chapter7

[] __**Chapter 7- Energy/Conservation of Energy**__

**__General Concepts__**

**Energy** is the ability of a system to do work on other systems.Types of energy that all combine into total energy are mechanical energy, chemical energy, thermal energy, electrical energy, nuclear energy, magnetic energy, and several other types. It is measured in Joules (J), which are the equivalent of one kilogram meter squared per second, or one Newton meter.

**Mechanical Energy** is the sum of the __Kinetic Energy "K"__ and the __potential energy "U"__.

**Kinetic energy** is the energy a system has due to its motion. The equation for linear kinetic energy of a system is //K =// //½ mv^2//, where m is the mass of the system and v is the velocity of the system. True total kinetic energy is the sum of linear kinetic energy, which was just described, and rotational kinetic energy (see chapter 9). Rotational kinetic ene. rgy is represented by the equation Kr = ½ Iώ^2 where I is the rotational inertia of the object, and ώ is the rotational velocity of the object. However, if an object in motion is not rotating, there is no rotational kinetic energy, and thus all of the kinetic energy is the linear kinetic energy. **Potential energy** is the energy stored in a system due to its position in a force field (such as gravitational field- see Chapter 11) or its configuration (such as attached to a oscillating spring- see Chapter 14).

__**Types of Potential Energy** __ **- Gravitational Potential Energy-** The potential energy of a system //h// meters above its original position. The equation is // Ug = mgh, // where m is the mass of the system and g is the constant acceleration due to gravity on Earth up to a certain height. (g = 9.8 meters/second squared)

**- Elastic Potential Energy-** The potential energy of an elastic system or an object attached to an elastic object that oscillates back and forth. The equation is //Ue= ½ kx^2//, where k  is the spring constant of the elastic object and x is the distance of the object from the equillibrium of the oscillating system.

<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">**- <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">Electrical Potential Energy- **<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"> The potential energy <span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"> of a charge due to its position in an electric field. For more about electrical potential energy, see Chapters 21-23. <span style="font-family: Arial,Helvetica,sans-serif;">There are also other kinds of potential energy such as chemical potential energy and nuclear potential energy.

__**<span style="font-family: Arial,Helvetica,sans-serif;">Examples **__ <span style="font-family: Arial,Helvetica,sans-serif;">This case is an example of gravitational potential energy, thus the equation // Ug = mgh //<span style="font-family: Arial,Helvetica,sans-serif;">is used. <span style="font-family: Arial,Helvetica,sans-serif;">So, since m = 2 and h = 20, <span style="font-family: Arial,Helvetica,sans-serif;">Ug = (2)(9.8)(20) = 392 Joules.
 * <span style="font-family: Arial,Helvetica,sans-serif;">1) Find the potential energy of a 2kg block at rest on a platform 20 meters above the ground. **

<span style="font-family: Arial,Helvetica,sans-serif;">This case is an example of elastic potential energy, so the equation //Ue= ½ kx^2// <span style="font-family: Arial,Helvetica,sans-serif;">is used. <span style="font-family: Arial,Helvetica,sans-serif;">Since k = 100, and x = 0.5, <span style="font-family: Arial,Helvetica,sans-serif;">Ue = (1/2)(100)(0.5)^2 = 25/2 or 12.5 Joules
 * <span style="font-family: Arial,Helvetica,sans-serif;">2) Find the potential energy of an oscillating block attached to a spring of spring constant 100 when it is 0.5 meters away from the equillibrium position of the system. **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 140%;">__**Conservation of Energy**__ <span style="font-family: Arial,Helvetica,sans-serif;">The law of __Conservation of Energy__ is one of the fundamental laws of science. Energy can be changed from form to form, but it can never be created or destroyed.



<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Sources: <span style="background-color: transparent; color: #0000ff; font-family: Times New Roman; font-size: 16px; text-align: left; vertical-align: auto;">__[]__ <span style="background-color: transparent; color: #0000ff; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">__[]__

__**Conservation of Total Mechanical Energy**__ Mechanical energy is always conserved if the total work done by external forces and internal nonconservative forces is zero. Thus, if mechanical energy is conserved, the sum of the kinetic and potential energy will always remain constant. As a result, if one quantity increases or decreases, the other must decrease or increase. This is demonstrated in the animations below, where a cart/sledder is moving down a frictionless incline, and there is no air resistance. With no external forces doing work on the cart, the total mechanical energy (TME) is conserved at all times. In the right animation, the sledder begins with both kinetic and potential energy, and since it stops (momentarily) with only potential energy, he/she ends up higher up on the slope than the starting position. This is all because of the conservation of total mechanical energy.

