Conservation of Energy Chapter 11 Conservation of Energy

Conservation of Energy Chapter 11

Conservation of Energy • The Law of Conservation of Energy simply states that: 1. The energy of a system is constant. 2. Energy cannot be created nor destroyed. 3. 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)]

Conservation of Energy Mechanical Kinetic Nonmechanical Potential Gravitational Elastic

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.

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

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 mv 2 PEfinal = 0 at bottom of slide. – Therefore: • • PEinitial = KEfinal mgh = ½ mv 2 v = √ 2 gh V = (2)(9. 81 m/s 2)(3 m) = 7. 67 m/s

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

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.

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.

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

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 100 kg. A B

Diagram the Problem A B Before Collision: p. A 1 = mv. A 1 p. B 1 = mv. B 1 = 0 After Collision: p. A 2 = mv. A 2 = 0 p. B 2 = mv. B 2

Solve the Problem • pbefore = pafter 0 0 • m. Av. A 1 + m. Bv. B 1 = m. Av. A 2 + m. Bv. B 2 • m. Av. A 1 = m. Bv. B 2 • (50 kg)(30 m/s) = (100 kg)(v. B 2) • v. B 2 = 15 m/s • Is kinetic energy conserved? • KEi =? KEf

Solve the Problem • m. A = 50 kg v. A 1 = 30 m/s • m. B = 100 kg v. B 2 = 15 m/s • • Is kinetic energy conserved? KEi =? KEf KEi = Sum(½ mivi 2) KEf = Sum(½ mfvf 2)

Example 5 Per 7 • 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 100 kg. A B

Diagram the Problem A B Before Collision: p. A 1 = mv. A 1 p. B 1 = mv. B 1 = 0 After Collision: p. A 2 = mv. A 2 p. B 2 = mv. B 2 Note: Since the carts stick together after the collision, v. A 2 = v. B 2 = v 2.

Solve the Problem • pbefore = pafter • m. Av. A 1 + m. Bv. B 1 = m. Av. A 2 + m. Bv. B 2 • m. Av. A 1 = (m. A + m. B)v 0 2 • (50 kg)(30 m/s) = (50 kg + 100 kg)(v 2) • v 2 = 10 m/s • Is kinetic energy conserved? • KEi =? KEf

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

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.

Simple Harmonic Motion & Springs • Simple Harmonic Motion: – An oscillation around an equilibrium position in which a restoring force is proportional 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.

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

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: 1. 2. This formula is true for only small angles of θ. The period of a pendulum is independent of its mass.

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 mv 2 MET = PE + KE MET = Constant Note: 1. 2. 3. 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.