Instructor: Dr. Tatiana Erukhimova Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lectures 19, 20, 21

Quiz A block of mass m is attached to a vertical spring, spring constant k. The spring is unstretched at y=0. A If the spring is compressed an amount A and the block released from rest, how high will it go?

All forces are CONSERVATIVE or NON-CONSERVATIVE

or A force is conservative if: The work done by the force in going from to is independent of the path the particle follows or The work done by the force when the particle goes from to around a closed path, back to , is zero.

Non-conservative: doesn’t satisfy the above conditions

If a force can be written as the derivative of some function, that force is conservative. 1D case: U(x) is called the potential energy function for the force If such a function exists, then the force is conservative

does NOT depend on path! 1D case:

does NOT depend on path! If Fx(x) is known, you can find the potential energy function as

Force of gravity: Potential energy function: Spring: Potential energy function:

A particle moves in one dimension under the influence of a single conservative force given by where  is a given constant. Take the potential energy reference to be at xi=0 such that: and calculate the potential energy function U(x).

or

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

ENERGY DIAGRAMS

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?

A particle moves along the x-axis while acted on by a single conservative force parallel to the x-axis. The force corresponds to the potential-energy function graphed in the Figure. The particle is released from rest at point A. What is the direction of the force on the particle when it is at point A? At point B?

c) At what value of x is the kinetic energy of the particle a maximum? d) What is the force on the particle when it is at point C? e) What is the largest value of x reached by the particle during its motion? f) What value or values of x correspond to points of stable equilibrium? g) Of unstable equilibrium?

2 or 3D cases: If or then

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

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 http://reynolds.asu.edu/topo_gallery/topo_gallery.htm

Given the potential energy function find the x and y components of the corresponding force.

Have a great day! Reading: Chapter 9 Hw: Chapter 8 problems and exercises