[G12] GEN-PHY 1.pdf

Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 The metric system, on the other hand, uses a Introduction to Physics few core units and adds prefixes to denote bigger Physics is the study of t...

Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 The metric system, on the other hand, uses a Introduction to Physics few core units and adds prefixes to denote bigger Physics is the study of the basic principles of the or smaller measurements. universe, such as motion, forces, energy, and ➔ Core Units – meter (m), gram (g), second more. It’s the most basic of the natural sciences. (s), liter (L) Some common prefixes (together with their The word “physics” comes from the Greek word numerical meanings) are: 9 physikos (φυσικός). ➔ giga- (G) – 10 6 ➔ mega- (M) – 10 Physics can generally be divided into classical 3 ➔ kilo- (k) – 10 physics and modern physics. This course will 2 focus on classical physics. ➔ hecto- (h) – 10 1 ➔ deka- (da) – 10 −1 Measurement ➔ deci- (d) – 10 −2 ➔ centi- (c) – 10 Measurement is the process of comparing an −3 unknown quantity with a standard. ➔ milli- (m) – 10 −6 - It involves assigning a quantitative value ➔ micro- (µ) – 10 to some data or phenomenon. −9 ➔ nano- (n) – 10 Measurement includes magnitude, units, and uncertainty using standard instruments. Conversion There are two systems of measurement. Conversion involves dimensional analysis. ➔ The English units, also known as the imperial units, are historical units that To convert between units, we use conversion were used. They are generally in use in factors. In general, the following rule is used. 𝑢𝑛𝑖𝑡𝑠 𝑦𝑜𝑢 𝑤𝑎𝑛𝑡 the USA. 𝑢𝑛𝑖𝑡𝑠 𝑦𝑜𝑢 ℎ𝑎𝑣𝑒 * 𝑢𝑛𝑖𝑡𝑠 𝑦𝑜𝑢 ℎ𝑎𝑣𝑒 = 𝑢𝑛𝑖𝑡𝑠 𝑦𝑜𝑢 𝑤𝑎𝑛𝑡 ➔ The SI units, also known as the Systéme International, are the modern system of The fraction is what’s known as a conversion units used around the world. It’s based on factor, which will depend on the units used. increments of 10. Below are the conversion factors for English units and SI units. The English system uses the following units. ➔ English ➔ Length – inch (in), foot (ft), yard (yd), mile 12 in = 1 ft (mi) 3 ft = 1 yd ➔ Mass – ounce (oz), pound (lb), ton (ton) 5280 ft = 1 mi ➔ Capacity – teaspoon (tsp), cup (c), pint 16 oz = 1 lb (pt), quart (qt), gallon (gal) 2000 lbs = 1 ton 48 tsp = 1 c 2 c = 1 pt 2 pt = 1 qt 4 qt = 1 gal Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 ➔ SI 2 𝑔 = 9. 8 𝑚/𝑠 – unlimited SF Multiply the unit with the relevant 1 𝑓𝑡 = 12 𝑖𝑛 – unlimited SF numerical meaning. For example, 1000 mg = 1 g because milli- tells us the Rule 2: All nonzero digits are significant. −3 number is 10. 22 𝑚 – 2 SF 5423 – 4 SF Let’s go through an example. 13 𝑖𝑛 – 2 SF 3 The density of the femur bone is 1. 85 𝑔/𝑐𝑚. Rule 3: Zeros at the end of a number are not Given this, what is the density of the femur in significant. 3 100 – 1 SF 𝑘𝑔/𝑚 ? 1000 𝑔 – 1 SF 𝑔 1 𝑘𝑔 100 𝑐𝑚 100 𝑐𝑚 100 𝑐𝑚 2400 𝑚 – 2 SF 1. 85 3 * 1000 𝑔 * 1𝑚 * 1𝑚 * 1𝑚 𝑐𝑚 3 𝑘𝑔 10 6 3 Rule 4: Zeros between nonzero numbers are = 1. 85 * 10 3 – Note: 3 = 10 𝑚 10 significant. 3 = 1850 𝑘𝑔/𝑚 101 – 3 SF 101201 𝐿 – 6 SF Note that there are three 𝑐𝑚/𝑚 units to cancel 403 𝑚 – 3 SF 3 the 𝑐𝑚 (𝑐𝑚 * 𝑐𝑚 * 𝑐𝑚). Rule 5: Zeros to the right of a decimal point are Given the same situation, what is the density of 3 significant. the femur in 𝑚𝑔/𝑚𝑚 ? 10. 10 – 4 SF 105. 40 𝑔 – 5 SF 𝑔 1 𝑚𝑔 1 𝑐𝑚 3 1. 85 3 * 0.001 𝑔 * ( 10 𝑚𝑚 ) 101. 240 𝑚 – 6 SF 𝑐𝑚 1 𝑚𝑔 1 = 1. 85 * * 0.001 1000 𝑚𝑚 3 Rule 6: Zeros at the end of a number and to the = 1. 85 𝑚𝑔/𝑚𝑚 3 right of the decimal point are significant. 