Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lecture 22

Examples Strategy: write down the total mechanical energy, E, E = K + U at the initial and final positions of a particle:

Then use or

A gun shoots a bullet at angle θ with the x axis with a velocity of magnitude V m. What is magnitude of the velocity when the bullet returns to the ground? How high it will go?

L H Spring constant is k; L, H, m are given How close the block will get to the floor? y=0

Conservative Forces If there are only conservative forces in the problem, then there is conservation of mechanical energy Conservative: Can go back and forth along any path and the potential energy and kinetic energy keep turning into one another –Good examples: Gravity and Springs Non-Conservative: As you move along a path, the potential energy or kinetic energy is turned into heat, light, sound etc… Mechanical energy is lost. –Good example: Friction (like on Roller Coasters)

Law of Conservation of Energy Mechanical Energy NOT always conserved If you’ve ever watched a roller coaster, you see that the friction turns the energy into heating the rails, sparks, noise, wind etc. Energy = Kinetic Energy + Potential Energy + Heat + Others… –Total Energy is what is conserved! K 1 +U 1 = K 2 +U 2 +E Heat…

Total Energy is what is conserved! K 1 +U 1 = K 2 +U 2 +E Heat …

Initial E 1 =KE 1 +U 1 …

Final E 2 =KE 2 +U 2 Mechanical energy is lost! What about the total energy? IT IS CONSERVED!

or

If or then 2 or 3D cases:

Several dimensions: U(x,y,z) Compact notation using vector del, or nabla: Another notation: Partial derivative is taken assuming all other arguments fixed

Geometric meaning of the gradient Direction of the steepest ascent; Magnitude : the slope in that direction Direction of the steepest descent Magnitude : the slope in that direction

Potential Energy Diagrams For Conservative forces can draw energy diagrams Equilibrium points –Motion will move “around” the equilibrium –If placed there with no energy, will just stay (no force)

Stable vs. Unstable Equilibrium Points The force is zero at both maxima and minima but… –If I put a ball with no velocity there would it stay? –What if it had a little bit of velocity?

Have a great day! Reading: Chapter 9 Hw: All Chapter 9 problems and exercises