Engineering 1182: Roller Coaster Dynamics-1: Energy Conservation Measuring Quantities Systems of Units International System (SI) Base quantity Name Symbol length meter
m mass kilogram kg time second s English System (FPS) Name foot
pound (old:slug) second Symbol ft lb s For the rest of this lecture, well be using SI units (metric) 2 Physics Concepts Definitions DISPLACEMENT- A measure of HOW FAR and in
WHAT DIRECTION an object has MOVED relative to a starting point (Units: m). S VELOCITY- Change in displacement per unit time (Units: m/s). dS v dt ACCELERATION- Change in velocity per unit time (Units: m/s2). dv d 2 S a 2 dt dt
MASS- A physical property of an object that identifies its resistance to having a velocity change (Units: kg). m 3 Physics Concepts Definitions FORCE 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=mg Where g is the acceleration due to the Earths gravitational force For Standard Gravity use g = 9.81 m/sec2 4 Physics Concept - Energy ENERGY 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. E
Units: Joules or N-m (Newton-meter). There are different formulas describing different forms of energy. 5 Mechanical Energy When a force, F, is applied to an object, the energy that is transferred to the E Fd by d object isgiven where
is the distance over which the force is applied. 6 Law of Conservation of Energy (COE) Energy can neither be created nor destroyed. Energy can only be changed from one form to another. 7 For A Roller Coaster
Main Elements of Roller Coaster System = Ball + Rails + Structure For 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. 8 Forms of Energy in a Rolling Ball Energy of the
Ball Potential Energy (PE) Kinetic Energy (KE) Total Mechanical Energy of the ball = PE + KE 9 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) KEinitial + PEinitial = KEfinal + PEfinal 10 Potential Energy Gravitational Potential Energy is the energy Mass= 2 kg stored in a body due to its
PE = ? height (h). The height is always measured relative h = 1.52 meters to some reference level (here the ground) An object of mass m at a vertical height h above PE of the ball shown = mgh the ground has a potential = 2(9.81)(1.52) = 30 energy of mgh Joules 11 Potential Energy ?? If we wanted the
ball to have 75 joules of energy, what height should it be raised to? 75J Mass= 2 kg h = ? meters h = PE/(m*g) = 75J / (2kg*(9.81m/s2)) = 3.82 m 12 PE Examples PEA = PEB
PEB > PEC > PED > PEE (Assuming that all the balls PEE = 0 Joules have the same mass) 13 Kinetic Energy in a Rolling Ball Kinetic Energy (KE) Translational Kinetic Energy (TKE)
Rotational Kinetic Energy (RKE) Kinetic Energy of the ball = TKE + RKE A rolling ball has both forms of Energy! 14 Translational Kinetic Energy An object has Translational Kinetic Energy (TKE) when it is undergoing linear displacement TKE = mv2
m = mass of object v = velocity of object 15 Translational Kinetic EnergyExample A 50 gram ball is moving in a straight line with a velocity v= 20 m/s. What is its TKE ? Watch out for the Units! TKE = mv2 = (50 x10-3 )(20)2
= 10 Joules 16 Rotational Kinetic Energy (RKE) An object spinning about an axis is said to have Rotational Kinetic Energy. RKE = I2 I: Moment of Inertia : Angular Velocity (radians/sec)
17 Moment of Inertia (I) The moment of Inertia (I) of an object Measures the resistance an object has to rotating about a particular axis, similar to the way that mass is the objects resistance to changing its velocity. Depends on its mass, shape and axis of rotation.
18 Rotational KE Example A solid sphere of radius 0.4 m and weighing 2kg is rolling with an angular velocity of 62.5 radians/s. Find its Rotational KE. I = (2/5) x M x R2 = 0.128 kg-m2 RKE = I 2 = 250 joules 19 Angular Velocity () vs Linear Velocity (V ) Linear Velocity (V ) Change in Linear Displacement ( Meters) Time ( Seconds)
Angular Velocity ( ) Change in Anglular Displaceme nt ( Radians ) Time ( Seconds ) R v V R This relationship between linear and angular velocities holds if and only if the ball is not slipping 20 Effective Rolling Radius Rails R The ball sits down between the tracks
making the rolling radius smaller. The angular velocity is increased. If the rails are not supported and split further apart, the ball will sit farther down. 21 Energy Transfers As the ball rolls down the roller coaster track, some energy of the moving ball is:
Unwanted Lost to friction and dissipated as heat Spent in overcoming Air Resistance Energy Losses ! Lost to Structural Deformation Converted to Sound Energy 22 Energy Transfers (continued) In general, energy transferred away from the ball will NOT come back, and so the total mechanical
energy of the ball will be always decreasing. In the real world, we cannot avoid losses but can only MINIMIZE and/or ALLOW for them. 23 Lets put it together ! For the ball rolling along the roller coaster track, between any two subsequent points: PE1+ TKE1+ RKE1 = +2 TKE +2 RKE+2 Energy Losses PE
24 Design Considerations You will be estimating the velocity of the ball at selected points along your roller coaster track using energy calculations to: Make sure the velocity into turns is not too high (making banking difficult) Make sure that the ball can reach the top of vertical loops Make sure that the ball will not fly off the
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