100. 00 – 5 SF 1. 0 𝑔 – 2 SF Significant Figures 500. 00 𝐿 – 5 SF Significant figures (SF) are digits of measurement that represent meaningful Rule 7: Leading zeros are not significant. measurements of data. 00023 – 2 SF There are a few rules we should follow when 000000001 𝑚𝐿 – 1 SF considering significant figures. 0. 00000000000023 𝑚𝑔 – 2 SF Rule 1: Numbers obtained through counting, Rule 8: Sums and differences are expressed with constants, and conversion factors have unlimited the least number of SFs in the original numbers. significant figures. 23. 24 − 20. 1 = 3. 1 (NOT 3. 14) 12 𝑏𝑜𝑦𝑠 – unlimited SF 23. 245 + 20. 20 = 43. 45 (NOT 43. 445) Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 23. 24 + 20. 1 = 43. 3 (NOT 43. 34) usually due to instrumental limitations, environmental factors, or variations in Rule 9: Products and quotients are expressed procedure. with the least number of SFs in the original ➔ A systematic error causes each numbers. measurement to vary by a predictable 20. 2 * 3 = 60 (NOT 60. 6) amount. It’s usually due to observational 20. 42 * 3. 23 = 66. 0 (NOT 66. 9566) errors, calibration errors, or environmental interference. Extension When multiplying with conversion factors, Scientific Notation express the answer with the least number of SFs. There are many cases where measurements are 12 𝑖𝑛 either too big or too small to write completely 5 𝑓𝑡 * = 60 𝑖𝑛 1 𝑓𝑡 without tiring out your hand or wasting space. 12 𝑖𝑛 4 𝑓𝑡 * 1 𝑓𝑡 = 50 𝑖𝑛 (NOT 48 𝑖𝑛) This is where scientific notation comes in. Why do significant figures matter? They help Scientific notation is a way of writing numbers determine whether a measurement is accurate or in a way that only indicates significant figures. It’s precise. a shorthand of sorts Accuracy refers to how close the measurement Scientific notation takes the form of: is to the real value. 𝑛 Precision refers to how consistent the 𝑐 × 10 measurements are. 𝑐 – any number from 1 to 10, not including 10 In the test below, two students were asked to 𝑛 – some integer measure the mass of a 50.0 g sample. Let’s see two examples to understand how scientific notation works. We can say that student A’s scale is precise, but Convert 156000 to scientific notation. not accurate. The values are close together, but they’re far off from the original measured mass. First, imagine a decimal point at the end. 156000. Student B’s scale, meanwhile, is accurate, but not precise. The values are closer to the original Now, move that decimal point until there is only 1 mass, but they’re fairly spread out. digit before the decimal point. 1. 56000 – The point moved 5 places to the left. Now, if there are errors in accuracy or precision, we call that an error. There are two types of That decimal is now 𝑐. As for 𝑛, count how many errors. places the decimal point moved. Since the ➔ A random error causes each decimal point moved left, 𝑛 is positive. measurement to differ from other 5 measurements by a random amount. It’s 156000 = 1. 56 * 10 Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 Convert 0. 000053 to scientific notation. Vectors Physical Quantities can be classified as either Here, we just go straight to moving the decimal vector or scalar point. 0000005. 3 – The decimal point moved 6 places Vector quantity is a quantity that is fully to the right. expressed by both magnitude and direction. The decimal is 𝑐. As for 𝑛, it’s still the number of places moved, however it must be negative. Scalar quantity is a quantity that is Since the decimal point moved right, 𝑛 is expressed by using magnitude alone. negative. −6 0. 