Today’s lecture – review Chapters 9-14, problem solving techniques Announcements Remember -- Tuesday, Oct. 28th, 9:30 AM – Second exam (covering Chapters 9-14 of HRW) – Bring the following: 1 equation sheet Calculator Pencil Clear head Note: If you have kept up with your HW, you may drop your lowest exam grade Today --Thursday, Oct. 23th, 4 PM – Physics Colloquium by Professor Bernd Schüttler, Dept. of Physics, U. Ga – will discuss the analysis of biological systems in terms of a physical and mathematical model Today’s lecture – review Chapters 9-14, problem solving techniques 12/7/2018 PHY 113 -- Lecture 14R

Gravitational forces and energy m r v0 Energy needed to escape Earth’s gravitational field, assuming an initial velocity : v0 12/7/2018 PHY 113 -- Lecture 14R

Energy needed to go from one stable circular orbit to another: 12/7/2018 PHY 113 -- Lecture 14R

Energy needed to go from one stable circular orbit to another -- Example: How much energy is needed to take a satellite of mass m=100kg from the international space station (R1=RE+390 km) to its usual orbit (R2=RE+600 km)? R1 R2 12/7/2018 PHY 113 -- Lecture 14R

Problem solving skills Equation Sheet Math skills Advice: Keep basic concepts and equations at the top of your head. Practice problem solving and math skills Develop an equation sheet that you can consult. 12/7/2018 PHY 113 -- Lecture 14R

Visualize problem – labeling variables Problem solving steps Visualize problem – labeling variables Determine which basic physical principle applies Write down the appropriate equations using the variables defined in step 1. Check whether you have the correct amount of information to solve the problem (same number of knowns and unknowns. Solve the equations. Check whether your answer makes sense (units, order of magnitude, etc.). 12/7/2018 PHY 113 -- Lecture 14R

Center of mass ri rj 12/7/2018 PHY 113 -- Lecture 14R

Position of the center of mass: Velocity of the center of mass: Acceleration of the center of mass: 12/7/2018 PHY 113 -- Lecture 14R

Physics of composite systems: Center-of-mass velocity: Note that: 12/7/2018 PHY 113 -- Lecture 14R

A new way to look at Newton’s second law: Define linear momentum p = mv Consequences: If F = 0   p = constant For system of particles: 12/7/2018 PHY 113 -- Lecture 14R

Statement of conservation of momentum: If mechanical (kinetic) energy is conserved, then: 12/7/2018 PHY 113 -- Lecture 14R

Snapshot of a collision: Pi Impulse: Pf 12/7/2018 PHY 113 -- Lecture 14R

angular “displacement”  q(t) angular “velocity”  Angular motion angular “displacement”  q(t) angular “velocity”  angular “acceleration”  s “natural” unit == 1 radian Relation to linear variables: sq = r (qf-qi) vq = r w aq = r a 12/7/2018 PHY 113 -- Lecture 14R

Special case of constant angular acceleration: a = a0: v1=r1w w r2 v2=r2w Special case of constant angular acceleration: a = a0: w(t) = wi + a0 t q(t) = qi + wi t + ½ a0 t2 ( w(t))2 = wi2 + 2 a0 (q(t) - qi ) 12/7/2018 PHY 113 -- Lecture 14R

Newton’s second law applied to center-of-mass motion Newton’s second law applied to rotational motion ri mi di Fi 12/7/2018 PHY 113 -- Lecture 14R

Object rotating with constant angular velocity (a = 0) v=Rw v=0 Kinetic energy associated with rotation: “moment of inertia” 12/7/2018 PHY 113 -- Lecture 14R

Kinetic energy associated with rolling without slipping: Distance to axis of rotation Rolling: 12/7/2018 PHY 113 -- Lecture 14R

Torque and angular momentum Define angular momentum: For composite object: L = Iw Newton’s law for torque:  If ttotal = 0 then L = constant In the absence of a net torque on a system, angular momentum is conserved. 12/7/2018 PHY 113 -- Lecture 14R

Center-of-mass Torque on an extended object due to gravity (near surface of the earth) is the same as the torque on a point mass M located at the center of mass. mi ri rCM 12/7/2018 PHY 113 -- Lecture 14R

Notion of equilibrium: Notion of stability: T- mg cos q = 0 -mg sin q = -maq F=ma  r q T t=I a  r mg sin q = mr2 a = mraq Example of stable equilibrium. mg(-j) 12/7/2018 PHY 113 -- Lecture 14R

Analysis of stability: 12/7/2018 PHY 113 -- Lecture 14R

12/7/2018 PHY 113 -- Lecture 14R

12/7/2018 PHY 113 -- Lecture 14R

12/7/2018 PHY 113 -- Lecture 14R

12/7/2018 PHY 113 -- Lecture 14R

12/7/2018 PHY 113 -- Lecture 14R