Energy

system

A system just being whatever section of the universe you happen to be talking about at the time

A 0.3 kg box is held against a spring causing the spring to compress 0.15 m. The spring has a spring constant of 400 N/m. When released, the spring will launch the box vertically. What is the speed of the box when it has reached a height of 0.5 m?

All of the energy is initially stored in the spring as spring potential energy. At the end, the energy has been transferred to the box. Some of this energy is gravitational potential energy and some of the energy is kinetic.

A student attaches a 2.0 kg mass to a 1.2 m string and pulls the string back to an angle of 25° relative to the vertical. The student releases the mass, and it swings back and forth. What is the speed of the mass when it reaches the bottom of the swing?

All the energy at the beginning is gravitational potential energy. This is converted entirely to kinetic energy at the bottom of the swing.

A 0.5 kg box is pressed against a spring causing the spring to compress 25 cm. When the spring is released, the box will be launched vertically into the air. What spring constant is required if the maximum height the box will reach is 1.5 m?

At the start of the process, all the energy is stored in the spring as spring potential energy. At the end of the process, all the energy has been transferred to the box, which is stored as gravitational potential energy.

A 2.0 kg box is held against a spring causing the spring to compress 20 cm. How fast will the box move if the spring is released? The spring constant is 300 N/m.

At the start of this process, all of the energy is stored in the spring as spring potential energy. At the end of the process, all of the energy has been transferred to the box, which is stored as kinetic energy.

Calculating Power from Energy

Because work is energy transfer, power is also the rate at which energy is expended. A 60-W light bulb, for example, expends 60 J of energy per second. Great power means a large amount of work or energy developed in a short time. For example, when a powerful car accelerates rapidly, it does a large amount of work and consumes a large amount of fuel in a short time.

Energy

Energy is a scalar measurement which means that energy does not require horizontal or vertical energy components. However, some questions still require force and displacement vector components.

energy definition

Energy is the ability to change a system and is measured in units of Joules.

Work

Formally, the work done on a system by a constant force is defined to be the product of the component of the force in the direction of motion times the distance through which the force acts. For one-way motion in one dimension, this is expressed in equation form as

Converting Between Potential Energy and Kinetic Energy

Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to 𝑚𝑔ℎ on it, thereby increasing its kinetic energy by that same amount (by the work-energy theorem). We will find it more useful to consider just the conversion of PEg to KEsize without explicitly considering the intermediate step of work. (See Example 7.7.) This shortcut makes it is easier to solve problems using energy (if possible) rather than explicitly using forces.

horsepower

It is impressive that this woman's useful power output is slightly less than 1 horsepower (1 hp=746 W)size 12{ \( 1" hp"="746"" W" \) } {}! People can generate more than a horsepower with their leg muscles for short periods of time by rapidly converting available blood sugar and oxygen into work output. (A horse can put out 1 hp for hours on end.) Once oxygen is depleted, power output decreases and the person begins to breathe rapidly to obtain oxygen to metabolize more food—this is known as the aerobic stage of exercise. If the woman climbed the stairs slowly, then her power output would be much less, although the amount of work done would be the same.

power definition.

Power is the rate that energy is transferred in or out of a system.

instantaneous power & instantaneous speed

Rate of power or speed at a given point in time

A student starting at a height of 6.0 m carries an 8.0 kg box down a flight of stairs to a final height of 2.0 m. What was the box's change in gravitational potential energy?

Since the box lost energy in this example, the box performed work on the student (the surroundings). In other words, the box lost energy, and the surroundings gained energy.

A 1200 kg car travels from rest to a final speed of 30 m/s in 4.0 seconds. What is the power rating of the car's engine?

The car started with no energy and ended with kinetic energy. The motor performed work on the car.

A student places a box on top of an inclined ramp that has a height of 0.8 m. The ramp's angle of incline is 20° and friction is negligible. What is the speed of the box when the box reaches the bottom of the ramp?

The energy starts in the box as gravitational potential energy. Even though the box is on a surface, it is still above zero height. This energy is converted to kinetic energy at the end.

A student applies a horizontal force to a box causing the box to move. The student starts to get tired, and the applied force decreases. The graph of force vs. displacement is shown below. How much work was performed on the box?

