1/10 th so final answer is 101.8 When multiplying or dividing numbers, the final answer has the same number of significant figures as the measurement having the smallest number of significant figures. 9. 8 1 3 significant figures x 0. 0 0 5 3 2 significant figures 0. 0 5 1 9 9 3 rounded to 2 significant figures = 0.052 Note that significant figures rules are a guideline to determine precision of calculations. Write the following in standard scientific notation. Show all work. Use another sheet of paper if necessary. a. 20,000 g. 543.6 b. 0.000005 c. 0.00055 d. 40,230,000 e. 34.5x10 5 f. 180x10-1 h. 34.1 i. 0.00345 j. 628,000 k. 0.004x10 2 l. 0.72x10-2 Solve the following problems, paying close attention to significant figures. a. (2x10 9 ) (4x10 3 ) b. (0.10x10 5 ) (4.9x10-2 ) c. (6.2x10-3 ) (1.5x10 1 ) d. (88x10 2 ) (0.15x10 4 ) e. 0.40x10-4 5.7x10 1 f. 10 8 x10-3 10-4 x10-5 g. 753x10 6 0.0300x10-8 h. 8.8x10 6 2.2x10 1 lgebra asics Solve the following: a. (x-3) 2 + 7 = 16 b. ¼ + 1/x = 1/3 c. 3x 2y = 8 y x = 2 Geometry asics d. 5x + y = -8 2x 2y = 4 e. x 2 + 5x + 4 = 0 f. 2x 2 3x 4 = 0 g. log (x/3) = 0.301 h. 10 x = 31.62 a. Line touches the circle at a single point. Line extends through the center of the circle. i. What is line in reference to the circle? ii. How large is the angle between lines and?
P PHYSICS SUMME SSIGNMENT 3 b. What is angle C? C 30 o 45 o c. What is angle? 30 o d. The radius of a circle is 5.5 cm, i. What is the circumference in meters? ii. What is its area in square meters? e. What is the area under the curve at the right? 4 i. Using the generic triangle to the right, ight Triangle Trigonometry and Pythagorean Theorem solve the following. Your calculator must be in degree mode. 12 20 a. = 55 o and c = 32 m, solve for a and b. b. = 45 o and a = 15 m/s, solve for b and c. c. b = 17.8 m and = 65 o, solve for a and c. d. a = 250 m and b = 180 m, solve for and c. e. a =25 cm and c = 32 cm, solve for b and. f. b =104 cm and c = 65 cm, solve for a and. Vectors Most of the quantities in physics are vectors. This makes proficiency in vectors extremely important. Magnitude: Size or extent. The numerical value. Direction: lignment or orientation of any position with respect to any other position. Scalars: physical quantity described by a single number and units. quantity described by magnitude only. Examples: time, mass, and temperature
P PHYSICS SUMME SSIGNMENT 4 Vector: physical quantity with both a magnitude and a direction. directional quantity. Examples: velocity, acceleration, force Notation: or Length of the arrow is proportional to the vectors magnitude. Negative Vectors Direction the arrow points is the direction of the vector. Negative vectors have the same magnitude as their positive counterpart. They are just pointing in the opposite direction. Vector ddition and subtraction Think of it as vector addition only. The result of adding vectors is called the resultant. + = So if has a magnitude of 3 and has a magnitude of 2, then has a magnitude of 3+2=5. When you need to subtract one vector from another think of the one being subtracted as being a negative vector. Then add them. is really + = negative vector has the same length as its positive counterpart, but its direction is reversed. So if has a magnitude of 3 and has a magnitude of 2, then has a magnitude of 3+(-2)=1. This is very important. In physics a negative number does not always mean a smaller number. Mathematically 2 is smaller than +2, but in physics these numbers have the same magnitude (size), they just point in different directions (180 o apart). There are two methods of adding vectors Parallelogram + Tip to Tail - - + - - It is readily apparent that both methods arrive at the exact same solution since either method is essentially a parallelogram. It is useful to understand both systems. In some problems one method is advantageous, while in other problems the alternative method is superior.