000053 = 5. 3 * 10 magnitude refers to a number plus the unit of measurement. (e.g. 5.5m, 20kg) PRACTICE EXERCISE: The answers are in white text. Highlight to reveal them. Scalars Vectors I. Convert the following units. distance displacement mass force 1. 48 𝑖𝑛 → 𝑓𝑡 | Answer: 4 ft 2. 1 𝑘𝑔 → 𝑔 | Answer: 1000 g speed velocity 3. 9 𝑘𝑚/ℎ → 𝑚/𝑠 | Answer: 2.5 m/s time acceleration II. Convert the following to scientific notation. work torque 1. 0.000023 | Answer: 2. 3 * 10 −5 density momentum 6 2. 5601000 | Answer: 5. 601 * 10 Vector Representation III. How many significant figures are in the – vector quantities are often represented by numbers below? scaled vector diagrams. 1. 25000 | Answer: 2 – vector diagrams depict a vector by use of an 2. 502.60 | Answer: 5 arrow drawn to scale in a specific direction. 3. 0.000341 | Answer: 3 Components of Vector Diagram IV. Express the following numbers in 1. A vector is represented by an arrow. The scientific notation with 3 significant figures. arrow has 3 important parts: arrowhead – indicated the −1 direction of the vector. 1. 0.56815 | Answer: 5. 68 * 10 3 the length of the arrow – 2. 1079.97 | Answer: 1. 08 * 10 represents the magnitude of the vector. Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 tail – represents the origin of the vector. Vector Addition 2. Devise a suitable scale. Resultant Vector (R) - the sum of two or more vectors that are PRACTICE EXERCISE: represented by a singel vector. A student walks 20 m from his classroom to the next building in a direction of 30° West of North. 1. Graphical Method How will you represent his displacement using a Head-to-Tail Method vector diagram? Parallelogram Method Solution: 2. Analytical/Mathematical Method Pythagorean Theorem To represent 20 m long arrow in a paper, we can Component Method use a scale of 1 cm = 4 m. We can use dimensional analysis, to arrive at the actual 3. Experimental Method length. Vector Addition - Graphical Method From here, we PRACTICE EXERCISE can see that a 20 HEAD-TO-TAIL METHOD m long arrow can be represented by A person walks 9 blocks east and 5 blocks north. a 5 cm long arrow. Using the head-to-tail method justifies that the resultant displacement/vector is 10.3 blocks at an angle of 29.10 north of east. Step 1. Draw an arrow to represent the first vector (9 blocks to the east) using a ruler and The magnitude and direction of protractor. the vector must be clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North) Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 Step 2. Now draw an arrow to represent the Step 5. To get the magnitude of the resultant, second vector (5 blocks to the north). Place the measure its length with a ruler. (Note that in most tail of the second vector at the head of the first calculations, we will use the Pythagorean vector. theorem to determine this length.) Step 6. To get the direction of the resultant, measure the angle it makes with the reference frame using a protractor. (Note that in most calculations, we will use trigonometric relationships to determine this angle.) Remember that the reference direction is the last direction in the phrase *direction* of *direction*. (e.g. you started at the south and went to the west, therefore the direction is “west of south”) Step 3. If there are more than two vectors, continue this process for each vector to be ANSWER: Using the graphical method, the added. Note that in our example, we have only head-to-tail method, the