The energy transferred to the box (work) is equal to the area under the above graph. You can break this area into a series of shapes and solve for those areas.

Hooke's Law

The equation for this force vs. spring displacement graph is called Hooke's law and is written as:

MAKING CONNECTIONS: USEFULNESS OF THE ENERGY CONSERVATION PRINCIPLE

The fact that energy is conserved and has many forms makes it very important. You will find that energy is discussed in many contexts, because it is involved in all processes. It will also become apparent that many situations are best understood in terms of energy and that problems are often most easily conceptualized and solved by considering energy.

Force of Gravity

The force of gravity and the normal force acting on the package are perpendicular to the displacement and do no work. Moreover, they are also equal in magnitude and opposite in direction so they cancel in calculating the net force. The net force arises solely from the horizontal applied force 𝐅app and the horizontal friction force 𝐟. Thus, as expected, the net force is parallel to the displacement, so that 𝜃=0º and cos𝜃=1size 12{"cos"q=1} {}, and the net work is given by

kinetic energy (KE)

The quantity 1/2mv² in the work-energy theorem is defined to be the translational kinetic energy (KE) of a mass 𝑚 moving at a speed 𝑣. (Translational kinetic energy is distinct from rotational kinetic energy, which is considered later.) In equation form, the translational kinetic energy, is the energy associated with translational motion. Kinetic energy is a form of energy associated with the motion of a particle, single body, or system of objects moving together.

THE WORK-ENERGY THEOREM

This expression is called the work-energy theorem, and it actually applies in general (even for forces that vary in direction and magnitude), although we have derived it for the special case of a constant force parallel to the displacement. The theorem implies that the net work on a system equals the change in the quantity 1/2mv² This quantity is our first example of a form of energy.

A particular computer has a power rating of 130 W. How much will it cost to run the computer for 24 hours if energy costs $0.26 for every kW-hr?

This is a unit conversion problem. Cancel out all of the units that are not related to cost.

How much work was performed on the box?

W= area W = (40 N)(10m) + (20 N)(5 m) +¹/₂ (20 N)(5m) W = 550 J

Net Work and the Work-Energy Theorem

We know from the study of Newton's laws in Dynamics: Force and Newton's Laws of Motion that net force causes acceleration. We will see in this section that work done by the net force gives a system energy of motion, and in the process we will also find an expression for the energy of motion.

Some of the Many Forms of Energy

What are some other forms of energy? You can probably name a number of forms of energy not yet discussed. Many of these will be covered in later chapters, but let us detail a few here. Electrical energy is a common form that is converted to many other forms and does work in a wide range of practical situations. Fuels, such as gasoline and food, carry chemical energy that can be transferred to a system through oxidation. Chemical fuel can also produce electrical energy, such as in batteries. Batteries can in turn produce light, which is a very pure form of energy. Most energy sources on Earth are in fact stored energy from the energy we receive from the Sun. We sometimes refer to this as radiant energy, or electromagnetic radiation, which includes visible light, infrared, and ultraviolet radiation. Nuclear energy comes from processes that convert measurable amounts of mass into energy. Nuclear energy is transformed into the energy of sunlight, into electrical energy in power plants, and into the energy of the heat transfer and blast in weapons. Atoms and molecules inside all objects are in random motion. This internal mechanical energy from the random motions is called thermal energy, because it is related to the temperature of the object. These and all other forms of energy can be converted into one another and can do work.

The three methods systems use to store mechanical energy

When an object or system experiences a change in mechanical energy, the change in energy is called "work." When work (the change in energy) is positive, work is done on the system. When work (the change in energy) is negative, work is done by the system.

Power Equation

Where P is power in units of watts, W is the work transferred in or out of the system, and t is the time it took to transfer the energy.

Work as the transfer of energy

Work done on an object is the amount of energy it gains or loses.

Potential Energy and Conservative Forces

Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy (PE) for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative. That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.

work

Work is the amount of energy that is transferred into or out of a system by application of a force over some displacement.

What It Means to Do Work

⚛ The scientific definition of work reveals its relationship to energy—whenever work is done, energy is transferred. ⚛ For work, in the scientific sense, to be done, a force must be exerted and there must be displacement in the direction of the force.

