Instructor: Dr. Tatiana Erukhimova Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lecture 15

Quiz This is a one-dimensional problem. Suppose a particle is attracted to the origin with a force Find the potential function.

Mechanical energy is conserved! Work-energy theorem: Mechanical energy is conserved!

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

Initial E1=KE1+U1…

Final E2=KE2+U2

Then use or

H

Who hits the bottom with a faster speed? Water Slide Who hits the bottom with a faster speed?

Roller Coaster You are in a roller coaster car of mass M that starts at the top, height H, with an initial speed V0=0. Assume no friction. What is the speed at the bottom? How high will it go again? Would it go as high if there were friction? H

Roller Coaster with Friction A roller coaster of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2). Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path?

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! K1+U1 = K2+U2+EHeat…

Total Energy is what is conserved! K1+U1= K2+U2+EHeat…

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

Spring problem revisited A block of mass M is on a horizontal surface and is attached to a spring, spring constant k. If the spring is compressed an amount A and the block released from rest, how far from unstretched position will it go before stopping if there is no friction between the block and the surface? How will this answer change is the block is not attached to the spring??

Block of mass m has a spring connected to the bottom Block of mass m has a spring connected to the bottom. You release it from a given height H and want to know how close the block will get to the floor. The spring has spring constant k and natural length L. H y=0