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Graph a given record of force vs end location, for a certain spring, find its force constant. Develop F(x) and PE(x).

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On a frictionless table, one end of a spring is fixed. A cord attached to the free end
pulls it along a meter scale beginning from 0. A record is made of the position of the free end, x, and the resisting force, F, exerted by the spring on the cord.
SEE ATTACHMENT #1 for the parameters and a record of F vs x, showing position x= .4 meter, and force by the spring, F= -16 nt. (Note that as the free end moves in +x direction the spring's force toward -x direction becomes more negative.)
PART a. Make a graph of F(x) from x=0 to x= .8 m. From your graph of F vs x, find the slope of the line. This slope is defined as the 'force constant', k, a property of the spring.
PART b. With the force constant, k, the slope of the line, express F(x), the force F as a function of x.
PART c. From the function F(x), calculate the force the spring would exert when the free end of the spring is at the 1.4 m mark.
PART d. Develop an equation expressing the potential energy contained in a spring whose free end was moved from 0 to Xf.

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Solution Preview

<br>For the most benefit to you, make your graph of the data points, then SEE ATTACHMENT #2 to check your work.
<br>From the (x, F) coordinates of any two points on the straight line, (.2, -4) and (.8, -16) say, the slope is expressed by:
<br>(1) k =(F2 - F1)/(x2 - x1) = (-16 -[-4])/(.8 - .2) = -20 nt/m.
<br>PART b.
<br>From the general equation of a straight line, 'y= m x + b', or with variables x and F on this ...