P PHYSICS SUMME SSIGNMENT 5 1. Draw the resultant vector using the parallelogram method of vector addition. Example b. d. a. c. e. 2. Draw the resultant vector using the tip to tail method of vector addition. Label the resultant as vector Example 1: + - Example 2: a. X + Y d. C D X Y C D b. T S T S e. + + C C c. P + V f. C P V C
P PHYSICS SUMME SSIGNMENT 6 In physics a coordinate axis system is used to give a problem a frame of reference. Positive direction is a vector moving in the positive x or positive y direction, while a negative vector moves in the negative x or negative y direction (This also applies to the z direction, which will be used sparingly in this course). What about vectors that don t fall on the axis? You must specify their direction using degrees measured from East. Component Vectors resultant vector is a vector resulting from the sum of two or more other vectors. Mathematically the resultant has the same magnitude and direction as the total of the vectors that compose the resultant. Could a vector be described by two or more other vectors? Would they have the same total result? This is the reverse of finding the resultant. You are given the resultant and must find the component vectors on the coordinate axis that describe the resultant. + y + y + x or + x ny vector can be described by an x axis vector and a y axis vector which summed together mean the exact same thing. The advantage is you can then use plus and minus signs for direction instead of the angle. 3. For the following vectors draw the component vectors along the x and y axis. a. c. b. d. Obviously the quadrant that a vector is in determines the sign of the x and y component vectors. Trigonometry and Vectors Given a vector, you can now draw the x and y component vectors. The sum of vectors x and y describe the vector exactly. gain, any math done with the component vectors will be as valid as with the original vector. The advantage is that math on the x and/or y axis is greatly simplified since direction can be specified with plus and minus signs instead of degrees. ut, how do you mathematically find the length of the component vectors? Use trigonometry.
P PHYSICS SUMME SSIGNMENT 7 cos adj hyp sin opp hyp 40 o 10 40 o 10 x y adj hyp cos opp hypsin x hyp cos y hyp sin x o o 10cos 40 y 10sin 40 x 766. y 643. 4. Solve the following problems. You will be converting from a polar vector, where direction is specified in degrees measured counterclockwise from east, to component vectors along the x and y axis. emember the plus and minus signs on you answers. They correspond with the quadrant the original vector is in. Hint: Draw the vector first to help you see the quadrant. nticipate the sign on the x and y vectors. Do not bother to change the angle to less than 90 o. Using the number given will result in the correct + and signs. The first number will be the magnitude (length of the vector) and the second the degrees from east. Your calculator must be in degree mode. Example: 250 at 235 o x hyp cos 235 o x 250cos 235 x 143 o d. 7.5x10 4 at 180 o 250 a. 89 at 150 o y hyp sin y 250sin 235 o y 205 e. 12 at 265 o b. 6.50 at 345 o f. 990 at 320 o c. 0.00556 at 60 o g. 8653 at 225 o
P PHYSICS SUMME SSIGNMENT 8 5. Given two component vectors solve for the resultant vector. This is the opposite of number 11 above. Use Pythagorean Theorem to find the hypotenuse, then use inverse (arc) tangent to solve for the angle. 2 2 2 Example: x = 20, y = -15 x y tan opp adj 20-15 x y 25 2 2 2 2 20 15 tan tan 1 1 opp adj y x o o o 360 36.9 323.1 a. x = 600, y = 400 d. x = 0.0065, y = -0.0090 b. x = -0.75, y = -1.25 e. x = 20,000, y = 14,000 c. x = -32, y = 16 f. x = 325, y = 998
