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Unformatted text preview: by Hooke’s law, |Fi | = ki xi ,
where |Fi | is the magnitude of the force exerted on (or by) spring i when the length of that spring changes
by an amount xi .
(a) Relate the magnitudes of the forces F1 and F2 exerted by each spring individually, and the magnitude
of the force F exerted on the crate. Explain. (Hint: Sketch separate free body diagrams for the springs
and the crate.) Then, ﬁnd the relationship between the stretches x1 and x2 of the individual springs and
the stretch x of the original (single) spring. Explain your reasoning. Based on these results, determine
an expression for the spring constant of the original spring (k) in terms of the spring constants (k1 and
k2 ) of the two individual “ spring pieces”. Based on this problem, if you chop an ideal spring in half,
what happens to its spring constant? Show all work. Figure 4:
(b) Now consider the case in which you replace the original spring with two new springs (3 and 4) that
connect directly to the crate and to the ceiling. (See Figure.) Assume that the crate ends up at the
same equilibrium height in this new situation as when there was just a single spring (k ). Following the
same logic as you did above, determine an expression for the spring constant k of the original spring in
terms of the spring constants k3 and k4 of the new springs. Explain your reasoning. Figure 5:
NOTE: There is an extra-credit survey online - just go to our course main page, you’ll see it linked
prominently on our “Special notes” section, under the current fun picture. Or go directly to
http://www.colorado.edu/physics/phys2210/phys2210 sp11/preﬂights/survey midterm 2210.html
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