MATH 131 Conversion Factors; Force and Stress Resource Sheet Conversion Factors Length Mass Area/Volume 1 yd = 3 ft 1 lb = 16 oz 1 ha = 2.47 acres 1 cup = 8 fl oz 1 N = 1 kgοm/s2 1 ft = 12 in 1 kg = 2.205 lb 1 acre = 43,560 ft 2 1 pt = 2 cups 1 Pa = 1 N/m2 1 mi = 5,280 ft 1 lb = 454 g 1 ft 2 = 144 in2 1 qt = 2 pt 1 MPa = 1 N/mm2 1 gal = 4 qt 1 psi = lbs/in2 1 gal = 4.5461 L 1 MPa = 1,000,000 Pa 1 US gal = 3.785 L 1 MPa = 145.038 psi 1 L = 1000 cm3 1 psi = 0.0689 bar 1 mi = 1.6093 km 1 US ton = 2000 lb 1 in2 = 6.4516 cm2 1 in = 2.54 cm 1 tonne = 1000 kg 1 m2 = 10,000 cm2 1 m = 3.28 ft 3 3 1 ft = 1,728 in 1 ft3 Force/Pressure = 7.48 US gal 1 US gal = 231 in3 Metric Prefixes SMALLER than Base Unit LARGER than Base Unit Prefix Symbol Relationship to Base tera T 1 tera = 1012 base = 1 000 000 000 000 base giga G 1 giga = 109 base = 1 000 000 000 base mega M 1 mega = 106 base = 1 000 000 base kilo k 1 kilo = 103 base = 1 000 base hecto h deca da 3 Prefix Symbol Relationship to Base deci d 10−1 base = 1 deci or 10 deci = 1 base centi c 10−2 base = 1 centi or 102 centi = 100 centi = 1 base milli m 10−3 base = 1 milli or 103 milli = 1000 milli = 1 base micro ο 10−6 base = 1 micro or 106 micro = 1 000 000 micro = 1 base 1 hecto = 102 base = 100 base nano n 10−9 base = 1 nano or 109 nano = 1 000 000 000 nano = 1 base 1 deca = 10 base = 1 deca pico p 10−12 base = 1 pico or 1012 pico = 1 000 000 000 000 pico = 1 base 3 3 BASE UNIT g m L N Pa 1 1 1 T G M k h da tera giga mega kilo hecto deca Force and Stress Formulas 1 BASE g, m, L N, Pa 1 1 3 3 3 d c m µ n p deci centi milli micro nano pico Common Materials: Maximum Tensile Stress (TS) & Compressive Stress (CS) Values πΉ =πβπ ππ = πΉπ π΄ π =πβπ ππΆ = πΉπΆ π΄ g (Earth) = 9.81 m/s2 π= πΉπ π΄ Material TS (MPa) CS (MPa) 305 Stainless Steel Wire 585 --- Copper 220 --- Concrete 3 30 White Pine 40 35 MATH 131 Geometry Resource Sheet (Surface Area, Volume) Key to Symbols used Some Useful Conversions Area 2 1 ππ‘ = 144 ππ2 1 ππ2 = 6.4516 ππ2 1 π2 = 10, 000 ππ2 Volume 3 1 ππ‘ = 1, 728 ππ3 Dimensions l = length w = width t, c, e = other sides (not equal) Hexagon Distances a = side length f = face to face D = corner to corner P = Perimeter A = Area C = Circumference (circles) s = slant height H = height of 3-D shape V = Volume LSA = Lateral Surface Area TSA = Total Surface Area Circles r = radius d = diameter h = height (2-D) b = base Two-Dimensional Bases Name/Shape Rectangle Formulas l c C = 2πr or d r 2 A = πr or A= b Trapezoid t c A=bβh Circle c h A=lβw P = 2b + 2c h P=b+c+e e Parallelogram Formulas Triangle P = 2l + 2w w b Name/Shape P=b+c+t+e e h A= b Hexagon C = πd (t + b) β h 2 P = 6a f D πd2 A= 4 bβh 2 D = 2a a A= f = √3 β a 3√3 2 a 2 Three-Dimensional Shapes Name Shape Example Prism (Rectangular, Triangular, Hexagonal, etc.) H w l Cylinder (Circular Prism) Pyramid (Square-based, Triangle-based, Hexagonal-based) H r s H Lateral Surface Area Total Surface Area Volume LSA = Pbase β H TSA = LSA + 2 β Abase V = Abase β H LSA = Pbase β H TSA = LSA + 2 β Abase V = Abase β H LSA = 2πr β H TSA = 2πrh + 2πr 2 V = πr 2 H LSA = Pbase β s 2 TSA = LSA + Abase LSA = Pbase β s 2 TSA = LSA + Abase Cone (Circular-based Pyramid) s H TSA = πr 2 + πrs LSA = πrs r Surface Area (Total) Sphere r SA = 4πr 2 or SA = πd2 V= Abase β H 3 V= Abase β H 3 Volume V= 4ππ 3 πd3 ππ V = 3 6 Math131 Module 1 Topic 1 (Surface Area & Volume) Surface Area and Volume Surface Area • defined as: ______________________________________________________________________ • applies to three-dimensional shapes • units are ____________________ which means unit conversion factors are too! for example: 1 ππ‘ 2 = 144 ππ2 Example: Find the area of a flat piece of sheet metal that is 36" × 120" What if we take that metal and create a duct: Lateral Surface Area (LSA): area of all surfaces excluding bases Total Surface Area (TSA): area of all surfaces including bases Page 1 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Volume • defined as: ______________________________________________________________________ • applies to three-dimensional shapes • units are ____________________ which means unit conversion factors are too! for example: 1 ππ‘ 3 = 1728 ππ3 Example: How much air can fit inside the duct made from the 36" × 120" piece of sheet metal? Review: Rectangular Prism & Cylinder Important formulas to calculate the surface area and volume of simple 3-D shapes can be found in the Geometry Resource Sheet on D2L. Remember to print and use this resource for assessments! Name Shape Rectangular Prism l h w Cylinder h r Lateral Surface Area Total Surface Area Volume LSA = Pbase β h TSA = LSA + 2 β Abase V = Abase β h LSA = (2l + 2w) β h TSA = 2lh + 2wh + 2lw V = lwh LSA = Pbase β h TSA = LSA + 2 β Abase V = Abase β h LSA = 2πr β h TSA = 2πrh + 2πr 2 V = πr 2 h These are just two specific examples of prisms. Page 2 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Prisms • three dimensional shapes that have a common face throughout the length/height of the prism. • common face is also known as the base of the shape. • distance from face to face is the height of the prism. Lateral Surface Area Total Surface Area Volume πππ = ππππ¬π β π‘ πππ = π³πΊπ¨ + π β ππππ¬π π = ππππ¬π β π‘ Page 3 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Let’s try a few examples. If necessary, round your final answers to 2 decimal places. Find the lateral surface area (LSA), total surface area (TSA) and volume (V) of: Cylinder 24 in 22 in Triangular Prism 6 ¾ ft 9 ft 15 ft 5 ft 12 ft Page 4 of 17 Math131 Module 1 Trapezoidal Prism Topic 1 (Surface Area & Volume) Find the lateral surface area (LSA), total surface area (TSA) and volume (V) 14 in 16 in 18 in 6 in 20 in Page 5 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Right Pyramids and Cones Right Pyramids • base is a regular polygon (rectangle, triangle) • three or more lateral faces that taper to a single point (apex) centered over the base Cones • pyramid-like figures with a circular base (radius and diameter refer to the base) • altitude/height (h) is the perpendicular distance from the apex to the base • slant height (s) is the distance from the apex to base along the surface/face. o how can we calculate h or s? what math (that we already know) can help us out? Lateral Surface Area πππ = ππππ¬π β π¬ π Total Surface Area πππ = π³πΊπ¨ + ππππ¬π Volume π= ππππ¬π β π‘ π Page 6 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Right Pyramid Example: A square based pyramid has base lengths of 8 feet and an altitude of 10 feet. Find the lateral surface area, total surface area and volume. Page 7 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Cone Example: Find lateral surface area, total surface area and volume of a cone given that the slant height is 18 inches and the radius is 10 inches height of pyramid (H) r Page 8 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Spheres • does a sphere have a base? o lateral surface area (LSA) = total surface area (TSA) • like a circle – defined by radius (r) and diameter (d) Surface Area and Volume using radius (r) using diameter (d) ππ = πππ« π ππ = πππ πππ« π π= π πππ π= π Sphere 3 Example: Find the surface area and volume of a spherical storage bin with a circumference of 11 ft 4 r d Page 9 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Steps to Solving Surface Area and Volume Problems Read the question & make a sketch Identify your 3D shape (prism, pyramid, cone, sphere) Determine the appropriate formula Complete remaining calculations. • Do all measurements have the same units? If not, convert • Find any missing measurements (e.g. radius, slant height, etc.) • What is the shape of the base? • Calculate Area (A) and Perimeter (P) of the base • Are you calculating LSA, TSA, V? • Write down the formula you are going to use • Does the final answer seem reasonable? • Check for correct final units, rounding Page 10 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Application Problems 1) A welder must create a spherical fuel tank that has a