P PHYSICS SUMME SSIGNMENT 9 Graphing and Graph Interpretation You should be familiar with graph construction (by hand and on a calculator). This is a topic that often appears on P exams and is an easy way to score points on any assignment. Fill in the following table and plot the points of the grid below as distance versus time. e sure to correctly label the graph (axis labels, units, and title). Draw the best fit line through your data points. Use a calculator to plot the graph. ecord the equation of the best-fit line. What is the slope of the line that you plotted? Time, t(s) Distance, d (m) 0.0 0 1.0 5.1 2.0 9.9 3.0 15.2 How are vectors used in Physics? They are used everywhere! Speed Speed is a scalar. It only has magnitude (numerical value). v s = 10 m/s means that an object is going 10 meters every second. ut, we do not know where it is going. Velocity Velocity is a vector. It is composed of both magnitude and direction. Speed is a part (numerical value) of velocity. v = 10 m/s north, or v = 10 m/s in the +x direction, etc. There are three types of speed and three types of velocity Instantaneous speed / velocity: The speed or velocity at an instant in time. You look down at your speedometer and it says 20 m/s. You are traveling at 20 m/s at that instant. Your speed or velocity could be changing, but at that moment it is 20 m/s. verage speed / velocity: If you take a trip you might go slow part of the way and fast at other times. If you take the total distance traveled divided by the time traveled you get the average speed over the whole trip. If you looked at your speedometer from time to time you would have recorded a variety of instantaneous speeds. You could go 0 m/s in a gas station, or at a light. You could go 30 m/s on the highway, and only go 10 m/s on surface streets. ut, while there are many instantaneous speeds there is only one average speed for the whole trip. Constant speed / velocity: If you have cruise control you might travel the whole time at one constant speed. If this is the case then you average speed will equal this constant speed. trick question Will an object traveling at a constant speed of 10 m/s also always have constant velocity? Not always. If the object is turning around a curve or moving in a circle it can have a constant speed of 10 m/s, but since it is turning, its direction is changing. nd if direction is changing then velocity must change, since velocity is made up of speed and direction.
P PHYSICS SUMME SSIGNMENT 10 ate Constant velocity must have both constant magnitude and constant direction. Speed and velocity are rates. rate is a way to quantify anything that takes place during a time interval. ates are easily recognized. They always have time in the denominator. 10 m/s 10 meters / second The very first Physics Equation Velocity and Speed both share the same equation. emember speed is the numerical (magnitude) part of velocity. Velocity only differs from speed in that it specifies a direction. v x t v stands for velocity x stands for displacement t stands for time Displacement is a vector for distance traveled in a straight line. It goes with velocity. Distance is a scalar and goes with speed. Displacement is measured from the origin. It is a value of how far away from the origin you are at the end of the problem. The direction of a displacement is the shortest straight line from the location at the beginning of the problem to the location at the end of the problem. How do distance and displacement differ? Supposes you walk 20 meters down the + x axis and turn around and walk 10 meters down the x axis. The distance traveled does not depend on direction since it is a scalar, so you walked 20 + 10 = 30 meter. Displacement only cares about you distance from the origin at the end of the problem. +20 10 = 10 meter. 6. ttempt to solve the following problems. Take heed of the following. lways use SI Units: must be in kilograms, meters, seconds. On the all tests, including the P exam you must: 1. List the original equation used. 2. Show correct substitution. 3. rrive at the correct answer with correct units. Distance and displacement are measured in meters Speed and velocity are measured in meters per second Time is measured in seconds (m) (m/s) Example: car travels 1000 meters in 10 seconds. What is its velocity? x m v v 1000 t 10s v 100 m s a. car travels 35 km west and 75 km east. What distance did it travel? b. car travels 35 km west and 75 km east. What is its displacement? c. car travels 35 km west, 90 km north. What distance did it travel? d. car travels 35 km west, 90 km north. What is its displacement? e. bicyclist pedals at 10 m/s in 20 s. What distance was traveled? f. n airplane flies 250.0 km at 300 m/s. How long does this take? g. skydiver falls 3 km in 15 s. How fast are they going? h. car travels 35 km west, 90 km north in two hours. What is its average speed? i. car travels 35 km west, 90 km north in two hours. What is its average velocity? (s)