Gravitational potential energy!

the energy due to an object's position in a gravity field

kinetic energy

the energy of motion

Spring energy

the energy stored in a spring due to a stretch or compression of that spring

force

⚛ If the applied force that is transferring the work is constant, then W = Fdcosθ. ⚛ If the force is not constant, then the work can be found by solving for the area under a graph of force vs. displacement.

A 5.0 kg box is held at a height of 2.0 m. The box is released. What is the velocity of the box just before it hits the ground?

⚛ It's possible to solve this problem using traditional kinematic equations; however, this example will approach the problem from an energy perspective. ⚛ At the start of this process, all of the energy in the system is stored as gravitational potential energy in the box. At the end of the process, all of the energy in the system is stored as kinetic energy. energy start = energy end

Kinetic Energy

⚛ Kinetic energy is the energy of motion. Any object in motion has kinetic energy. If an object speeds up, it has gained kinetic energy. If an object slows down, it has lost kinetic energy. Objects with more inertia (measured by mass) require more energy to move. ⚛ Where K is the amount of kinetic energy measured in Joules, m is the mass of the moving object, and v is the speed of the moving object.

Lesson Sections

⚛ 7.1 Work & Mechanical Forms of Energy ⚛ 7.2 Conservation of Energy ⚛ 7.3 Power

Book

⚛ 7.1 Work: The Scientific Definition ⚛ 7.2 Kinetic Energy and the Work-Energy Theorem ⚛ 7.3 Gravitational Potential Energy ⚛ 7.4 Conservative Forces and Potential Energy ⚛ 7.6 Conservation of Energy ⚛ 7.7 Power

A 6.0 kg box moving with an initial speed of 15 m/s slows to a final speed of 8 m/s over a distance of 4.5 m due to friction. What is the friction force on the box?

⚛ Again, this can be solved using a combination of forces and kinematics or through energy analysis. ⚛ The box has kinetic energy at the start and less kinetic energy at the end. Where did this energy go? The box performed work on the surroundings through friction. energy start +/- work = energy end

Gravitational potential energy

⚛ Another mechanical form of energy is potential energy. Potential energy is a way of saying "stored energy that might potentially become kinetic energy." Potential energy is notated by a U followed by a subscript that indicates the specific type of potential energy. ⚛ One form of potential energy is gravitational potential energy, or energy that an object has due to its height in a gravitational field. The greater the height, the greater the gravitational energy. ⚛ Where Ug is the amount of gravitational energy, m is the mass of the object, g is the gravitational field strength, and h is the height of the object.

Work Done Against Gravity

⚛ Climbing stairs and lifting objects is work in both the scientific and everyday sense—it is work done against the gravitational force. When there is work, there is a transformation of energy. The work done against the gravitational force goes into an important form of stored energy that we will explore in this section. ⚛ Let us calculate the work done in lifting an object of mass 𝑚 through a height ℎ, such as in Figure 7.5. If the object is lifted straight up at constant speed, then the force needed to lift it is equal to its weight 𝑚𝑔. The work done on the mass is then 𝑊 = 𝐹𝑑 = 𝑚𝑔 "W = Fd = mgh. We define this to be the gravitational potential energy (PEg) put into (or gained by) the object-Earth system. This energy is associated with the state of separation between two objects that attract each other by the gravitational force. For convenience, we refer to this as the PEg gained by the object, recognizing that this is energy stored in the gravitational field of Earth. Why do we use the word "system"? Potential energy is a property of a system rather than of a single object—due to its physical position. An object's gravitational potential is due to its position relative to the surroundings within the Earth-object system. The force applied to the object is an external force, from outside the system. When it does positive work it increases the gravitational potential energy of the system. Because gravitational potential energy depends on relative position, we need a reference level at which to set the potential energy equal to 0. We usually choose this point to be Earth's surface, but this point is arbitrary; what is important is the difference in gravitational potential energy, because this difference is what relates to the work done. The difference in gravitational potential energy of an object (in the Earth-object system) between two rungs of a ladder will be the same for the first two rungs as for the last two rungs.

