USING INTEGRATION TO CALCULATE WORK, ENERGY, ETC

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Presentation transcript:

USING INTEGRATION TO CALCULATE WORK, ENERGY, ETC

BASIC FORMULA WORK = (FORCE).(DISTANCE) ENERGY = CAPACITY TO DO WORK

NO ENERGY

Lots of energy

Example: Spring Spring stretched ---> Work done ---> Energy is stored (Elastic Potential Energy) Spring released ---> Energy is released ----> Spring can do work

Time to compute! What is the amount of work done when a spring with spring constant 10 is stretched 6 inches beyond its natural position? 60? 360? 5? 2.5?

Answer NONE OF THE ABOVE! Need to Integrate!! Why?

Why indeed? Force versus amount of Because the force acting at different positions of the spring is different! In fact it changes continuously If x is amount of stretch, Force = 10.x (here k = spring constant = 10) Force versus amount of stretch

Approximate calculation of work done This chart shows work done over intervals of 0.3 length. Example: Work done when the spring is stretched from 2.1 inches to 2.4 inches is approximately equal to (Force at 2.1 inches)(0.3) 10(2.1)(0.3) = 6.3 units Shown by brown column

Now for the exact answer Work= Lim ∑ (work done over n intervals) = = 180

Other forms of potential energy Potential energy stored in an object at a certain height above earth Potential energy in atoms

The Catenary The catenary is the shape taken by a chain (like in a suspension bridge such as golden gate bridge (actual shape is parabola)) or an elastic rope suspended between two poles The shape minimizes potential energy Segue alert

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