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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.
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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
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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
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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
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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.
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Conservation of Energy

- Individual energy levels change.
- Sum of all individual energies is constant.
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Ex. Conservation of Mechanical Energy: Object dropped from height h above floor.

Example: The smaller the frictional force fk, the larger the distance, s, it will travel before stopping.

5

2

4

3

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.

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

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

Summary

- Potential Energy function & force
- The Conservation of Mechanical Energy
- The Conservation of Energy
- Mass and Energy
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