## Posts Tagged ‘work-energy theorem’

### The Math Behind the Work-Energy Theorem

Friday, January 1st, 2016
 As an engineering expert I’ve applied the Work-Energy Theorem to diverse situations, but none as unique as its most recent application, the progress of Santa’s sleigh.   Last week we saw how Santa and his reindeer team encountered a wind gust which generated enough force to slow them from an initial velocity of v1 to a final velocity, v2, over a distance, d.   Today we’ll begin using the Work-Energy Theorem to see if Santa was able to keep to his Christmas delivery schedule and get all the good boys and girls their gifts in time.     Before we can work with the Work-Energy Theorem, we must first revisit the formula it’s predicated upon, de Coriolis’ formula for kinetic energy, KE = ½ × m × v2                            (1) where, KE is kinetic energy, m is the moving object’s mass, and v its velocity.     The equation behind the Work-Energy Theorem is, W = KE2 – KE1                                    (2) where W is the work performed, KE1 is the moving object’s initial kinetic energy and KE2 its final kinetic energy after it has slowed or stopped.   In cases where the object has come to a complete stop KE2 is equal to zero, since the velocity of a stationary object is zero.     In order to work with equation (2) we must first expand it into a more useful format that quantifies an object’s mass and initial and final velocities.   We’ll do that by substituting equation (1) into equation (2).   The result of that term substitution is, W = [½ × m × v22 ] – [½ × m × v12]      (3)     Factoring out like terms, equation (3) is simplified to, W = ½ × m × [v22 – v12]                        (4)     Now according to de Coriolis, work is equal to force, F, times distance, d.   So substituting these terms for W in equation (4), the expanded version of the Work-Energy Theorem becomes, F × d = ½ × m × [v22 – v12]                 (5)     Next time we’ll apply equation (5) to Santa’s delivery flight to calculate the strength of that gust of wind slowing him down. Copyright 2015 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________

### The Work-Energy Theorem Affects Santa’s Sleigh!

Friday, December 25th, 2015
 I hope all the good little boys and girls got Santa’s presents, because his sleigh was spotted yesterday on radar moving at an alarmingly slow pace.     As the radar screen here shows, the progress of Santa and his reindeer had slowed from an initial velocity of v1 to a much slower velocity of v2 over the distance, d, he traveled from his workshop.   Santa’s Sleigh Tracked On Radar         Upon closer examination, the radar operator noted that the North Wind was especially powerful this year, which caused Rudolph and the gang to double their efforts to overcome its negative force.     Santa and His Reindeer Fighting The Wind        Did Santa make it in time to deliver Christmas presents?   Find out next time, when we put the developing drama into the form of mathematical equations. Copyright 2015 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________

### The Work-Energy Theorem — Background

Friday, December 18th, 2015
 My work as an engineering expert sometimes involves computations of energy expended, as when I must determine how much energy is required to move something.   But sometimes the opposite needs to be calculated, that is, how much energy is required to stop something already in motion.   That’s the subject of today’s discussion, which we’ll approach by way of the Work-Energy Theorem.       The Work-Energy Theorem states that the work required to slow or stop a moving object is equal to the change in energy the object experiences while in motion, that is, how its kinetic energy is reduced or completely exhausted.   Although we don’t know who to attribute the Theorem to specifically, we do know it’s based on the previous work of Gaspard Gustave de Coriolis and James Prescott Joule, whose work in turn built upon that of Isaac Newton’s Second Law of Motion.       Consider the example shown here.  A ball of mass m moves unimpeded through space at a velocity of v1 until it is met by an opposing force, F.   This force acts upon the ball over a travel distance d, resulting in the ball’s slowing to a velocity of v2.   The Work – Energy Theorem Illustrated         Does the illustration make clear the Work-Energy Theorem dynamics at play?   If not, return for the second part of this blog, where we’ll clarify things by getting into the math behind the action. Copyright 2015 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________