circumference of no more than 30.5 ft. a) How much steel, in sq.ft, would be needed to create this tank? Round the final answer to 1 decimal. b) What is the maximum volume (in cu.ft) that this tank can hold? Round the final answer to 1 decimal place. Page 11 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) 2) A vent cap for a smokestack is to be fabricated in the shop. Before the holes are punched, the vent cap looks like the diagram below. How much material is needed, in square inches (sq.in), to create this end cap if the cylinder height is 10 in and the diameter of the cylinder is 15 in? The total height of the completed end cap is 33 in. Keep in mind this vent cap needs to slip over the end of the existing smokestack. Round the final answer to the nearest square inch to ensure you will have enough material for the structure. Page 12 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) 3) How many litres of paint are needed for the outside walls of a building 7,700 mm high by 12,000 mm by 7,860 mm if there are 41 m2 of windows? One litre covers 9.8 m2. Assume that you cannot buy a fraction of a litre (i.e. round to the minimum number of whole litres you should purchase). Page 13 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) 4) The roof of a building is in the shape of a square pyramid 23 m on each side and the slant height of the pyramid is 14 m, a) How much will roofing material cost at $6.00/m2? Report your answer with two decimal places. b) What is the volume of air trapped in the square pyramid roof? Report your answer with two decimal places. Page 14 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) Application Problems-Extra Practice 1) A spherical tank has a diameter of 4,865 mm. If a litre of paint will 8.0 m2, how many litres are needed to cover the tank? Round final answer to the nearest litre that ensures you will have enough paint to cover the tank. Answer: 10 L 2) A welder has created a funnel in the shape of a cone (see below). How much stainless steel, in square feet (sq.ft), was used to create this funnel? The radius of the funnel opening is 17.0 in, the height of the cone is 24.0 in. Round your final answer to 1 decimal place. Answer: 10.9 sq.ft 3) A gazebo roof has the shape of a hexagonal pyramid with the following dimensions: • Slant Height (s) = 10ft • Length of one side (a) = 5ft • Half of the distance across the flats is f/2 = (√3 β a)/2 (in feet) How many square feet of shingles are needed to cover the roof? Express the answer to the nearest square foot. Answer: 150 sq ft 4) A cylindrical tank with top and bottom has a radius of 7 m and an altitude of 5.95 m. If a litre of paint will cover 10.75 m2 of surface, how much paint is needed to put two coats of paint on the entire surface of the tank? Round to the nearest litre to ensure you will have enough paint. Answer: 106 L 5) The Last Chance gambling casino is designed in the shape of a square pyramid with a side length of 75 ft. The height of the building is 32 ft. An air filtering system can circulate air at 25,000 cu ft per hr. How long will it take the system to filter all the air in the room? Round to one decimal place. Answer: 2.4 hours Page 15 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) 6) In creating a custom storage tank for fuel, a welder has drawn the following sketch of what is to be created. How much fuel, in cubic feet (cu.ft), can the tank hold if S = 12 in, G = 60 in, M = 44 in and R = 28 in. Round your final answer to the nearest cubic foot. Answer: 15 cu.ft 7) You have made the forms to create four rectangular columns in the building you are working on. A diagram of what each forms looks like can be found below. How many tons of concrete are required to make these columns if L = 20 ft, W = 2.00 ft and H = 1.50 ft. Note: 13.5 cubic feet of concrete weighs approximately 1 ton. Round your final answer to 1 decimal place. Report units as tons. Answer: 17.8 tons 8) A welder has created a funnel in the shape of a cone (see below). How much fluid, in