Density There are various kinds of density: Mass density = Mass/Volume

1. Density • There are various kinds of density: Mass density = Mass/Volume • Energy density = Energy/Volume • Charge density = Charge/Volume • (What do all these densities have in common?) • The most common is mass density. If you see the word “density” all by itself—no other word(s) in front of it—it means mass density: • Density = Mass/Volume ρ = m/V • ρ (pronounced “roh”) is the Greek “r.” Don’t confuse it with p or P! • What are the SI units of density? kg/m3 OSU PH 212, Before Class #10

2. You can use density, mass and volume together: If you know any two of these, you know the other.... Example: A spherical tank of mass mtank is filled with liquid of density ρ. The tank is sitting at rest on an accurate scale that shows a value FN. Find an expression for the inside radius, r, of the tank. [Vsphere = (4/3)πr3] Solution: DFy = may FN – FG = may FN – mtotalg = 0 But mtotal = mtank + mfl : FN – (mtank + mfl)g = 0 And mfl = rflVfl : FN – (mtank + rflVfl)g = 0 And Vfl = (4/3)πr3: FN – [mtank + (4/3)rflπr3]g = 0 (Now solve algebraically for r.) OSU PH 212, Before Class #10

3. Pressure Pressure is defined simply as force per unit area: P = F/A What are the SI units of pressure? N/m2 or Pascals Example: A 10-kg brick has dimensions 14 cm x 10 cm x 7 cm. What pressure does it exert as it rests flat (on its largest face) on a level table? Pflat = mg/Aflat = 10(9.80)/[(.14)(.10)] = 7.00 x 103 Pa (Is this a lot of pressure? Compare: The earth’s atmospheric pressure at sea level is about 101 x 103 Pa. So the brick increases the total pressure on the table by about 7%.) What about when you set the brick on its side—or end? Pside = mg/Aside = 10(9.80)/[(.14)(.07)] = 10.0 x 103 Pa Pend = mg/Aend = 10(9.80)/[(.10)(.07)] = 14.0 x 103 Pa Notice how you can increase pressure either by increasing the force or (in this case) by decreasing the area. OSU PH 212, Before Class #10

4. Fluid Pressure In solid materials, the particles are chemically bound in a structure. In a solid object feeling a force, the particles respond nearly “as one.” In fluids (liquids or gases), the particles flow freely.This means they can individually respond to forces, push back, or move and collide —with each other and other objects—transmitting forces of their own, in all directions. We can’t measure each tiny force exerted during a collision, but we see the collective effect as fluid pressure. Envision a tiny cube immersed in a fluid. The pressure on each surface of the solid would result from collective force exerted (perpendicularly) by all the fluid particles colliding with that surface. OSU PH 212, Before Class #10

5. Incompressible Fluids In this unit, we’ll look at simple examples of the effects of such fluid pressure—often simplified by idealizing the fluid as incompressible: Its density does not change significantly with pressure. Liquids can be nearly incompressible; gases can’t. For the remainder of this unit, unless otherwise indicated, assume that all fluids are incompressible. OSU PH 212, Before Class #10

6. Pressure at depth in a static fluid How does water “hold itself up?” How does water “float” in itself? This may seem like a strange question, but it must still have a valid physics explanation. Otherwise, any given subset of water would sink through the surrounding water—we’d have infinite fluid flow, with every bit of water sinking through every other bit (not possible). Look carefully at Figure 14.10 on page 363 of the textbook, which takes just a random volume of water (in the shape of a cylindrical column) and analyzes the forces that must sum to zero in order for that much water to be at rest. In particular: SFy = may = 0 The inescapable conclusion: Pdeep = Pshallow + rgDh where Dh is the difference in depth between the two points. Notice that this result was derived for static (motionless) fluid only. OSU PH 212, Before Class #10

7. Archimedes’ Principle: Buoyancy The most useful and widely known consequence of static pressure in a fluid is simple buoyancy: “A solid object immersed—or partially immersed—in a fluid will experience a force due to the fluid pressure, called the buoyant force. This force is directed opposite to the force of gravity, and its magnitude is equal to the gravitational force on the fluid displaced by the object.” FB = FG.displaced fluid FB = (mdisplaced fluid )g FB = (V)displaced fluid g Notice: In incompressible fluids, FBdoes not depend on depth. OSU PH 212, Before Class #10

8. How do we know FB = FG.displaced fluid ? The proof is startlingly simple—look at page 369, especially Fig. 14.17. The buoyant force is simply the difference between the pressure forceacting upward on the object’s bottom and the pressure force acting downward on object’s top. (We already know: The pressure on the bottom is greater because the bottom is deeper in the fluid.) And note: This works even if the pressure force on the top is zero (the object is only partly immersed). OSU PH 212, Before Class #10

9. Check your understanding: Which object has the greatest buoyant force magnitude acting on it? • A. A bowling ball, sitting completely submerged at the bottom • of a swimming pool. • B. A basketball, floating 25% immersed in that swimming pool. • C. A ping-pong ball, floating 5% immersed in that swimming pool. • D. There is not enough information. OSU PH 212, Before Class #10

10. Check your understanding: Which object has the greatest buoyant force magnitude acting on it? • A.A bowling ball, sitting completely submerged at the bottom • of a swimming pool. • B. A basketball, floating 25% immersed in that swimming pool. • C. A ping-pong ball, floating 5% immersed in that swimming pool. • D. There is not enough information. • The largest buoyant force is exerted on the object that is displacing the most fluid (“making the biggest hole” in the fluid). That’s the bowling ball here. • A buoyant force presence or magnitude is not indicated by whether the object is floating! “Not floating” simply means that the gravitational force on the bowling ball is even larger than the (large) buoyant force. OSU PH 212, Before Class #10

11. Other implications of buoyancy (and see also pp. 370-372): ･ If an object floats in a fluid, avg.object ≤ fluid. ･ If the object sinks, avg.object > fluid. ･ The fraction of a floating object’s volume that is immersed in the fluid is equal to the ratio of its density to that of the fluid: Vobj.immersed / Vobj.total = avg.object / fluid ･ Even when an object is not floating on its own, so long as it is displacing any fluid, a buoyant force is still exerted. Consider, for example, the rest weight (as indicated by a scale) of a sunken stone or a brick suspended in water by a wire. The scale reading is reduced by FB. OSU PH 212, Before Class #10