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. E T = 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 Energy Mechanical Kinetic Potential Gravitational Elastic Nonmechanical
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 KE initial + PE initial = KE final + PE final
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? KE initial + PE initial = KE final + PE final KE initial = 0 because v is 0 at top of slide. PE initial = mgh KE final = ½ mv 2 PE final = 0 at bottom of slide. –Therefore: PE initial = KE final mgh = ½ mv 2 v = 2gh V = (2)(9.81 m/s 2 )(3 m) = 7.67 m/s √
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: 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 100kg. A B
Diagram the Problem A B Before Collision: p B1 = mv B1 = 0 After Collision: p A2 = mv A2 = 0 p A1 = mv A1 p B2 = mv B2
Solve the Problem p before = p after m A v A1 + m B v B1 = m A v A2 + m B v B2 m A v A1 = m B v B2 (50 kg)(30 m/s) = (100 kg)(v B2 ) v B2 = 15 m/s Is kinetic energy conserved? 0 0
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. A B Per 7
Diagram the Problem A B Before Collision: p B1 = mv B1 = 0 After Collision: p A1 = mv A1 p B2 = mv B2 p A2 = mv A2 Note: Since the carts stick together after the collision, v A2 = v B2 = v 2.
Solve the Problem p before = p after m A v A1 + m B v B1 = m A v A2 + m B v B2 m A v A1 = (m A + m B )v 2 (50 kg)(30 m/s) = (50 kg + 100 kg)(v 2 ) v 2 = 10 m/s Is kinetic energy conserved? 0
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 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.This formula is true for only small angles of θ. 2.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 –ME T = PE + KE –ME T = Constant Note: 1.Maximum kinetic energy is achieved at the lowest point of the pendulum swing. 2.The maximum potential energy is achieved at the top of the swing. 3.When PE is max, KE = 0, and when KE is max, PE = 0.