Classical Mechanics Lecture 8 Today's Concepts/Examples: a) Potential Energy b) Mechanical Energy Mechanics Lecture 8, Slide 1 Midterm 2 will be held on March 13. Covers units 4-9

Schedule Mechanics Lecture 8, Slide 2 Midterm 2 will be held on March 13. Covers units 4-9

Potential & Mechanical Energy Mechanics Lecture 8, Slide 3

Work is Path Independent for Conservative Forces Mechanics Lecture 8, Slide 4

Work is Path Independent for Conservative Forces Mechanics Lecture 8, Slide 5

Conservative Forces. Work is zero over closed path Mechanics Lecture 8, Slide 6

Potential Energy Mechanics Lecture 8, Slide 7

Gravitational Potential Energy Mechanics Lecture 8, Slide 8

Mechanical Energy Mechanics Lecture 8, Slide 9

Mechanical Energy Mechanics Lecture 8, Slide 10

Conservation of Mechanical Energy Mechanics Lecture 8, Slide 11

Relax. There is nothing new here Mechanics Lecture 8, Slide 12 It’s just re-writing the work-KE theorem: everything except gravity and springs If other forces aren't doing work

Conservation of Mechanical Energy Mechanics Lecture 8, Slide 13 Pumped Storage Hydropower: Store energy by pumping water into upper reservoir at times when demand for electric power is low….Release water from upper reservoir to power turbines when needed... Energy “battery” using conservation of mechanical energy Upper reservoir hh

Gravitational Potential Energy Mechanics Lecture 8, Slide 14

Earth’s Escape Velocity Mechanics Lecture 8, Slide 15 What is the escape velocity at the Schwarzchild radius of a black hole?

Potential Energy Function Mechanics Lecture 8, Slide 16 U 0 : can adjust potential by an arbitrary constant…that must be maintained for all calculations for the situation.

Finding the potential energy change: Mechanics Lecture 8, Slide 17 Use formulas to find the magnitude Check the sign by understanding the problem…

Clicker Question A. B. C. Mechanics Lecture 8, Slide 18

Spring Potential Energy: Conserved Mechanics Lecture 8, Slide 19

Vertical Springs Mechanics Lecture 8, Slide 20 Massless spring

Vertical Spring in Gravitational Field Mechanics Lecture 8, Slide 21 Formula for potential energy of vertical spring in gravitational field has same form as long as displacement is measured w.r.t new equilibrium position!!!

Vertical Spring in Gravitational Field Mechanics Lecture 8, Slide 22

Non-conservative Forces Mechanics Lecture 8, Slide 23 Work performed by non-conservative forces depend on exact path.

Summary Mechanics Lecture 8, Slide 24 Work done by any force other than gravity and springs will change E Lecture 7 Work – Kinetic Energy theorem Lecture 8 For springs & gravity (conservative forces) Total Mechanical Energy E = Kinetic + Potential

Summary Mechanics Lecture 8, Slide 25

Energy Conservation Problems in general Mechanics Lecture 8, Slide 26 For systems with only conservative forces acting E mechanical is a constant

Energy Conservation Problems in general Mechanics Lecture 8, Slide 27  conservation of mechanical energy can be used to “easily” solve problems. (for conservative forces) ALWAYS!  Identify important configurations i.e where potential is minimized  U=0.  Define coordinates: where is U=0?  Identify important configurations, i.e starting point where mass is motionless  K=0  Problem usually states the configurations of interest!

Pendulum Problem Mechanics Lecture 8, Slide 28 Using Work Formalism Using Conservation of Mechanical energy Conserve Energy from initial to final position

Pendulum Problem Mechanics Lecture 8, Slide 29 Don’t forget centripetal acceleration …required to maintain circular path. At bottom of path: Tension is …”what it has to be!”

Pendulum Problem Mechanics Lecture 8, Slide 30

Pendulum Problem Mechanics Lecture 8, Slide 31 Kinetic energy of mass prior to string hitting peg is conserved. Set h=0 to be at bottom equilibrium position

Pendulum Problem Mechanics Lecture 8, Slide 32 Radius for centripetal acceleration has been shortened to L/5 !

Loop the Loop Mechanics Lecture 8, Slide 33 To stay on loop, the normal force, N, must be greater than zero.

Mechanics Lecture 8, Slide 34  Mass must start higher than top of loop

Mechanics Lecture 8, Slide 35

Mechanics Lecture 8, Slide 36

Mechanics Lecture 8, Slide 37

Mechanics Lecture 8, Slide 38

Mechanics Lecture 8, Slide 39

Gravity and Springs…Oh my! Mechanics Lecture 8, Slide 40 Solve Quadratic Equation for x! Conservation of Energy Coordinate System Initial configuration Final configuration

Pendulum 2 Mechanics Lecture 8, Slide 41 Given speed and tension at certain point in pendulum trajectory can use conservation of mechanical energy to solve for L and m….

Pendulum 2 Mechanics Lecture 8, Slide 42

Pendulum 2 Mechanics Lecture 8, Slide 43

Pendulum 2 Mechanics Lecture 8, Slide 44