cubic feet (cu.ft), could the funnel hold at one time? The radius of the funnel opening is 31.0 in, the height of the cone is 36.0 in. Round your final answer to 1 decimal place. Answer: 21.0 cu.ft Page 16 of 17 Math131 Module 1 Topic 1 (Surface Area & Volume) 9) Dirt must be excavated for the foundation of a building 28 yards by 10 yards to a depth of 6 yards. How many trips will it take to haul the dirt away if a truck with a capacity of 3 cu yd is used? Express the answer to the nearest whole number. Answer: 560 trips 10) How many cubic metres of dirt are there in a pile, conical in shape, 10 m in diameter and 4 m high? Round the final answer to 1 decimal place. Answer: 104.7 m3 11) How much air, in cubic inches, does a section of duct contain that is in the form a rectangular prism with dimensions 7.75 ft by 39 in and 3.5 ft high? Express the answer to the nearest cubic inch. Answer: 152,334 cu in 12) A gazebo roof has the shape of a hexagonal pyramid with the following dimensions: • Slant Height (s) = 11ft • Length of one side (a) = 9ft • Half of the distance across the flats is f/2 = (√3 β a)/2 (in feet) How many cubic feet of air space is contained beneath the roof? Express the answer to 1 decimal. Answer: 544.5 cu. ft 13) The concrete footings for piers for a raised foundation are measuring 2.1 ft by 1.3 ft by 1.4 ft. How many cubic yards of concrete are needed for 27 footings? Round the final answer to 1 decimal. Answer: 3.8 cu. yd 14) A conical oil cup with a radius of 2.7 cm must be designed to hold 55 cm3 of oil. What should be the altitude of the cup? Round the final answer to 1 decimal place. Answer: 7.2 cm Page 17 of 17 Math131 Module 1 Topic 2 (Scientific Notation) One second of time represents how much of a 365-day calendar year? According to Statistics Canada, non-residential construction accounted for $ 201, 492, 100 worth of capital expenditures in 2022. Is there an easier way of writing or reporting these values? Scientific Notation • a method of writing either very small or very large decimal numbers • written as the product of a number between 1 and 10 (called a coefficient) and a power of 10. The coefficient must have only one number in front of the decimal place. Example: 2.43 × 106 Writing a Decimal Number in Scientific Notation 1) Starting from the left, place an arrow ο where the decimal point will appear in the coefficient (between the first and second nonzero numbers) 0.0000689 1235000 2) Count the number of places the original decimal point had to move to reach the arrow (from 1). The number of moves will become the value (exponent) of the power of 10. • If the original number is smaller than 1, the exponent is a NEGATIVE number • If the original number is larger than 1, the exponent is a POSITIVE number. 0.0000689 = 1235000 = Page 1 of 6 Math131 Module 1 Topic 2 (Scientific Notation) Writing a Number in Scientific Notation as a Decimal Number • POSITIVE power of 10 means a LARGE number o move the decimal point in the coefficient to the RIGHT the same number of places as the exponent o add zeros (if there is not a number already in a place value) 6.25 x 102 = 1.5 x 104 = 9.153 x 106 = • NEGATIVE power of 10 means a SMALL number o move the decimal point in the coefficient to the LEFT the same number of places as the exponent o add zeros if you need to (no number in a place value already) 1.06 x 10−1 = 3.0x 10−3 = 4.68 x 10−5 = Page 2 of 6 Math131 Module 1 Topic 2 (Scientific Notation) Calculations using Scientific Notation (Exponent laws can be used on questions involving scientific notation if you feel comfortable with them. However, it is easy to make mistakes this way and can be time-consuming.) Use the EXP button on your calculator to enter scientific notation values. Exp Notes: Using 10 x or “ x 10 ^ ” or “ x 10 yx ” for scientific notation does NOT work properly if the calculations involve multiplying or dividing. To enter a negative sign in an exponent, use either the “+/−” or “(−)” key NOT “minus” (like for subtraction). Some calculators may require changing the MODE to display scientific notation. Check the calculator manual/instructions for how to. Practice (look at the display – how does your calculator show scientific notation?) 