1. USSC2001 Energy Lecture 2 Elastic Energy and Work Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6874-2749

2. COMPRESSION OF A SPRING show that if an object is placed on the top end of a vertically positioned spring whose bottom end is fixed, the spring will be compressed by a distance that is proportional to the mass of the object Empirical measurements

3. INCREMENTAL COMPRESSION If a small amount of mass is added on to the mass on the spring, then the spring compresses by a small (incremental) amount, the gravitational potential energy has been reduced by

4. INCREMENTAL COMPRESSION If a small amount of mass is added on to the mass on the spring, then the spring compresses by a small (incremental) amount, the gravitational potential energy has been reduced by

5. INCREMENTAL DECOMPRESSION If an small amount of mass is subtracted from the mass on the spring, then the spring decompresses by a small (incremental) amount, the gravitational potential energy has been increased by

6. REVERSIBILITY If mass M is divided into N equal small masses and added incrementally to the top of the spring the total change in gravitational potential energy is If mass M is divided into N equal small masses and subtracted incrementally from the top of the spring the total change in gravitational potential energy is The net change in gravitational potential energy is

7. ELASTIC POTENTIAL ENERGY The empirical facts together with the preceding argument shows that a compressed spring can be used to create, through decompression, gravitational potential energy exactly equal to the gravitational potential energy required to compress it by slowly adding matter to the top of the spring. The amount of gravitational potential energy that can be obtained from decompressing a spring is called the elastic energy.

8. TUTORIAL 2 1. Show that the elastic energy of a spring having stiffness k that is compressed by a distance d is 2. Explain the elastic energy in an elastic band that is stretched. What happens if it is compressed ? How high can you shoot an elastic band ?

9. FORCE AND WORK To do work on a static system (consisting of massive objects and springs), such as lifting objects or compressing springs, means to increase the net potential energy. This requires force. The work, which measures the increase in potential energy, is related to the force and distance (for one dimensional motion) by

10. WORK TO COMPRESS A SPRING The figure below illustrates a spring being compressed Initial (Relaxed) StateCompressed State Hook’s Law states thattherefore k = spring constant

11. WORK AND FULCRUMS Lifting mass is a form of work. It requires energy. One source of this energy is to lower another mass. These ‘toys’ for children are examples of reversible machines – they can be used to lift and then lower the heavier weights using an arbitrarily small extra force that is sufficient to overcome the friction. arm or lever fulcrum 1m 3kg 3m 1kg

12. In the balance shown below, the heavier/lighter mass may be lifted by lowering the lighter/heavier mass. Here, as in the balance, the objects move in opposite directions by distances that are inversely proportional to their masses ? WORK AND PULLEYS 2kg 2m 1m 1kg

13. TUTORIAL 2 3. Compute the mass of the object on the side of the block that has length 2m. Hint: use that fact that if one object is moved down and the other is moved up the total gravitational potential energy remains the same.

14. TUTORIAL 2 4. Compute the required spring constant of a spring gun that is is to be compressed by 0.1m and capable of shooting a 0.002kg projectile to a height of 100m. Assume that the mass of the spring is zero and that no frictional forces are present. 5. Compute the energy required to compress 1 cubic meter of gas to one half of its original volume at constant temperature if the original pressure equals 101300N / square meter. Hint: use the fact that the pressure is inversely proportional to the volume (and therefore increases as the gas is compressed).

15. DEFINITIONS OF ENERGY 1 The capacity for work or vigorous activity, strength 2 Exertion of vigor or power ‘a project requiring a great deal of time and energy’ 3 Usable heat or power ‘Each year Americans consume a high percentage of the world’s energy’ 4 Physics. The capacity of a physical system to do work -attributive. energy – conservation, efficiency [1] The American Heritage Dictionary of the English Language, Houghton Mifflin, Boston, 1992.

16. ENERGY-WORK-TOOL CONCEPT (old form 5.5-7ky) Werg – to do derivatives handiwork,boulevard,bulwark, energy, erg, ergative,-urgy; adrenergic,allergy,argon,cholinergic,demiurge, dramaturge,endergonic, endoergic,energy,ergograph,ergometer, ergonomics,exergonic,exergue, exoergic,georgic,hypergolic,lethargy,liturgy,metallurgy,surgery,synergids ynergism,thaumaturge,work [1] Appendix: PIE (suffixed form) Werg-o Greek: ergon  energos  energeia  Latin: energia  French:energie Germanic: werkam  Old High German: werc, Old English: weorc,werc http://www.bartleby.com/61/roots/IE577.html (zero-grade form) Wig derivatives wrought, irk, wright (o-grade form) Worg derivatives organ, organon (= tool), orgy

17. OPTIONAL: POTENTIAL AND KINETIC ENERGY is conserved (constant function of t) Theorem: For a dropping weight, the total energy The quantity is called the kinetic energy. Proof. Let E = E(t) denote the total energy. Then since and the fundamental theorem of calculus implies that

18. OPTIONAL: HARMONIC OSCILLATIONS For an object attached to a spring that moves horizontally, the total energy is conserved, therefore where is the angular frequency is the phase, and is the period. is the amplitude

19. OPTIONAL: HARMONIC OSCILLATIONS Consider a pendulum - an object on a swinging lever. Then for small L