Chapter 6, Continued

1. Chapter 6, Continued

2. Summary so Far • Work (constant force): • W = F||d =Fd cosθ • Work-Energy Principle: • Wnet = (½)m(v2)2 - (½)m(v1)2 KE • Total work done by ALLforces! • Kinetic Energy: • KE  (½)mv2

3. Sect. 6-4: Potential Energy A mass can have aPotential Energydue to its environment Potential Energy (PE)  Energy associated with the position or configuration of a mass. Examples of potential energy: A wound-up spring A stretched elastic band An object at some height above the ground

4. Potential Energy (PE)  Energy associated with the position or configuration of a mass. Potential work done! Gravitational Potential Energy: PEgrav mgy y = distance above Earth m has the potential to do work mgy when it falls (W = Fy, F = mg)

5. Gravitational Potential Energy We know that for constant speed ΣFy = Fext – mg = 0 So, in raising a mass m to a height h, the work done by the external force is Fexthcosθ So we define the gravitational potential energy at a height y above some reference point (y1) as (PE)grav

6. Consider a problem in which the height of a mass above the Earth changes from y1to y2: • The Change in Gravitational PE is: (PE)grav= mg(y2 - y1) • Work done on the mass: W = (PE)grav y = distance above Earth Where we choose y = 0 is arbitrary, since we take the difference in 2 y’s in (PE)grav

7. Of course, thispotential energy can be converted to kinetic energy if the object is dropped. Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces). If PEgrav = mgy, from where do we measure y? It turns out not to matter, as long as we are consistent about where we choose y = 0. Because only changes in potential energy can be measured.

8. Example 6-7: Potential energy changes for a roller coaster A roller-coaster car, mass m = 1000 kg, moves from point 1 to point 2 & then to point 3. ∆PEdepends only on differences in vertical height. a. Calculate the gravitational potential energy at points 2 & 3 relative to point 1. (That is, take y = 0 at point 1.)b. Calculate thechangein potential energy when the car goes from point 2 to point 3. c. Repeat parts a. & b., but take the reference point (y = 0) at point 3.

9. Many other types of potential energy besides gravitational exist! Consider an IdealSpring AnIdeal Spring, is characterized by a spring constant k, which is a measure of it’s “stiffness”. The restoring force of the spring on the hand: Fs = - kx (Fs >0, x <0; Fs <0, x >0) This is known as Hooke’s “Law”(but, it isn’t really a law!)It can be shown that the work done by the person is W = (½)kx2 (PE)elastic We use this as the definition of Elastic Potential Energy

10. Work Done to Compress Spring

11. Elastic Potential Energy (PE)elastic≡(½)kx2 Relaxed Spring The work to compress the spring a distance x is W = (½)kx2 (PE)elastic The spring stores potential energy! When the spring is released, it transfers it’s potential energy PEe = (½)kx2to the mass in the form of kinetic energy KE = (½)mv2

12. In a problem in which compression or stretching distance of spring changes from x1 to x2. • The change in PE is: (PE)elastic= (½)k(x2)2 - (½)k(x1)2 • The work done is: W = - (PE)elastic The PE belongs to the system, not to individual objects