Conservation of Energy

1. Conservation of Energy Chapter 11

2. Conservation of Energy • The Law of Conservation of Energy simply states that: • The energy of a system is constant. • Energy cannot be created nor destroyed. • Energy can only change form (e.g. electrical to mechanical to potential, etc). • True for any system with no external forces. ET = KE + PE + Q • KE = Kinetic Energy • PE = Potential Energy • Q = Internal Energy [kinetic energy due to the motion of molecules (translational, rotational, vibrational)]

3. Conservation of Energy Energy Mechanical Nonmechanical Potential Kinetic Elastic Gravitational

4. Conservation of Mechanical Energy • Mechanical Energy: • If Internal Energy is ignored: ME = KE + PE • PE could be a combination of gravitational and elastic potential energy, or any other form of potential energy. • The equation implies that the mechanical energy of a system is always constant. • If the Potential Energy is at a maximum, then the system will have no Kinetic Energy. • If the Kinetic Energy is at a maximum, then the system will not have any Potential Energy.

5. Conservation of Mechanical Energy ME = KE + PE KEinitial + PEinitial = KEfinal + PEfinal

6. Example 4: √ • A student with a mass of 55 kg goes down a frictionless slide that is 3 meters high. What is the student’s speed at the bottom of the slide? KEinitial + PEinitial = KEfinal + PEfinal • KEinitial = 0 because v is 0 at top of slide. • PEinitial = mgh • KEfinal = ½ mv2 • PEfinal = 0 at bottom of slide. • Therefore: • PEinitial = KEfinal • mgh = ½ mv2 • v = 2gh • V = (2)(9.81 m/s2)(3 m) = 7.67 m/s

7. Example 5: • A student with a mass of 55 kg goes goes down a non-frictionless slide that is 3 meters high. • Compared to a frictionless slide the student’s speed will be: • the same. • less than. • more than. • Why? • Because energy is lost to the environment in the form of heat (Q) due to friction.

8. Example 5 (cont.) • Does this example reflect conservation of mechanical energy? • No, because of friction. • Is the law of conservation of energy violated? • No, some of the “mechanical” energy is lost to the environment in the form of heat.

9. Energy of Collisions • While momentum is conserved in all collisions, mechanical energy may not. • Elastic Collisions: Collisions where the kinetic energy both before and after are the same. • Inelastic Collisions: Collisions where the kinetic energy after a collision is less than before. • If energy is lost, where does it go? • Thermal energy, sound.

10. Collisions • Two types • Elastic collisions – objects may deform but after the collision end up unchanged • Objects separate after the collision • Example: Billiard balls • Kinetic energy is conserved (no loss to internal energy or heat) • Inelastic collisions – objects permanently deform and / or stick together after collision • Kinetic energy is transformed into internal energy or heat • Examples: Spitballs, railroad cars, automobile accident

11. Example 4 • Cart A approaches cart B, which is initially at rest, with an initial velocity of 30 m/s. After the collision, cart A stops and cart B continues on with what velocity? Cart A has a mass of 50 kg while cart B has a mass of 100kg. B A

12. Diagram the Problem pA1 = mvA1 pB2 = mvB2 B A Before Collision: pB1 = mvB1 = 0 After Collision: pA2 = mvA2 = 0

13. Solve the Problem 0 0 • pbefore = pafter • mAvA1 + mBvB1 = mAvA2 + mBvB2 • mAvA1 = mBvB2 • (50 kg)(30 m/s) = (100 kg)(vB2) • vB2 = 15 m/s • Is kinetic energy conserved? • KEi =? KEf

14. Solve the Problem • mA = 50kg vA1 = 30m/s • mB = 100kg vB2 = 15m/s • Is kinetic energy conserved? • KEi =? KEf • KEi = Sum(½ mivi2) • KEf = Sum(½ mfvf2)

15. Example 5 • Cart A approaches cart B, which is initially at rest, with an initial velocity of 30 m/s. After the collision, cart A and cart B continue on together with what velocity? Cart A has a mass of 50 kg while cart B has a mass of 100kg. Per 7 B A

16. Diagram the Problem pA1 = mvA1 pB2 = mvB2 pA2 = mvA2 B A Before Collision: pB1 = mvB1 = 0 After Collision: Note: Since the carts stick together after the collision, vA2 = vB2 = v2.

17. Solve the Problem 0 • pbefore = pafter • mAvA1 + mBvB1 = mAvA2 + mBvB2 • mAvA1 = (mA + mB)v2 • (50 kg)(30 m/s) = (50 kg + 100 kg)(v2) • v2 = 10 m/s • Is kinetic energy conserved? • KEi =? KEf

18. Key Ideas • Conservation of energy: Energy can be converted from one form to another, but it is always conserved. • In inelastic collisions, some energy will be lost as heat • ET = KE + PE + Q

19. Key Ideas • Gravitational Potential Energy is the energy that an object has due to its vertical position relative to the Earth’s surface. • Elastic Potential Energy is the energy stored in a spring or other elastic material. • Hooke’s Law: The displacement of a spring from its unstretched position is proportional the force applied. • Conservation of energy: Energy can be converted from one form to another, but it is always conserved.

20. Simple Harmonic Motion & Springs • Simple Harmonic Motion: • An oscillation around an equilibrium position in which a restoring force is proportional the the displacement. • For a spring, the restoring force F = -kx. • The spring is at equilibrium when it is at its relaxed length. • Otherwise, when in tension or compression, a restoring force will exist.

21. Simple Harmonic Motion & Springs • At maximum displacement (+ x): • The Elastic Potential Energy will be at a maximum • The force will be at a maximum. • The acceleration will be at a maximum. • At equilibrium (x = 0): • The Elastic Potential Energy will be zero • Velocity will be at a maximum. • Kinetic Energy will be at a maximum

22. Harmonic Motion & The Pendulum • Pendulum: Consists of a massive object called a bob suspended by a string. • Like a spring, pendulums go through simple harmonic motion as follows. T = 2π√l/g Where: • T = period • l = length of pendulum string • g = acceleration of gravity • Note: • This formula is true for only small angles of θ. • The period of a pendulum is independent of its mass.

23. Conservation of ME & The Pendulum • In a pendulum, Potential Energy is converted into Kinetic Energy and vise-versa in a continuous repeating pattern. • PE = mgh • KE = ½ mv2 • MET = PE + KE • MET = Constant • Note: • Maximum kinetic energy is achieved at the lowest point of the pendulum swing. • The maximum potential energy is achieved at the top of the swing. • When PE is max, KE = 0, and when KE is max, PE = 0.