Lesson Objectives

⚛ Define and calculate kinetic, gravitational, and spring energy ⚛ Apply the law of conservation of energy to solve for unknown variables in a system (velocity, height, mass, spring constant, etc.) ⚛ Calculate the change in energy of a system when work is done on or by the system ⚛ Calculate the rate of energy transferred into or out of a system

Potential Energy of Spring

⚛ Energy can be created nor destroyed ⚛ A conservative system is one that doesn't lose energy through work

Work and Mechanical Forms of Energy

⚛ Energy is commonly defined as the ability to do work, where work is given by an equation that will be explored later in the lesson. ⚛ This definition often causes some confusion among entry- level physics students; it can be difficult to wrap one's head around an equation as a definition. ⚛ Perhaps a more approachable definition of energy might be that energy is the ability to create change. Energy is a scientific construct that allows us to determine how much change any particular system can experience. The change could be movement, a spring stretch, a temperature increase, a chemical reaction, the emission of light, or any of a number of different possibilities. Energy is a method of keeping track of what's possible. ⚛ Energy can be stored in several different ways. Some of the more familiar ways to store energy might include: chemical, solar, hydro-electric, electromagnetic, gravitational, kinetic, and nuclear, among many others. In this lesson you will focus on three different methods commonly used to store energy. These three methods are often referred to as mechanical forms of energy.

Introduction

⚛ Energy is one of the most pivotal concepts in science; every branch of science includes a study of energy in its various forms. It is also one of the most elusive scientific concepts for some students to understand. ⚛ This lesson will introduce the scientific definition of energy and explore the three forms of mechanical energy in more detail.

Law of Conservation of Energy

⚛ Energy, as we have noted, is conserved, making it one of the most important physical quantities in nature. The law of conservation of energy can be stated as follows: ⚛ Total energy is constant in any process. It may change in form or be transferred from one system to another, but the total remains the same.

Lab Objectives

⚛ Lab: Use principles of conservation of energy to explain the relationship between the mass of the skater and the time to complete the track ⚛ Lab: Use principles of conservation of energy to explain the relationship between the vertical position of the ramp and the time to complete the track ⚛ Lab: Use principles of conservation of energy to explain the relationship between the depth of the ramp and the skater's maximum kinetic energy ⚛ Lab: Design and conduct an experiment and use principles of conservation of energy to explain the results

New Spring

⚛ Now repeat the same process but with a stiffer spring that is more difficult to stretch. The new spring would require more force to achieve the same stretch distance. The force vs. stretch graph would now look like: ⚛ The slope of this graph is related to the stiffness of the spring; the stiffer the spring, the steeper the slope. Specifically, the value of the slope is the amount of force required to stretch or compress a spring by 1.0 m. This value is called the spring constant and is represented by the symbol k.

Power

⚛ Power is the rate of change of energy experienced by a system. Power is represented by the symbol P and has units of Joules / second, which has been combined into the unit of a watt (W). Another common set of units for power is horsepower (hp). The conversion factor between watts and horsepower is: ⚛ 746 W = 1 hp ⚛ Since a system's change in energy is the work done on or by the system, this means that power is work / time or:

What is Power?

⚛ Power—the word conjures up many images: a professional football player muscling aside his opponent, a dragster roaring away from the starting line, a volcano blowing its lava into the atmosphere, or a rocket blasting off, as in Figure 7.22. ⚛ These images of power have in common the rapid performance of work, consistent with the scientific definition of power (𝑃size 12{P} {}) as the rate at which work is done.

A student plans to use a motor to pull a 10.0 kg bucket of water out of a 100 m well. If the motor has a power rating of 2.0 hp, how long will it take to lift the water out of the well?

⚛ Set the bottom of the well to be 0 height. This means the bucket starts with no energy and ends with gravitational potential energy. The motor performs work on the bucket. ⚛ First solve for the work done.

A 65 kg bungee jumper starts from rest at a height of 12.0 m. She jumps off the platform and reaches the lowest point of her motion (maximum bungee cord stretch) at 2.5 m above the ground. If the bungee cord has a spring constant of 250 N/m and an un-stretched length of 3.0 m, how much energy was lost to the surroundings during the initial fall?

