PHYS 172: Modern Mechanics Fall 2011 DEMO – ball-spring models of matter. Lecture 7 – Speed of Sound in a Solid, Buoyancy Read 4.9 – 4.13

Derivative form of the Momentum Principle Works only if force is constant during t The rate of the momentum change is equal to force If force changes introduce instantaneous rate of change: The derivative of the momentum with respect to time is equal to the net force acting on an object The momentum principle

Newton’s Second Law Newton’s original formulation: The rate of change of amount of body’s motion is proportional to force momentum Momentum principle is the second Newton’s law Assume nonrelativistic case: (Assume m = const) (definition of acceleration) You have probably wondered for some time – where is the second Newton’s law which we used to talk about in high school? Why did we only talk about first and third one? Etc… Note: there are different translations of Newton’s formulation of 2nd law (it was written in Latin), but key point is that he talked about “amount of body motion”, i.e. quantity we call momentum today. http://www2.dsu.nodak.edu/users/edkluk/public_html/crt_aphys/m_newt2_h.html Newton’s second law Traditional form of 2nd Newton’s law

Spring-mass system: horizontal x y z 1. System: block 2. Apply momentum principle: =0 Equilibrium

Spring-mass system: Analytical solution Motion along x: px = p nonrelativistic Mention that one can solve it using iterative approach and computer, but this problem could be also solved analytically. Differential equation:

Spring-mass system: Analytical solution amplitude Search solution in form: Angular frequency

Spring-mass system: period and frequency Period T: T 2T A [s] t Frequency: -A [s-1 ][Hz] A Angular frequency: [radian/second]

Static equilibrium (system never moves) System is at rest: y x Can predict s:

Spring-mass system: vertical Choose origin at equilibrium position Apply momentum principle: x s0 y Details: 4.14 (p. 167) Skip derivation – just mention that it is easy to show, they can look at derivation later themselves. Bottom line – differential equation is the same, and thus the solution will be the same. The same equation and motion in the presence of gravity if you choose origin at equilibrium!

Speed of sound in solids Qualitatively: Larger , larger v Larger d, larger v Detailed derivation: Speed of sound in a solid

Buoyancy 11

Microscopic view: Pressure Colliding ball exerts force on wall. Balloon: Air molecules constantly hit surface and bounce off, exerting forces on surface. Pressure: force per unit area: P = F/A At 50 km – essentially no air. Pressure is a result of collisions between objects. 12

Buoyancy: microscopic view Air is denser closer to the earth. (Why? We’ll see in Chapter 12!) There are more air molecules here than here Microscopic view – hard to account for all interactions! Thus Pbottom > Ptop OR Net force pushing up > Net force pushing down buoyancy 13

Buoyancy: macroscopic view Archimedes principle: any body partially or completely submerged in a fluid or gas is buoyed up by a force equal to the weight of the fluid/gas displaced by the body. At STP: 0 °C and 101.325 kPa 14

Pressure and suction Pressure: ~ 15 psi (pound/inch2) Mair Example: brick on a table no air between surfaces mg = 1 kg × 9.8 N/kg =9.8 N Air pressure Area A Fair = PA = 105 N/m2 × 0.1 m × 0.2 m Fair = 2 × 103 N (~450 lb) ? Why doesn’t the table collapse? Suction cup: how does it work? 1 cm2 – Fair = 10 N 15

Clicker Question A boat carrying a large boulder is floating in a lake. The boulder is thrown overboard and sinks. The water level in the lake (with respect to the shore): Rises Falls Remains the same Explanation: When it is inside the boat, the boulder displaces its weight in water. When it is thrown overboard, it only displaces its volume in water, so the water level of the lake with respect to the shore falls. 16