Understanding Mass, Weight, Volume, and Density in Physics

1. Plan • Unit systems (base units / derived units) • Mass, Weight, Volume, Density • Calculations with units (dimensional analysis)

2. * *Current would seem to be a derived quantity: charge / time (C/s)

3. All other quantities are related to (derived from) these “fundamental” ones • Volume = length3  cm3(= mL) • Density = mass/volume  g/mL • Concentration = amount/volume  mol/L

4. Mass, Weight, Volume, Density • Mass is basically: • the amount of “fundamental stuff” (i.e., protons, neutrons, etc.) present in an object or sample. • Mass is independent of where the object/sample is located. • Mass is also the quantity that determines how hard it is to change the motion of an object. • It is harder to accelerate [or decelerate!!] a truck than it is a subcompact because there is more mass in the truck than in the car.

5. Mass, Weight, Volume, Density(continued) • Weight, on the other hand, is a reflection of • the gravitational force or “pull” (of a planet or moon, for example) on something that has mass. • A bowling ball will weigh less on the moon than it does on the earth, even though the object’s mass is the same. • This is because the force of gravity depends on the mass of both objects. The moon has less mass than the earth, so its "pull" is less strong on a given object.

6. Mass, Weight, Volume, Density(continued) • Volume (of a sample) is • the amount of space occupied by the “stuff” (in that sample). • Volume is not a measure of an "amount of matter". It is a measure of "space“. • 1000 cm3 of Styrofoam has a lot less mass in it than does 1000 cm3 of lead, but these two samples occupy the same amount of space.

7. Mass, Weight, Volume, Density(continued) • Density (of an object or sample) reflects • how much mass is present in a given volume (of the object or sample) • Density is a measure of the "compactness" of matter. • A high value of density means "very compact" matter (a lot of mass in a given amount of space). A low value means "very spread out" matter. • Popcorn kernel before popping is more dense than the “fluffy” piece of popcorn that remains afterwards (mass gets “spread out” upon popping).

8. Mass, Weight, Volume, Density(continued) • Density determines whether a substance “sinks” or “floats” in a liquid (once it is submerged) • If dsub > dliq substance sinks • If dsub < dliq substance floats • Velocity of object doesn’t matter • Neither does surface area or surface tension (that can affect something whether something on the surface of a liquid submerges; discussed later)

9. Caution: Mass  Density! • Consider these two samples of matter: • 1) a cruise ship • 2) a cup of water • Which has the greater mass? • Which has the greater density? • How can the massive ship have a density that is smaller than the cup of water? • It’s volume is also larger than the cup of water, by an even greater factor than the mass

10. Caution: Mass  Density! mship mcanon ball mcup of water vs. Vship vs. Vcup of water Vcanon ball Ball also has greater mass & greater V than water… dwater Ship has greater mass & greater V than water… …but V is “less” bigger (increases by a smaller factor). …but V is “more” bigger (increases by a greater factor).  d is greater, ball sinks  d is smaller, ship floats

11. Mass, Weight, Volume, Density(final qualitative comments) • “amount of matter”  “amount of space” (mass) (volume) • How do you experimentally assess? • Volume can be determined by “liquid displacement” • & can be estimated [roughly] by sight • Mass can be determined with a balance. • & can be estimated [roughly] by “feel” • Density is usually calculated rather than measured directly • & can be estimated [roughly] by “sight” and “feel”

12. Demo/Exercise(s) • Can’t assess mass “visually” • Try “feeling”, but sometimes brain is fooled! • Can(if shapes same) assess volume visually • Water displacement can be used to measure volume (if non-absorbent, and substance sinks in water!)

13. Basic Calculations involving Physical Quantities (& Dimensional Analysis [DA]) • SI system of units (next slide) • Unit conversions • Other calculations

14. *Can be used with any SI unit of measurement **The 8 prefixes with an arrow indicate those you are responsible for on Exam 1a

15. Assertions • Units are treated like a algebraic variables during calculations • It is often useful to turn “equivalences” into “conversion factors” (fractions) to do many calculations. • “this for that” concept • Procedure called “Dimensional Analysis” (or “factor label”)

16. Dimensional Analysis uses “conversion factors” • 1 kg = 1000 g  • Note: You can do the math with the numbers and combine the units, without loss of info: • 1000 g/kg or (1/1000) kg/g = 0.001 kg/g • 2.2046 lb = 1 kg  • / means “per”

17. Conversions can be done by starting with one qty and multiplying by one or more “factors” • If you are looking for an “amount”, start with an “amount”; if you are looking for a “this for that”, start with a “this for that” • See board (next slide) • Be careful to construct factors properly • Can’t just “make them up” to fit your needs!!! • The factors are “what they are” (determined by equivalences)

18. Basic Calculations involving Physical Quantities (& Dimensional Analysis [DA]) • Unit conversion calculations • What is the mass of a 154 lb person expressed in grams? • 1 kg = 1000 g (this is an exactqty; discussed later) • 2.205 lb = 1 kg (this is not an exact qty) Many approaches. How would you do it? (Some use explicit proportions; in US, most use DA) Online resource with examples: http://www.alysion.org/dimensional/fun.htm (NOTE: I have some issues with some the work on this site, but overall, the examples themselves are good ones)

19. Equivalences within a system are typically exact (defined) (e.g., 1 L = 1000 mL). Those between systems are usually NOT exact. Exceptions should be indicated as here, with “exact”. The “1” in any equivalence is always exact; any uncertainty in an inexact equivalence is found in the quantity that is not “1”. →(See Table at the back (or front?) of Trofor more equivalences)

20. Example • Convert 45 pm into km • One way (not shortest, but generalizable!) • Write equivalences: • 1 pm = 10-12 m; 1 km = 1000 m • Convert from pm to m (the “base” unit) first • Use/create appropriate conversion factor • Then convert from m to km • Use/create appropriate conversion factor • SEE BOARD

21. 2nd Example (w/ squares and cubes) • For long (multistep) conversion calculations, use the “dimensional analysis” approach to guide you, BUT NEVER STOP THINKING! • Be careful with squares and cubes: • Instructions for a fertilizer suggest applying 0.206 kg/m2. Convert into lb/ft2 • 2.205 lb = 1 kg; 2.54 cm = 1 in (exact) • See board and/or next slide for setup and solution

22. Convert 0.206 kg/m2 into lb/ft2 2.205 lb = 1 kg; 2.54 cm = 1 in (exact) I’ll convert kg to lb in the numerator first; then convert m2 to ft2 in denom. Note: lb/ft2 is a “this for that”, so started with a “this for that” (kg/m2)

23. Dimensional Analysis • Useful tool, but very easy to stop thinking…DON’T! • See Ppt03 slide; you already know about “amounts” and “this for thats”! • For single step calculations in particular, think about the “big guys” and “little guys” and reason first (use DA to check work) • See board, next Ppt for idea and examples