Determining Density Introduction to Engineering Design © 2012 Project Lead The Way, Inc.

Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Density Density is a measure of the amount of matter per unit of volume Objects more dense than water sink Objects less dense than water float High Density Low Density Think about identically sized samples of steel and Styrofoam – for instance small cubes the size of the wooden cubes used for the puzzle cube. Which is more dense? Steel or Styrofoam?

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Matter: Mass vs. Weight Mass is the amount of matter in an object or the quantity of the inertia of the object Weight is the force of gravity on mass W = weight m = mass g = acceleration of gravity W = mg Many materials are purchased by weight Project Lead The Way® Copyright 2006

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Matter: Mass vs. Weight Mass and Weight are often confused An example using SI units A man has a mass of 100 kg He weighs 980 N W = mg W = (100 kg)(9.8 m/sec2) = 980 Newton Project Lead The Way® Copyright 2006

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Matter: Mass vs. Weight Mass and Weight are often confused US Customary units example A woman weighs 100 pounds Her mass is 3.1 slugs W = mg m = W g Slugs are not commonly used because the pound is a more practical unit for commerce. As a result, pounds are often used to indicate mass instead of slugs. But since “pound” is used as a unit of force as well, there is often confusion about what the unit “pound” represents. = 100 lb 32.2 ft s 2 = 3.1 slugs Project Lead The Way® Copyright 2006

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Mass vs. Weight Pound-mass (lbm) is a unit of mass 1 lbm = 0.45359237 kg (by definition) 1 kg = 2.205 lbm (formula sheet) 1 slug = 32.2 lbm (formula sheet) Formula Sheet The Avoirdupois pound, or pound-mass, is defined as 0.45359237 kg. It is a unit of mass. The slug is another accepted unit of mass in the US Customary system. However, because slugs are not well understood, it has become common to use the pound as a unit of mass (as well as a unit of weight) in the US Customary system. Unfortunately, as a result, mass and weight are often confused in the US system, and many people do not understand the difference between the meaning of “pound-mass” and “pound-force”. Often times the context determines to which “pound” a person is referring. On the formula sheet, the pound-force unit is indicated only with the letters lb. The pound-mass unit is indicated with a subscript of m attached to the letters lb. Remember that mass and weight are not the same. Mass of an object is constant no matter what the environment. The force of weight changes with the acceleration of gravity (which can change as you move across the surface of the earth). Project Lead The Way® Copyright 2006

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Mass vs. Weight Pound-force (lb) is a unit of force The gravitational force exerted on a mass of one lbm on the surface of the Earth 1 lb = 4.44822162 N 1 N = 0.225 lb Formula Sheet Pound-force is a unit of force. Weight is a force and is the result of the acceleration of gravity on the mass of an object. On the Earth’s surface, we can generally say (although not exactly) that 1 pound-mass will weigh 1 pound-force. This assumes and acceleration of gravity of 9.8066 m/sec2. However, the acceleration of gravity will vary from this value at different locations on Earth. Therefore, an object’s weight will vary depending on where the object is located. Inconsistencies in weight measurements will occur among different locations – thus, mass is preferred in scientific work. Project Lead The Way® Copyright 2006

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Mass vs. Weight How are pound-mass and pound-force related? On Earth (g = 32.174 ft/s2) 1 pound-mass object weighs1 pound-force In outer space (no gravity) 1 pound-mass object has no weight On the moon (g = 5.32 ft/s2) 1 pound-mass object weighs 0.166 pound-force Unfortunately, in common practice we have come to use the word “pound” to refer to a mass and a force, and the meaning of the word can be confusing. On Earth the mass of an object in units of pound-mass can change slightly due to differences in the acceleration of gravity but is roughly equivalent to the force of gravity, or weight of the object in units of pound-force. However, because the mass of an object does not change but the weight (force of gravity) of an object is directly related to the acceleration of gravity imposed on the object, the weight of an object will change if the acceleration of gravity changes (for instance an object is taken into outer space or to the moon). Project Lead The Way® Copyright 2006

Density Density can be indicated in multiple ways m = VDm W = VDw Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Density Density can be indicated in multiple ways Mass Density = mass per unit volume Weight Density = weight per unit volume Formula Sheet m = VDm D m = m V The context of the situation will determine whether the density is expressed in terms of mass or weight. We will not worry too much about differentiating among the two at this point. However it is important to be able to distinguish between the two terms of mass and weight and know the difference. W = VDw D w = W V

Weight Density (lb/in.3) Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Mass/Weight Density Mass Density commonly used for SI units g/cm3 or kg/m3 Weight Density commonly used for US Customary units lb/in.3 or lb/ft3 Material Mass Density (g/cm3) Weight Density (lb/in.3) Apples 0.64 0.023 Water (Pure) 1.00 0.036 Water (Sea) 1.03 0.037 Ice 0.92 0.034 Concrete 2.40 0.087 Aluminum 2.71 0.098 Steel (1018) 7.8 0.282 Gold 19.32 0.698 Although density is often defined as the mass per unit volume, weight density is commonly used to express density when using US Customary units. In the curriculum, density will refer to either mass density or weight density depending on the measurement device and the measurement system being used.

Mass and Weight: Measurement Balance – measures mass Uses objects of known mass to find the unknown mass of an object Measurement is unaffected by difference in gravity Scale – measures weight Measures force caused by gravity Measurement is affected by difference in the acceleration of gravity May display a “mass” measurement reading by using an assumed acceleration of gravity to convert from weight to mass

Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Volume: Measurement Volume is the amount of three dimensional space enclosed by an object Two methods to determine volume Calculate using geometry of object Measure indirectly using water displacement (or 3D scanner) A 3D scanner is available to purchase as an optional item as shown in the Purchase Manual.

Volume: Calculation Some object volumes can be calculated Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Volume: Calculation Some object volumes can be calculated What is the volume formula for each object? Students should refer to the Engineering Formula Sheet.

Volume: Indirect Measure Volume of an irregular object can be determined indirectly with fluid displacement Level Elevates

Volume: Indirect Measure Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Volume: Indirect Measure Record water level with only water Add samples Record new level Difference is sample volume Explain to students the need to read the water level based on the bottom of the meniscus. The meniscus might be very slight depending on the attraction between the water and the graduated cylinder material. Volume Change Read level here Meniscus shape is exaggerated for clarity

Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Example What is the density of a 1.12 in. diameter sphere of titanium that weighs 0.82 lb? V = 4 3 π r 3 V = 4 3 π 1.12 in. 3 V = 5.88 in3 W = VDw D w = W V = 0.82 lb 5.88 in. 3 D w = 0.14 lb/in.3 The formula for the volume of a sphere is on the formula sheet. McMaster-Carr published value for Titanium Grade 2: http://www.mcmaster.com/#=i6dpgt What volume of water (mL) would it displace? 5.88 in. 3 1 ft 3 1728 in. 3 1L 0.0353 ft 3 = 0.096 L or 96 mL