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**__Examples__** **1) A 10kg block 5 meters up on an incline is released from rest and falls down the incline and then slides into a spring with spring constant 200. If all the surfaces are frictionless, and there is no air resistance, how far is the spring compressed by the block during the instant that the block-spring system is at rest?**  Because there are no nonconservative forces (air resistance or friction), total mechanical energy is conserved. The initial total mechanical energy (which consisted of only gravitational potential energy) is equal to the final total mechanical energy (which consists only of elastic potential energy). Kinetic energy exists while the block is sliding down the incline and into the spring, but because it is at rest at the initial and final moments, kinetic energy can be neglected. Thus,

**__Nonconservative Forces and Loss of Total Mechanical Energy__**

Usually in the real world, dissipative nonconservative forces exist and often decrease the mechanical energy of a system. However, while mechanical energy is not conserved, total energy (mechanical energy, chemical energy, thermal energy, and any other types of energy) is still conserved. Thus, either chemical or thermal energy would increase if a system lost mechanical energy. Kinetic friction and air resistance are such forces that convert mechanical energy into thermal energy. Below is an example of how a plane with friction does work on a system (the skier) and causes it to lose mechanical energy at the bottom of the slope.

Source: [] __** Examples **__ **__1) A 4 kg block, originally sliding a horizontal frictionless surface with a velocity of 13m/s, slides over a section where the surface is rough. After the block exits the rough surface and resumes sliding on the frictionless surface, its velocity is 8 m/s. How much thermal energy was produced by the friction?__** After sliding through the rough section of the surface, the block was traveling at a slower velocity. That means it lost kinetic energy and subsequently lost total mechanical energy as it did not gain potential energy of any kind. However, according to the law of conservation of energy, the mechanical energy is not lost, but it is converted into another form of energy. In this case, that new form of energy is thermal energy. Thus, the thermal energy produced by the friction is equal to the amount of mechanical energy lost. Thus, Due to the friction, 210 Joules of energy are converted from mechanical energy to thermal energy.

__**2) A block is released from rest at the top of a frictionless, 8 meter high incline. After it reaches the bottom, it slides through a 4 meter long section of rough surface of which the coefficient of kinetic friction is 0.3. After the block exits the rough section of the surface and resumes sliding on the frictionless, flat surface, how fast is it sliding?**__ In this example, the block begins with potential energy, all of which is converted to kinetic energy at the bottome of the incline. Some mechanical energy is then lost due to the rough surface. Because the work done by the rough surface is equal to the amount of energy lost, we can conclude that the sum of the work done by the rough surface and the final kinetic energy is equal to the initial kinetic energy at the bottom of the incline (which is also equal to the potential energy at the top of the incline). Because the gravitational potential energy at the top of the incline is the same as the kinetic energy at the bottom of the incline, the kinetic energy at the bottom of the incline does not need to be calculated. The final velocity after the block exits the rough surface is roughly 9.6 meters per second.

__**Laboratory Experiment**__ The purpose of this lab is to use principles of energy learned in this chapter to calculate the kinetic friction constant of various surfaces.

**__Materials__** 2 Photogates 2 Consoles Several flat surfaces A smooth block

__**Procedure**__ This is a fairly simple lab. Begin by setting up the two photogates a distance x apart (they should not be too far apart) on the first flat surface. Slide the block on the surface through each of the photogates and record your results from the photogates, v1 from the first photogate and v2 from the second photogate. Then repeat this until you have results for three different surfaces. **__Lab Questions__** **__1)__** Calculate the kinetic friction constant of each surface with your recorded data for each surface. Keep your equations in variable form until you have the friction constant in terms of the other variables and fundamental constants. Then plug in the data for the friction constant in the chart. **__ ﻿Sample Experiment __** For the sample lab experiment, a 100g weight was used as the sliding object, and the surfaces used were a lab table, a waxed wooden table, and the top of a large textbook. The data obtained from the three trials was:  Now, maniputlate equations using your knoweldge of energy and loss of total mechanical energy to obtain the kinetic friction constant in terms of other variables and fundamental constants.  Now, substitute the data in for each and obtain the kinetic friction constant for each surface  Thus, the finished chart should be:  The coeffecient of kinetic friction has now been obtained for three different surfaces, and the lab is complete!


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