Chapter 7 Conservation of Energy

Recap – Work & Energy The total work done on a particle is equal to the change in its kinetic energy

Potential Energy The total work done on an object equals the change in its kinetic energy But the total work done on a system of objects may or may not change its total kinetic energy. The energy may be stored as potential energy.

Potential Energy – A Spring Both forces do work on the spring. But the kinetic energy of the spring is unchanged. The energy is stored as potential energy

Conservative Forces If the ski lift takes you up a displacement h, the work done on you, by gravity, is –mgh. But when you ski downhill the work done by gravity is +mgh, independent of the path you take

Conservative Forces The work done on a particle by a conservative force is independent of the path taken between any two points

Potential-Energy Function If a force is conservative, then we can define a potential-energy function as the negative of the work done on the particle

Potential-Energy Function potential-energy function associated with gravity (taking +y to be up) The value of U0 = U(y0) can be set to any convenient value

Potential-Energy Function of a Spring By convention, one chooses U0 =U(0) = 0

Force & Potential-Energy Function In 1-D, given the potential energy function associated with a force one can compute the latter using: Example:

7-1 Conservation of Energy

Conservation of Energy Energy can be neither created nor destroyed Closed System Open System

Conservation of Mechanical Energy If the forces acting are conservative then the mechanical energy is conserved

Example 7-3 (1) How high does the block go? Initial mechanical energy of system Final mechanical energy of system

Example 7-3 (2) Forces are conservative, therefore, mechanical energy is conserved Height reached

Example 7-4 (1) How far does the mass drop? Initial mech. energy Final mech. energy

Example 7-4 (2) Final mech. energy = Initial mech. energy

Example 7-4 (3) Solve for d Since d ≠ 0

Example 7-4 (4) Note is equal to loss in gravitational potential energy

Conservation of Energy & Kinetic Friction Non-conservative forces, such as kinetic friction, cause mechanical energy to be transformed into other forms of energy, such as thermal energy.

Work-Energy Theorem Work done, on a system, by external forces is equal to the change in energy of the system The energy in a system can be distributed in many different ways

Example 7-11 (1) Find speed of blocks after spring is released. Consider spring & blocks as system. Write down initial energy. Write down final energy. Subtract initial from final

Example 7-11 (2) Initial Energy Take potential energy of system to be zero initially Kinetic energy of system is zero initially

Example 7-11 (3) Final Energy Kinetic and potential energies of system have changed

Example 7-11 (4) Subtract initial energy from final energy But since no external forces act, Wext = 0, so Ef = Ei

Example 7-11 (5) And the answer is… Try to derive this.

E = mc2 E = mc2 In a brief paper in 1905 Albert Einstein wrote down the most famous equation in science E = mc2

Sun’s Power Output Power 1 Watt = 1 Joule/second 100 Watt light bulb = 100 Joules/second Sun’s power output 3.826 x 1026 Watts

Sun’s Power Output Mass to Energy Kg/s = 3.826 x 1026 Watts / (3 x 108 m/s)2 The Sun destroys mass at ~ 4 billion kg / s

Problems To go…

Ch. 7, Problem 19

Ch. 7, Problem 29

Ch. 7, Problem 74