⚛ The energy before the jump is all gravitational. The energy at the bottom of the jump is a combination of spring energy in the cord and gravitational potential energy for the jumper. Solve for the work done by the jumper on the surroundings. ⚛ The bungee cord experiences an amount of stretch equal to the difference between the total height and a combination of the unstretched length and final distance from the ground:

Conservation of Energy

⚛ The first law of thermodynamics states that all energy in the universe is conserved. All the energy that exists is all the energy that will ever exist; it cannot be created or destroyed. ⚛ This law means that when a system gains or loses energy, that energy has to come from somewhere or go to somewhere in equal amounts. If a student lifts a box up over her head, the amount of work she loses must be equal to the amount of energy the box gains. ⚛ In other words, the following is always true: ⚛ Energy a system has at the start +/- work = energy the system has at the end. ⚛ This seems simple enough, but complex systems can make this simple idea feel complicated.

Angle of Ramp

⚛ The given angle was not necessary in the solution of the last example problem. The angle of the ramp is not important when considering gravitational potential energy since it is based on the object's height and not the object's overall displacement.

Spring potential energy

⚛ The last form of mechanical energy is elastic or spring potential energy. This is a type of energy that is stored due to stretching or compressing a spring. ⚛ A key element of spring or elastic energy is that the object that has been stretched or compressed must be able to return to its original shape. For example, if a car crashes into a wall and the front end of the car crumples, this is not elastic energy because the car will not restore itself to its original shape when it is pulled away from the wall. Rubber bands, springs, bungee cords, etc., are good examples of objects that can store elastic or spring potential energy.

Law of Conservation of Energy

⚛ The law of conservation of energy notes that energy cannot be created or destroyed. ⚛ It can, however, be transferred in or out of a system or converted from one form of energy to another form of energy. ⚛ A mathematical way of stating this is that the energy a system has at the start +/- work done on or by the system = the energy the system has at the end.

Calculating Energy Costs

⚛ The motivation to save energy has become more compelling with its ever-increasing price. Armed with the knowledge that energy consumed is the product of power and time, you can estimate costs for yourself and make the necessary value judgments about where to save energy. Either power or time must be reduced. It is most cost-effective to limit the use of high-power devices that normally operate for long periods of time, such as water heaters and air conditioners. This would not include relatively high power devices like toasters, because they are on only a few minutes per day. It would also not include electric clocks, in spite of their 24-hour-per-day usage, because they are very low power devices. It is sometimes possible to use devices that have greater efficiencies—that is, devices that consume less power to accomplish the same task. One example is the compact fluorescent light bulb, which produces over four times more light per watt of power consumed than its incandescent cousin. ⚛ Modern civilization depends on energy, but current levels of energy consumption and production are not sustainable. The likelihood of a link between global warming and fossil fuel use (with its concomitant production of carbon dioxide), has made reduction in energy use as well as a shift to non-fossil fuels of the utmost importance. Even though energy in an isolated system is a conserved quantity, the final result of most energy transformations is waste heat transfer to the environment, which is no longer useful for doing work. As we will discuss in more detail in Thermodynamics, the potential for energy to produce useful work has been "degraded" in the energy transformation.

Transformation of Energy

⚛ The transformation of energy from one form into others is happening all the time. The chemical energy in food is converted into thermal energy through metabolism; light energy is converted into chemical energy through photosynthesis. In a larger example, the chemical energy contained in coal is converted into thermal energy as it burns to turn water into steam in a boiler. ⚛ This thermal energy in the steam in turn is converted to mechanical energy as it spins a turbine, which is connected to a generator to produce electrical energy. (In all of these examples, not all of the initial energy is converted into the forms mentioned. This important point is discussed later in this section.) Another example of energy conversion occurs in a solar cell. Sunlight impinging on a solar cell (see Figure 7.20) produces electricity, which in turn can be used to run an electric motor. Energy is converted from the primary source of solar energy into electrical energy and then into mechanical energy.

