5.3. Conservation of Energy

Transcription

1 5.3. Conservation of Energy

2 Conservation of Energy Energy is never created or destroyed. Any time work is done, it is only transformed from one form to another: Kinetic Energy Potential Energy Gravitational, elastic Electromagnetic, chemical potential (bonds) Thermal Energy The kinetic energy of molecules

3 Mechanical Energy Mechanical Energy is the sum of potential and kinetic energy: ME = PE + KE. In a system without friction (= loss to thermal energy), mechanical energy is conserved i.e. constant. Conservation of mechanical energy (or mechanical + thermal energy) can be used to solve various problems.

4 Pendulum demo Note PE can be mgh, 1 k x 2 or both! 2

5 Example You drop a 1.0-kg stone from a height if 2.0 m. What is its velocity when it hits the ground? Solution 1: Equations of Motion v f 2 = v i 2 + 2a x = 0 + 2gh v f = 2gh Solution 2: Conservation of Energy PE i + KE i = PE f + KE f mgh + 0 = mv2 v 2 = 2gh v = 2gh

6 Example 2 A batter hits a baseball, which leaves the bat at 36 m/s. A fan in the outfield bleachers, 7.2 m above, catches it. What was its speed when caught? v = PE i + KE i = PE f + KE f mv i 2 = mgh mv f 2 v f 2 = v i 2 2gh = 34 m/s

7 Example 3 If friction is negligible, which swimmer has greater speed at the bottom of the slide? The path did not matter! Gravity is called a conservative force. The change in PE grav depends only on the initial and final positions, not path. This allows us to solve problems with variable a.

8 Conservative and Nonconservative Forces Conservative force: the work done by or against it is stored in the form of potential energy that can be released (recovered) at a later time Examples of a conservative force: gravity, spring Nonconservative force: the work by or against it is lost, and not stored as potential energy Example of a nonconservative force: friction

9 Work Done by a Conservative Force Work done by a conservative force (e.g. gravity) around a closed path is zero.

10 Work Done by a Conservative Force The work done by a conservative force does not depend on the path: Consequently the work done or required can be computed from only the initial and final positions.

11 Work Done by a Nonconservative Force Work done by a nonconservative force (e.g. friction) around a closed path is not zero:

12 Non-conservative Forces The work done by friction is converted from kinetic energy into thermal energy (which is the KE of the molecules). Since the frictional force depends on the direction of motion, it depends on the path taken. Potential energy [fields] cannot be defined for non-conservative forces.

13 Conservative Forces and Work Which requires more work, lifting a box straight up (W 1 ) or sliding it up a frictionless ramp at constant velocity (W 2 )? W 2 = Fd = F 2 L = mg sin φ L = mg h L L = mgh = W 1 Why then is a ramp used?

14 Potential Energy Curves The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:

15 Potential Energy Curves The potential energy curve for a spring:

16 Conservation of Energy Energy can be transferred from one object to another, e.g. when a compressed spring transfers its potential energy to the kinetic energy of a ball. In any case, the total energy of the two-object system is conserved (constant).

17 Read/try the great examples in the book in the Conservation of Energy section! Thermal, electrical, nuclear and chemical energy can be described in terms of kinetic energy and potential energy. Nuclear reactions indicate that mass can be converted to energy (E=mc 2 ), and thus is a form of energy. Loop-the-loop demo: where did the PE go?

18 Hamilton s Principle I mgh mv2 = C mg h + 1 v m 2v t 2 t mgv + mva = 0 mg + ma = 0 F gravity + ma = 0 Conservation of Energy = 0 Take t of both sides. Simplify. Divide by v. Newton s 2 nd Law! Thus Newton s 2 nd Law (conservation of momentum) is a manifestation of conservation of energy.

19 Hamilton s Principle II 1 2 kx mv2 = C Conservation of Energy 1 x k 2x + 1 v m 2v 2 t 2 t kxv + mva = 0 kx + ma = 0 F spring + ma = 0 = 0 Take t. Simplify. Divide by v. Newton s 2 nd Law! Thus Newton s 2 nd Law (conservation of momentum) is a manifestation of conservation of energy.