7 Conservation of Energy Potential Energy The Conservation of Mechanical Energy The Conservation of Energy Mass and Energy Hk: 23, 27, 39, 47, 55, 65, 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 100N/m spring is compressed 0.2m. F = -kx = -(100N/m)(0.2m) = -20N U = ½kx2 = ½(100N/m)(0.2m)2 = 2J /

Gravitational Potential Energy

Ex. Gravitational Potential Energy Ex: A 2kg object experiences weight (2kg)(9.8N/kg) = 19.6N. At 3m above the floor it has a stored energy of mgy: (2kg)(9.8N/kg)(3m) = 48.8Nm = 48.8J. /

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

Conservation of Mechanical Energy

KE E Ug

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

Energy E1 E2 E3 Kinetic ½mv22 PE-g mgh PE-spring ½kx2 Totals 1 2 3

y y Energies and speeds are same at height y Energy E(h) E(y) Kinetic ½mv2 PE-g mgh mgy Totals ½mv2 + mgy Energies and speeds are same at height y Accelerations at y are not same y y

Work Energy with Friction

Energy Ei Ef Kinetic ½mvi2 PE-g Thermal fks Totals Example: The smaller the frictional force fk, the larger the distance, s, it will travel before stopping. s

A 2. 00kg ball is dropped from rest from a height of 1 A 2.00kg ball is dropped from rest from a height of 1.0m above the floor. The ball rebounds to a height of 0.500m. A movie-frame type diagram of the motion is shown below. 1 2 5 4 3 Type E1 E2 E3 E4 E5 gravita-tional mg(1) mg(1/2) kinetic ½ m(v2)2 ½ m(v4)2 elastic PE-elastic thermal E-thermal

Calculate v2: (use 1st and 2nd columns) mg(1) = ½ m(v2)2. g = ½ (v2)2. By energy conservation, the sum of all energies in each column is the same, = E1 = mg(1) = 19.6J Calculate v2: (use 1st and 2nd columns) mg(1) = ½ m(v2)2. g = ½ (v2)2. v2 = 4.43m/s Calculate PE-thermal: (use 1st and 5th columns) mg(1) = mg(1/2) + PE-thermal mg(1/2) = PE-thermal PE-thermal = 9.8J

Calculate PE-elastic: (use 1st and 3rd columns) PE-elastic + PE-thermal = mg(1) PE-elastic + 9.8 = 19.6 PE-elastic = 9.8J Calculate v4: (use 1st and 4th columns) ½ m(v4)2 + PE-thermal = mg(1) ½ m(v4)2 + 9.8 = 19.6 ½ m(v4)2 = 9.8 (v4)2 = 2(9.8)/2 v4 = 3.13m/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 /