9.153 x 106 on calculator: 3.0 x 10−3 on calculator: Examples: Express each value below in scientific notation, then calculate and report the answer in scientific notation as well. Round coefficients in the answers to the nearest hundredth. 9 070 × 0.806 = 798,000 × 0.0125 = Page 3 of 6 Math131 Module 1 Topic 2 (Scientific Notation) Review: Metric Prefixes Recall that the Metric System of Measurements is based on multiples of ten. Adding a prefix to the base unit (grams, metres, litres, etc.) changes its value (larger or smaller) as noted in the table below: LARGER than Base Unit Prefix Symbol tera T 1012 base = 1 000 000 000 000 base = 1 tera giga G 109 base = 1 000 000 000 base = 1 giga mega M 106 base = 1 000 000 base = 1 mega kilo k 103 base = 1 000 base = 1 kilo hecto h 102 base = 100 base = 1 hecto deca da 10 base = 1 deca BASE UNIT Relationship to Base grams (g), metres (m), litres (L) deci d 10 deci = 1 base centi c 102 centi = 100 centi = 1 base SMALLER milli m 103 milli = 1 000 milli = 1 base than Base Unit micro ο 106 micro = 1 000 000 micro = 1 base nano n 109 nano = 1 000 000 000 nano = 1 base pico p 1012 pico = 1 000 000 000 000 pico = 1 base Remember: it takes a LOT of very small units to make just one LARGE unit! The largest and smallest prefixes are highlighted at the top and bottom of table, respectively. Using these prefixes on base units are another way of indicating very large or very small measurements. Examples: How many kilobytes (kB) of data can be stored on a 512 MB flash drive? What about a 64 GB memory card? Page 4 of 6 Math131 Module 1 Topic 2 (Scientific Notation) To convert between two different prefixes, it is important to know how they relate to each other (not the base unit). Below, let’s list the relationships (conversion factors) between Tera, Giga, Mega and kilo: It is also important to consider very small measurements as well. What are the relationships between the prefixes: nano, micro, milli and centi? Examples: The diameter of a human hair is 87 οm. How many metres does this correspond to? A single grain of sand weighs about 13 mg. What would its mass be in kilograms? Page 5 of 6 Math131 Module 1 Topic 2 (Scientific Notation) Application Problems 1) It rains frequently in England, and a new tarp is needed to protect the soccer field at Anfield, where the Liverpool Football Club plays. If the rectangular pitch measures 101 metres by 69 metres, what is the area of the tarp required in square centimetres? Round your answer to the nearest whole number, and then report it in scientific notation. 2) The thickness of a wire is often referred to as its “gauge”. For example, 20-gauge wire has a diameter of 0.812 mm. What is the cross-sectional area of this 20-gauge wire? Round your answer in square millimetres to three decimal places. Convert your answer to m2, and report in scientific notation. Page 6 of 6 Math131 Module 1 Topic 3 (Force and Stress) Definition of Force Force can be described as: _______________________________________________________________ Some types of force are: Normal Force (FN): When two objects are in contact with one another, the normal force is the force that prevents one object from taking the others place. For example: standing on the floor. You and the floor are in contact with one another. The floor is exerting a force upon you, if it wasn’t, you would fall through the floor. Tension (FT) Force: When an object is being pulled from opposite ends, there is tension force. For example: a rope being pulled with an object on the end is experiencing tension force. Compression (FC) Force: When an object is being pushed from opposite ends, there is compression force. For example: a support column is experiencing compression from the weight above. Regardless of the type of force, its value (magnitude) can