resultant vector is 10.3 two vectors, so we have finished placing arrows units north of east. tip to tail. Vector Addition - Analytical Method Step 4. Draw an arrow from the tail of the first vector to the head of the last vector. This is the PRACTICE EXERCISE resultant, or the sum, of the other vectors. PYTHAGOREAN THEOREM USING PYTHAGOREAN THEOREM TO DETERMINE THE MAGNITUDE: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement. Given: d1 = 11km, North d2 = 11km, East Solution: 2 2 2 𝑐 =𝑎 +𝑏 Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 2 2 2 𝑐 = (11𝑘𝑚) + (11𝑘𝑚) 2 2 2 𝑐 = 121𝑘𝑚 + 121𝑘𝑚 2 2 𝑐 = 242 2 2 𝑐 = 242𝑘𝑚 𝑐 = 15. 6 𝑅 = 15. 6𝑘𝑚 Given: 𝑉𝑏𝑜𝑎𝑡 = 4𝑚/𝑠, 𝐸𝑎𝑠𝑡 Eric is 15.6 km away from his original position. 𝑉𝑟𝑖𝑣𝑒𝑟 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 = 3𝑚/𝑠, 𝑁𝑜𝑟𝑡ℎ USING TRIGONOMETRIC FUNCTIONS magnitude: TO DETERMINE THE DIRECTION 2 2 2 𝑐 =𝑎 +𝑏 2 2 2 𝑐 = (4. 0𝑚/𝑠) + (3. 0𝑚/𝑠) 2 2 2 2 2 𝑐 = 16𝑚 /𝑠 + 9𝑚 /𝑠 2 2 2 Continuation of Eric’s direction: 𝑐 = 25𝑚 /𝑠 2 2 2 𝑐 = 25𝑚 /𝑠 𝑐 = 5. 0 𝑅 = 5. 0𝑚/𝑠 direction: 𝑡𝑎𝑛θ = (𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒/𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡) 𝑡𝑎𝑛θ = (3/4) −1 Using the sine function, the resultant θ = 𝑡𝑎𝑛 (3/4) ◦ ◦ θ = 36. 9 displacement is 15. 6 km,45 North of East PRACTICE EXERCISE 2 Therefore the resultant velocity of the boat is 5.0m/s, 36.9° from the +x axis or North of A motorboat leaves the shore at a velocity of East. This is the effect of the river current upon 4m/s due east. The water current in the river the water boat. moves due north at a velocity of 3m/s. What would be the resultant velocity of the boat? Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 Vector Addition - Analytical Method 𝑐𝑜𝑠θ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑠𝑖𝑛θ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ◦ 𝑑1 𝑑1𝑦 𝑐𝑜𝑠 34 = 36𝑚𝑥 ◦ PRACTICE EXERCISE 𝑠𝑖𝑛 34 = 36𝑚 COMPONENT METHOD 𝑑1𝑥 = (3𝑔𝑚)(𝑐𝑜𝑠 34 ) ◦ ◦ 𝑑1𝑦 = (36𝑚)(𝑠𝑖𝑛 34 ) 𝑑1𝑥 = 29. 8𝑚 𝑑1𝑦 = 20. 1𝑚 used for adding two or more vectors ◦ this x-component is the y-component goes Given: 𝑑1 = 36𝑚, 34 𝑁 𝑜𝑓 𝐸 aimed at the right, so, up so it’s positive. ◦ it is positive. 𝑑2 = 23𝑚, 64 𝑊 𝑜𝑓 𝑁 Vector 𝑑2 Solution: Step 1: Determine the size and direction of the vectors that you want to add. x-components y-components Step 2: Find the x- and y-components for the first and second vectors. That is, by using SOH-CAH-TOA trigonometric functions. Vector 𝑑1 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑛θ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐𝑜𝑠θ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ◦ 𝑑2𝑥 𝑑2 ◦ 𝑠𝑖𝑛 64 = 23𝑚 𝑐𝑜𝑠 64 = 23𝑚𝑦 ◦ ◦ 𝑑2𝑥 = (23𝑚)(𝑠𝑖𝑛 64 ) 𝑑2𝑦 = (23𝑚)(𝑐𝑜𝑠 64 ) x-components y-components 𝑑2𝑥 = 20. 6𝑚 𝑑2𝑦 = 10. 0𝑚 this x-component the y-component goes points to the left so it up, so it is positive. is negative. Step 3: Add all the x- components and add all the y-components of your two vectors, but do not add an x- component to the y-component. Module 1: Semester 1 — General Physics 1 Mr. Romo Sanvictores | S12-12 Step 4: Use the Pythagorean theorem to get the magnitude of the total 2D displacement. 2 2 𝑑𝑇 = (𝑑𝑇𝑥) + (𝑑𝑇𝑦) 2 2 𝑑𝑇 = (9. 2𝑚) + (30. 1𝑚) 2 2 𝑑𝑇 = 84. 6𝑚 + 906. 0𝑚 2 𝑑𝑇 = 990. 6𝑚 𝑑𝑇 = 31. 5𝑚 Step 5: Use the tangent function to get the angle. 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 |Σ𝑦| 𝑡𝑎𝑛θ = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 or 𝑡𝑎𝑛θ = |Σ𝑥| |30.1𝑚| 𝑡𝑎𝑛θ = |9.2𝑚| −1 |30.1| θ = 𝑡𝑎𝑛 |9.2| ◦ θ = 73. 0 , 𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡1