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Conservation of Energy

Engineering 1182:Roller Coaster Dynamics-1: Energy Conservation

2Measuring QuantitiesSystems of UnitsInternational System (SI)English System (FPS)Base quantityNameSymbolNameSymbollengthmetermfootftmasskilogramkgpound (old:slug)lbtimesecondsseconds For the rest of this lecture, well be using SI units (metric)Stress on the importance of Units as they will be seeing a mixture of SI Units and English Units. EnglishUnits are still common in industry in the USA and so students should expect to see a mix of units after graduating.

SI (Systme International dUnits) is the official designation for the meter-kilogram-second metric system. It is sometimes called MKS based on the first letters of the primary units. The English system is sometimes called FPS fromthe first letters of foot-pound-second. You will also see an alternate metric system called CGS for centimeters-grams-seconds.

3Physics Concepts - DefinitionsDISPLACEMENT- A measure of HOW FAR and in WHAT DIRECTION an object has MOVED relative to a starting point (Units: m). SVELOCITY- Change in displacement per unit time (Units: m/s).

ACCELERATION- Change in velocity per unit time (Units: m/s2).

MASS- A physical property of an object that identifies its resistance to having a velocity change (Units: kg). m

Velocity describes how quickly an object changes its position. The higher the velocity the quicker an object travels between 2 locations. Phrases like, how fast or how quickly, are used to describe velocity. Often the word speed is substituted for the word velocity in common usage. However, technically the two are different. Velocity is actually speed with direction. For example, 60 mph west is a velocity. "West" is the direction and 60 mph is the speed.

We are intentionally trying to avoid teaching them vector concepts as this will complicate the physics involved. The approach of using Energy equations to design the roller coaster does not need these vector concepts.

Acceleration:Acceleration describes how quickly an object changes its velocity. Phrases like, slow down, speed up, change speed and change velocity are used to describe accelerations. If students want an easy way to determine if they are visualizing acceleration or a constant velocity along a straight line they only need to ask one question, "Is the object slowing down or speeding up?" If the answer is "Yes," then it is accelerating (decelerating). If the answer is "No" then it is moving with a constant velocity. Changes in acceleration greatly contribute to the thrill of a roller coaster ride. A rider may feel greater sensations in a low-speed coaster with sharp acceleration changes than on a faster coaster with a smoother ride. This is important because the magnitude of accelerations are limited by the physical limitations of typical riders. Pure speed is often not as recognizable as the surge of acceleration during a coaster ride.

Mass can be defined either inertially as above or in some sense as amount of stuff. There is no theoretical reason that these twodefinitions HAVE to be the same, but nobody has yet found any difference. Physics groups are working on this and on alternatemass definitions. In practice, mass is defined by comparison with a known standard mass (amount of stuff) by using something like a scale (equal resistance to having a velocity change under the force of gravity).4Physics Concepts - DefinitionsFORCE is a PUSH or a PULL that is defined by its effect on a mass (Units: Newtons, N).F=ma (1 Newton = 1 kg-m/s2)

WEIGHT- The force acting on a mass when it is subjected to gravity. F=mgWhere g is the acceleration due to the Earths gravitational forceFor Standard Gravity use g = 9.81 m/sec2

Example of the use of force: I applied 450 N force to push my car out of the garage Sir Isaac Newton 1642-1727 is famous for his work on Gravity. He discovered that gravitational force acts equally on all objects. The gravitational pull on the surface of the moon is 1/6 that of the earth because the moon has a different mass and different diameterthan the earthThe action of gravity causes an object to be pulled towards the earth At standard earth gravity, the force of gravity acting on a mass of 1kg is 9.8 N Weight is a measure of earths gravitational pull on an object. An object can only have weight if gravity is at work In space, an object will have the same mass as on the surface of the earth but little or no weight - because there is little or no gravity!In contrast, when skydiving, you have the impression of being weightless because there is little opposing force but you still have weight and are initially accelerating due to the force of gravity.

5ENERGY is a conserved property of an object that relates to its ability to do work. Energy can have a number of forms, for example mechanical, electrical, chemical, or nuclear. EUnits: Joules or N-m (Newton-meter).There are different formulas describing different forms of energy.

Physics Concept - EnergyIm trying to avoid the circular definition that Energy IS the ability to do Work, while Work is the transfer of Energy from one system to another or from one form to another. PE converts to KE by means of work done by gravity. Work and energy are different in that Work is a process while energy is a property or attribute, but for our purposes, we can talk about energy transfers and not specifically introduce the concept of work. A systems energy is the ability to produce work, regardless of whether any work actually happens. A raised boulder has energy even though its tied up and left that way and nothing ever happens.

Whenever possible, the concepts learned should be tied to the Roller Coaster. 6

When a force, F, is applied to an object, the energy that is transferred to the object is given by where is the distance over which the force is applied.

Mechanical Energy

Mechanical Energy is what well be concerned with during the roller coaster design.Ex. Energy is used in lifting a book from the table (a force is needed to overcome gravity)Ex. Energy is used in dragging an object along the floor (a force is needed to overcome friction).

7Law of Conservation of Energy (COE)Energy can neither be created nor destroyed.

Energy can only be changed from one form to another.

In Physics, the word conservation means that it does not change with time. Energy could certainly change form,(e.g. from P.E to K.E), but the overall energy of the system remains conserved.

8For A Roller CoasterFor our roller coaster we will represent the cars by a rolling ball.We only care about the energy stored in the rolling ball. This is only part of the energy of the complete system.

Main Elements of Roller Coaster System= Ball + Rails + Structure It is very important to talk about whose energy we are referring to. Is it the energy of the ball or the energy of the complete roller coaster or the energy contained in the room with the roller coaster?

The ball by itself is a system.The track by itself is a system.The structure by itself is a system.

When all these are put together, it forms a bigger system that is different from the individual systems. Main Elements of Roller Coaster System = Ball + Rails + Structure

When we speak about energy, we need to know which system we are referring to! The law of Conservation of Energy holds true for any closed system, but our roller coaster is NOT a closed system since energy will be transferred from the ball to the surrounding environment. We are only interested in the magnitude of these Losses so that we can keep track of the energy of our ball.9Forms of Energy in a Rolling BallPotential Energy (PE)Energy of the BallKinetic Energy (KE)Total Mechanical Energy of the ball = PE + KEThis is a very important breakdown of the various forms of energy associated with a rolling ball. Each form of energy will be discussed in detail in the following slides.10

KEinitial + PEinitial = KEfinal + PEfinal Energy Conservation (no friction!)At the top of a hill, the cars in a roller coaster possess a large quantity of potential energy. During the first drop, the cars lose much of their potential energy and consequently gain kinetic energy.Each change in height corresponds to a change of speed as potential energy (due to height) is transformed to and from kinetic energy (due to speed)TME is Total Mechanical EnergyAs the ride continues, the train of cars is continuously losing and gaining height. Each gain in height corresponds to a loss of speed as kinetic energy (due to speed) is transformed into potential energy (due to height). Each loss in height corresponds to a gain of speed as potential energy (due to height) is transformed into kinetic energy (due to speed). This transformation of mechanical energy from the form of potential to the form of kinetic and vice versa is illustrated in the animation.

TMEinitial =KEinitial + PEinitial = KEfinal + Pefinal = TMEfinal

K.E.(Joules) = * mass(kg) * Velocity2[(m/s)2] P.E.(Joules) = mass(kg) * g(m/s2) * height(m)http://surendranath.tripod.com/Coaster/Coaster.html

11Potential EnergyGravitational Potential Energy is the energy stored in a body due to its height (h). The height is always measured relative to some reference level (here the ground)An object of mass m at a vertical height h above the ground has a potential energy of mgh

h = 1.52 meters

Mass= 2 kgPE = ?PE of the ball shown = mgh = 2(9.81)(1.52) = 30 Joules

Gravitational Potential Energy is stored as the result of the gravitational attraction of the Earth for the object. The gravitational potential energy of the heavy ram of a pile driver is dependent on two variables - the mass of the ram and the height to whic