7 Conservation of Energy Potential Energy The Conservation

7 Conservation of Energy • • Potential Energy The Conservation of Mechanical Energy The Conservation of Energy Mass and Energy • Hk: 23, 27, 39, 47, 55, 69, 71

Potential Energy • Potential Energy is stored energy • Potential Energy is position dependent (KE is speed dependent) • Ex. object at higher height has more PE • Types of PE: gravitational, elastic, electric, magnetic, chemical, nuclear. • /

Conservative Forces • When the work done by a force moving from position 1 to 2 is independent of the path, the force is Conservative. • The work done by a Conservative Force is zero for any closed path. • Conservative Forces have associated Potential Energies • /

Non Conservative Forces • Produce thermal energy, e. g. friction • Work done by Non Conservative Forces is path dependent, e. g. longer path, more work required • /

Potential Energy Functions

Elastic Potential Energy

Ex. Elastic Potential Energy • • 100 N/m spring is compressed 0. 2 m. F = -kx = -(100 N/m)(0. 2 m) = -20 N U = ½kx 2 = ½(100 N/m)(0. 2 m)2 = 2 J /

Gravitational Potential Energy

Ex. Gravitational Potential Energy • Ex: A 2 kg object experiences weight (2 kg)(9. 8 N/kg) = 19. 6 N. • At 3 m above the floor it has a stored energy of mgy: • (2 kg)(9. 8 N/kg)(3 m) = 48. 8 Nm = 48. 8 J. • /

Conservation of Energy • Individual energy levels change. • Sum of all individual energies is constant. • /

Conservation of Mechanical Energy

Ex. Conservation of Mechanical Energy: Object dropped from height h above floor.

Energy E 1 E 2 E 3 Kinetic 0 ½mv 22 0 PE-g 0 0 mgh PEspring ½kx 2 0 0 Totals ½kx 2 ½mv 22 mgh

Energy Kinetic PE-g Totals E(h) E(y) 0 ½mv 2 mgh mgy mgh ½mv 2 + mgy Energies and speeds are same at height y Accelerations at y are not same

Work Energy with Friction

Example: The smaller the frictional force fk, the larger the distance, s, it will travel before stopping. s Energy Ei Ef Kinetic ½mvi 2 0 PE-g 0 0 Thermal 0 fks Totals ½mvi 2 fk s

A 2. 00 kg ball is dropped from rest from a height of 1. 0 m above the floor. The ball rebounds to a height of 0. 500 m. A movieframe type diagram of the motion is shown below. Type E 1 E 2 E 3 E 4 E 5 gravita mg(1) -tional 0 0 0 mg(1/2) kinetic 0 ½ m(v 2)2 0 ½ m(v 4)2 0 elastic 0 0 PEelastic 0 0 therma 0 l 0 Ethermal

By energy conservation, the sum of all energies in each column is the same, = E 1 = mg(1) = 19. 6 J Calculate v 2: (use 1 st and 2 nd columns) mg(1) = ½ m(v 2)2. g = ½ (v 2)2. v 2 = 4. 43 m/s Calculate PE-thermal: (use 1 st and 5 th columns) mg(1) = mg(1/2) + PE-thermal mg(1/2) = PE-thermal = 9. 8 J

Calculate PE-elastic: (use 1 st and 3 rd columns) PE-elastic + PE-thermal = mg(1) PE-elastic + 9. 8 = 19. 6 PE-elastic = 9. 8 J Calculate v 4: (use 1 st and 4 th columns) ½ m(v 4)2 + PE-thermal = mg(1) ½ m(v 4)2 + 9. 8 = 19. 6 ½ m(v 4)2 = 9. 8 (v 4)2 = 2(9. 8)/2 v 4 = 3. 13 m/s

Potential Energy & Force

Equilibrium • Stable: small displacement in any direction results in a restoring force toward Equilibrium Point • Unstable: small displacement in any direction results in a force away from Equilibrium Point • Neutral: small displacement in any direction results in zero force

Mass and Energy

Efficiency & Thermodynamics

Summary • • • Potential Energy function & force The Conservation of Mechanical Energy The Conservation of Energy Mass and Energy /