be determined by the formula: πΉ =πβπ where F _____________________________________________________ m ____________________________________________________ a _____________________________________________________ Example: How much force is required to move a skid of lumber with a mass of 160 kg across the floor at 0.4 π π 2 ? Ignore friction and air resistance. Round to nearest whole number of Newtons. Page 1 of 9 Math131 Module 1 Topic 3 (Force and Stress) Example: An air conditioning unit, being pushed with a force of 787.5 N, is moving at 1.75 π π 2 . What is the mass of the unit, in kg? Ignore friction and air resistance. Round to whole number of kilograms. Weight Weight is another type of force that _______________________________________________________ _____________________________________________________________________________________ The words “weight” and “mass” are often used to mean the same thing, but they are different. Weight: force, depends on gravity Mass: measure of the amount of matter, same no matter where in the universe Weight is just a specific force and can be determined by the formula: π =πβπ where W : Weight _____________________________ m : mass _______________________________ g : acceleration due to gravity (on Earth, g = 9.81 π π 2 ) Page 2 of 9 Math131 Module 1 Topic 3 (Force and Stress) Example: A steel drum, with a mass of 56 kg is sitting on a wooden support frame. What is the weight of the steel drum? What is the normal force exerted by the wooden support frame? Round to two decimal places and remember the units! 1 Example: You are supporting a large piece of steel with two identical " 305 stainless steel wires. 2 One wire is attached to each end of the piece of steel and then attached to the ceiling. You’ve been told the piece of steel “weighs” 500 lbs. What is the tension in each wire? Round to two decimal places and remember the units! Page 3 of 9 Math131 Module 1 Topic 3 (Force and Stress) Definition of Stress Stress can be described as: ______________________________________________________________ _____________________________________________________________________________________ Consider: the weight of a steel drum sitting on a wooden support frame with four legs. The force is being distributed to each of the four legs. However, the force on each leg is only being experienced by the area on the top of each leg. Just like forces, there are many types of stresses. The following diagram describes the different kinds of stresses we will be looking at (Compressive stress, Tensile stress, and Shear stress): Plane of expected breakage Tensile Stress (σT) Compressive Stress (σC) Shear Stress (τ) The formula to calculate stress is fundamentally the same no matter which kind of stress you are calculating for and is given by: ππ‘πππ π (π) = πΉππππ (πΉ ) π΄πππ (π΄) → ππ = πΉ π΄ ππΆ = πΉ π΄ π= πΉ π΄ where F : Force in Newtons (N), and A : cross-sectional Area, in m2, along the plane of expected breakage. To calculate a stress value, you must calculate the respective force. Page 4 of 9 Math131 Module 1 Topic 3 (Force and Stress) Units of Stress Metric System: Pascals (Pa) and MPa Formula to calculate stress: ππ‘πππ π (π) = πΉππππ (πΉ) π΄πππ (π΄) What are the units for Force (F) and Area (A)? Therefore the unit for stress will be: and this is equivalent to the metric unit of Pascal (Pa) In referencing stress values for safety levels, we sometime need to convert to other units, including the MegaPascal (MPa) Conversion factor between units of Pa and MPa: Think: What does the prefix “Mega” mean? Examples: Convert 2,125,500 Pa to MPa. Convert 35 MPa to Pa Non-Metric: Pounds per Square Inch (psi) We have been calculating stress in Pa or MPa (metric units). Another common unit of stress is pounds per square inch (psi). Conversion factor between units of psi and MPa: 1 πππ = 145.038 ππ π Examples: Convert 45 MPa to psi Convert 1000 psi to MPa Page 5 of 9 Math131 Module 1 Topic 3 (Force and Stress) Calculating Stress and Determining Safety