More on work

⚛ This change in mechanical energy (work) is usually performed through the application of a force for some distance. ⚛ For example, a driver applies the breaks of a car for some distance and the car slows as a result. A vertical upward force must be applied to a box in order to lift the box up. Forces must be applied to springs in order to cause the springs to stretch or compress. All of these systems experience work (an increase or decrease in mechanical energy) through a force. ⚛ If this applied force is a constant force (i.e., the force does not increase or decrease), then the work can be calculated using the equation: ⚛ Where W is the work (change in energy), is the component of the applied force that is parallel to the motion, and d is the distance traveled. ⚛ Again, it's very important to understand that this particular work equation is only accurate if the applied force is constant throughout the motion. If the force increases or decreases, this form of the work equation cannot be used.

A student applies a constant horizontal 50 N force to a 25 kg box that is initially at rest. The student moves the box a distance of 3.5 m. What is the speed of the box at the end of the motion? (Assume friction is negligible.)

⚛ This problem can be solved using a combination of Newton's second law and kinematics or through the use of energy analysis. ⚛ At the start, the box has no energy as it's motionless at zero height. At the end, the box has kinetic energy. Where did this energy come from? The student applied a force to the box, which means that she performed work on the box. ⚛ Energy start +/- work = energy end

Power Problems.

⚛ Total work done at a given time interval ⚛ Power is rate at which work is done

Work Transfers Energy

⚛ What happens to the work done on a system? Energy is transferred into the system, but in what form? Does it remain in the system or move on? The answers depend on the situation. For example, if the lawn mower in Figure 7.2(a) is pushed just hard enough to keep it going at a constant speed, then energy put into the mower by the person is removed continuously by friction, and eventually leaves the system in the form of heat transfer. In contrast, work done on the briefcase by the person carrying it up stairs in Figure 7.2(d) is stored in the briefcase-Earth system and can be recovered at any time, as shown in Figure 7.2(e). In fact, the building of the pyramids in ancient Egypt is an example of storing energy in a system by doing work on the system. Some of the energy imparted to the stone blocks in lifting them during construction of the pyramids remains in the stone-Earth system and has the potential to do work. ⚛ In this section we begin the study of various types of work and forms of energy. We will find that some types of work leave the energy of a system constant, for example, whereas others change the system in some way, such as making it move. We will also develop definitions of important forms of energy, such as the energy of motion.

A student applies a constant 200 N at an angle 30° below the horizontal to a 40.0 kg box for a distance of 3.0 m. How much work did the student do on the box?

⚛ What if the applied force is NOT a constant applied force (the force increases or decreases)? How can work on the system be determined? ⚛ Recall that the energy stored in a spring could be determined by solving for the area under the force vs spring displacement graph. It turns out that this trick works for all non-constant forces. ⚛ Work is always equal to the area under the appropriate applied force vs. displacement graph. As before, consider only the component of force that is parallel to the direction of the displacement.

Hooke Law

⚛ Where Fs is the force applied to a spring, k is the spring constant, and Δx is the distance the spring was stretched or compressed. It's also interesting to note that the area under the line on this graph is equal to the spring potential energy stored in the spring. The area under the line forms a triangle which has an area of... ⚛ For perspective, a slinky has a spring constant of about 2.0 N/m while most trampoline springs tend to be in the 4000 N/m range. In other words, a slinky takes about 2 N of force to stretch 1 meter, whereas a trampoline spring requires upwards of 4000 N of force to stretch 1 meter.

Spring Energy

⚛ Where is the amount of spring energy stored in a stretched or compressed spring, k is a constant called the spring constant, and Δx is the distance the spring was stretched or compressed. ⚛ Take a moment to consider the spring constant. ⚛ Picture connecting a spring to a sensor that measures applied force. The sensor can be used to measure the amount of force required to achieve various lengths of spring stretch. The graph of the force on the spring () vs. the amount of spring stretch (Δx) would look like:

A 700 kg car slows from an initial speed of 30 m/s to 20 m/s. What was the car's change in kinetic energy?

⚛ Work is a change in system energy. ⚛ If the system gains energy, then work is done by the surroundings on the system. ⚛ If a system loses energy, then work is done by the system on the surroundings In the last example, the car lost 175 kJ of energy. This means that 175 kJ of work was done by the car on its surroundings, or in other words, the car lost energy to its surroundings.