1. USING INTEGRATION TO CALCULATE WORK, ENERGY, ETC

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

3. NO ENERGY

4. Lots of energy

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

6. 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?

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

8. 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

9. 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

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

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

12. 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