6.0 Energy Conservation. 6.1 Work

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1 Phys 300/301 Physics: Algebra/Trig Eugene Hecht, 3e. Prepared 1/09/ Energy Conservation After Newtonian mechanics came a lull in the state of mechanical physics. In the beginning of the 1800 s, the concept of Energy began to be formulated. With an emphasis on Mechanics, we look at Energy as an all embracing medium. While there is no completely satisfactory definition of Energy, we quantify it justly using both the motion and position of a system. As usual, it is inept to consider any quantity, energy being the example in this case, without taking it in relation to another quantity. Thus it is frivolous to study energy on its own. As a result, we study energy as it changes, not as an individual quantity. There is no universal Energy measuring device. What we come to conclude is that a physical change in the system causes a change in Energy. A force is the agent of that physical change and energy is the measurement of change. Since energy is not an entity in and of itself, due to the fact that we observe it by observing changes in matter, it is safe to conclude that matter is the catalyst of energy s existence. Just as there cannot be a tide without the ocean, there cannot be energy without matter. Energy is a way for us to provide a means of accounting for physical change. We ask why did that tennis ball go back over the net and answer because the tennis racquet provided the force which gave it energy. This leads into the most important characteristic of energy: conservation. The energy stored in the strings of the tennis racquet (provided by elasticity) was transferred to the tennis ball. The amount of total energy always remains unchanged. The tennis balls initial speed caused the strings of the racquet to stretch; the tennis ball transferred its energy to the racquet, the strings then bounced back; the strings transferred their stored energy back to the tennis ball, the ball then traveled back in its original direction; the ball is now storing energy to be transferred to the next object it hits. Energy is conserved. 6.1 Work Gaspard Coriolis defined the product of force and distance work. Work is the change in the energy of a system resulting from the application of a force acting over a distance. Even better, work is a movement against resistance of force; gravity, friction, etc. Simplistically, if an object was moved a distance d by the application of a force F, the preliminary definition of work W is W = Fd (6.1) As long as an object moves along the line of action while the force acts on it, work is being done. The SI unit of work is a Newton-meter, more commonly known as a Joule, J. The work done by a 1-N force through 1m is defined as 1 joule (J) 1 J = 1 N-m A more rigorous definition of work can be stated by saying that the work done on an object is the product of the component of the force in the direction of motion multiplied by the distance the force acts on the object. W = F cos θ d (6.)

2 Any force that causes work to be done acts in the direction of motion or act against motion. Forces perpendicular to motion are not causing a change in position or motion, thus are not doing work. For example, if you support a heavy object above your head and walk along a flat floor at a constant speed you are not doing work on the object because the Force and the displacement are perpendicular to each other. Work & Gravity In general, work must be done to overcome some resisting force. The most common force to overcome is the force due to gravity. For example, if you want to raise a mass a distance h vertically, you must overcome the force of gravity acting on the mass. The force and the displacement are in the same direction: W = Fd W = Fh W = F g h W = mgh (6.3) In another way, you must consider the definition of work when considering paths that are not consistently straight. For example, who does more work a man who travels along a winding path to the top of a mountain or one who climbs straight up? Both men do the same amount of work. Since any motion perpendicular to the force (gravity) takes no work to complete, the work done is independent of the path taken. The work done is determined only by the weight of the man and his vertical displacement. Mechanical Energy If you recall, all mater interacts gravitationally with all other matter. This results in a huge web that binds everything in the universe together. It was once suggested that inertia is derived from that invisible web of forces. If that s true, all matter is affected by a body moving anywhere in the universe. If you ve ever heard of Chaos Theory, one hypothetical situation is that if a butterfly flaps its wings in North America that it rains in Beijing instead of being a sunny day. 6. Kinetic Energy Work that causes motion equals the resulting change in energy. That simply means that every time work is done on an object, the amount of that work is equal to the energy which results from that change. In 1849 Lord Kelvin defined the energy associate with moving objects as Kinetic Energy. Imagine a rigid body being pushed across a surface under a constant force F. W = Fd F = ma W = mad Under constant acceleration, v f = v i + ad - -

3 ad = ½ v f ½ v i Multiply both sides by m yields, Fd = ½ mv f ½ mv i W = ½ mv f ½ mv i Because of Lord Kelvin s energy association with work, we now define the kinetic energy of any object of mass m traveling at speed v as: KE = ½ mv (6.4) The unit of Kinetic Energy, as with all forms of energy, is a kg-m-m / s, or, a Joule (J). 6.3 Potential Energy Imagine continuously acting force acting on an object, gravitational force for example. If you applied work on a mass by raising it up against the downward pull of gravity, the force will continue to act even after the displacement. When raised in a gravitational field, an object will fall when released; back to where it started. Kinetic energy drives the process. Thus, work on the system transfers Energy into it. That being said, work done on the system is ultimately converted into Kinetic Energy. But what about a ball that is lifted high above the ground? The ball is given energy but has yet to have it appear as kinetic energy. This storing of energy is known as gain of potential energy. Potential energy is retrievable, based upon the position of the object relative to the force. Work done on an object against a force increases an objects potential energy (PE). More formally, a change in potential energy of an object is caused by moving it from one point to another overcoming a force (gravitational, electromagnetic, etc) and is equal to the work done to move it. The most common potential energy you will see in this course is gravitational PE. The gained potential energy is equal to the amount of gravity an object must overcome to get through a vertical height, h: PE = F h Given that the force is due to gravity and any object has a mass of m: PE = F g h PE = mg h (6.5) PE = mg(h f h i ) Much like Kinetic Energy, potential energy is relative. One must choose a zero reference point where it is convenient. It is important to note that since the zero reference point h i is being chosen arbitrarily, we can only measure the change in potential energy, not the amount of energy itself. One cannot disregard the fact that work must be done on a system or object in order for energy to be changed. The only way that potential energy is given to an object is that there had to be some work done to overcome a force

4 Conservation of Mechanical Energy The Law of Conservation of Energy states that: The total energy of any system that is isolated from the rest of the universe remains constant, even though energy may go from one form to another within the system. 6.4 Mechanical Energy We define the mechanical energy of a system as the sum of the kinetic and gravitational potential energies. If the system does a certain amount of work to overcome a force, such as friction, that work is subtracted from the total amount of mechanical energy since work transfers energy. Alternatively, if no outside forces are applied, the amount of work done by the system is zero thus no energy is transferred and mechanical energy is conserved. Mathematically, this concept is simple to develop. W = E E = KE + PE E = E f E i If W = 0 E = 0 E f = E i (6.6) If mechanical energy is to be conserved, at every instant in time the total mechanical energy must be constant. That is, if an object kinetic energy increases, its potential energy must decrease, and vice versa. E = KE + PE (6.7) E f = E i KE + PE = KE + PE ½ mv f + mgh f = ½ mv i + mgh i Given the example of a projectile which is launched straight up, this concept is easily recognized. Other than gravity, which is accounted for through potential energy, no work is expended after the ball is initially launched. As the velocity of the ball decreases as the height of the ball increases, causing a change in energy from kinetic to potential. At the bottom of the projection, h i = 0, the potential energy is 0 and the kinetic energy is maximized. At the highest point in the projection h f = h, v = 0, the kinetic energy is zero and the potential energy is maximized. As the projectile falls the kinetic energy increases as the velocity does and the potential energy decreases as the height does

5 6.6 Power Power is simply defined as the rate of doing work. P = W / t (6.8) From the definition of work, power can be defined as the rate at which energy is transferred into or out of a system. The SI unit for power is the watt (w) and it is equal to 1 joule of work done in 1 second: 1 W = 1 J / s You might have recognized that a joule is a Newton meter; put that over seconds and you find that a watt is easily defined as a Newton-m/s. Power is dependent on the speed at which the application of a force is moving. Alternatively, the power delivered to a moving object equals the product of the component of the force in the direction of motion and the speed. Consider a jogger moving at 1m/s. It seems much easier to accelerate at 1m/s than it is to accelerate when the jogger is already moving at 10m/s. P = W / t P = F cos θ d / t P = Fv cos θ (6.9) Clearly, from the previously stated example, the faster something is moving the more Force it takes to increase its power. Referring back to Newton s Third Law, fundamental interactions transfer energy everywhere, ensuring that a total amount of fixed energy is maintained and consistent