Consider: The tensile force on copper wire is 4448.979 N and the cross-sectional area of the wire is 0.00127 m2. Calculate the tensile stress on the wire. Will the wire break? Once we calculate the stress on the wire, we can use this information to make decisions about safety. We can determine if the wire is strong enough to support the load by comparing it to known stress limits for the material. Below are some common materials and their approximate Tensile Strength (TS), which is the maximum tensile stress it can take before it breaks, and their Compressive Strength (CS), the maximum compressive stress it can take before it breaks. Material TS (MPa) CS (MPa) 305 Stainless Steel Wire 585 --- Copper 220 --- Concrete 3 30 White Pine 40 35 One very important thing to note here is that although the values in the table are the breaking stress values, the materials can become severely deformed prior to breaking which may jeopardize their integrity. Page 6 of 9 Math131 Module 1 Topic 3 (Force and Stress) Force and Stress Application (Example 1) Three 4” square support columns are being used to support a medium sized deck. If the deck is 1400 lbs of mass, how much stress is each support column experiencing? Round final answer only to 2 decimals. Page 7 of 9 Math131 Module 1 Topic 3 (Force and Stress) Force and Stress Application (Example 2) Six concrete support columns are used in an office building to support the floor above. If each of the columns have cross sections with an area of 0.09 m2, what is the maximum downward force they can support together? Round to a whole number of Newtons. What is the maximum mass they can support together? Round to 2 decimal places. Remember units! Material TS (MPa) CS (MPa) 305 Stainless Steel Wire 585 --- Copper 220 --- Concrete 3 30 White Pine 40 35 Page 8 of 9 Math131 Module 1 Topic 3 (Force and Stress) Force and Stress: Application Problems 1) A large crate is being transported across the warehouse with a force of 482 N. If it has a mass of 1250 lb, how fast is the crate accelerating? Ignore friction and air resistance. Round to 2 decimals. Answer: 0.85 m/s2 2) A steel drum, with a mass of 75 lb is sitting on a wooden support frame. What is the weight of the steel drum? Round to one decimal place. Remember your units! Answer: 333.7 N 3) A rope, with a diameter of 2 cm, is being pulled with a force of 20 000 N. Calculate the tensile stress. Report your answer in MPa to one decimal place. Answer: 63.7 MPa 4) A square concrete column that has base length of 24 cm is required to support a 31 000 kg mass. If the maximum compressive stress that concrete can take is 30 MPa, will the column be safe? Answer: Yes, the column is safe (with stress = 5.3 MPa) 5) You need to construct a square column capable of supporting 281,500N of downward force. If you only have enough space for a 7-inch side length, what maximum compressive stress (in MPa) can the column support? Round the answer to 1 decimal place. Answer: 8.9 MPa 6) Four hexagonal support columns made from white pine are being created to support 13,000 lbs. What is the stress in each column if you know each side of the hexagon is 8 in? Will the support columns be able to support this mass? Assume the mass is evenly distributed to the four support columns. (Recall: maximum compression stress of white pine is 35 MPa) Answer: Yes, the columns will be able to support the mass (with stress of 0.1 MPa per column) 7) What is the maximum downward force that three white pine columns can support if the columns have a total cross-sectional area of 315 cm2 ? Round your answer to a whole number of Newtons. (Recall: maximum compression stress of white pine is 35 MPa) Answer: 1,102,500 N 8) A bridge is being supported by 6 concrete columns and the maximum suspected load that the bridge will have to carry is 15,000 lbs. What is the amount of force each column is experiencing if the load is perfectly distributed to all 6 columns? Round final answer to the thousand’s place. Answer: 11,000 N Page 9 of 9
