PhysicsUMSLc8

Gravitational Potential Energy

As long as you are going downhill, work from GPE is positive. Independent of path—only depends on change in height

Conservation of Mechanical Energy

Definition of mechanical energy: E = U + K Using this definition and considering only conservative forces, we find: Ef = Ei Or equivalently: E = constant

Gravitational Potential Energy (GPE or PE)

W =mgh0 -mghf gravity The gravitational potential energy PE is the energy that an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level: U = mgy 1N×m=1joule(J)

Nonconservative Forces

Work done by friction on a closed path is not zero

Elastic Potential Energy

• A body is elastic if it returns to its original shape after being deformed. • Elastic potential energy is the energy stored in an elastic body, such as a spring. • The elastic potential energy stored in an ideal spring is Uel = ½kx².

Conservative and Nonconservative forces

• A conservative force allows conversion between kinetic and potential energy. Gravity and the spring force are conservative. • The work done between two points by any conservative force a) can be expressed in terms of a potential energy function. b) is independent of the path between the two points. c) is 0 if the starting and ending points are the same. • A force (such as friction) that is not conservative is called a nonconservative force or a dissipative force

Conservative Forces

• A force is conservative when the work it does on a moving object is independent of the path between the object's initial and final positions. • A force is conservative when it does no work on an object moving around a closed path, starting and finishing at the same point. • A force is conservative if the work it does is stored in a form of energy that can be released and used later.

Energy diagrams

• An energy diagram is a graph that shows both the potential-energy function U(x) and the total mechanical energy E. • The figure on the right illustrates the energy diagram for a glider attached to a spring on an air track.

NC Force Example

• Consider kinetic friction: W=(Fcosθ)s=fk cos180°s=-fks ★Work is always negative ★Even if the object returns to its original position, work has been done from friction o Work is done on a closed path o Work is dependent on the path taken

Problem-Solving Strategy

• Decide whether the problem should be solved using Newton's Laws or Energy Cons., or a combo ★ Look for things like varying forces (energy) or elapsed time (Newton's Laws) • Energy approach ★ Identify initial and final positions; draw a sketch ★ Define the coordinate system. Usually +y is up ★ Identify forces that do work but cannot be described by potential energy. Draw FB diagram ★ List the known & unknown quantities, include coords & velocities at each point ★ Execute & Evaluate the solution

When Other Forces Do Work

• The work done by all of the forces other than the conservative force equals the change in the total mechanical energy Wother +Wgrav =K₂ −K₁ =ΔK Wgrav =−ΔUgrav Wother =ΔK+ΔUgrav

Situations with both gravitational and elastic forces

• When a situation involves both gravitational and elastic forces, the total potential energy is the sum of the gravitational potential energy and the elastic potential energy: U = Ugrav + Uel. Wother +Wgrav +Wel =K₂ −K₁ =ΔK Wgrav +Wel =−ΔUgrav −ΔUel Wother =ΔK+ΔU

Equilibrium

• When the force is zero, the particle in an energy diagram is said to be at equilibrium. • Remember that force is the derivative of the potential, or the slope of the graph ★F = 0 => slope = 0 ★Unstable equilibrium points are maxima in the potential-energy curve ★Stable equilibrium points are minima in the potential-energy curve

Conservation of Mechanical Energy (cont.)

★ The total mechanical energy of a system is the sum of its kinetic energy and potential energy. ★ A quantity that always has the same value is called a conserved quantity. ★ When only the force of gravity does work on a system, the total mechanical energy of that system is conserved. This is an example of the conservation of mechanical energy