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Suppose that in cases B, C, and D in problem 1, each combination of springs were replaced by a single spring such that the motion of the block is the same as before the replacement. We will call the spring constant of this new spring the effective spring constant of the original combination. Quantitatively, the effective spring constant, keff, may be defined by the relationship , where is the sum of all forces on the block by the springs, and x is the position of the block with respect to the equilibrium position. Rank the four cases according to keff, from largest to smallest. Explain. Use your ranking above and the relationship T = 2pi to rank the cases according to period of oscillation. Explain. Use your ranking from part a and the relationship Etotal = 1/2 keffA2 to rank the cases according to total energy. Explain.

Explanation / Answer

For figure B and D the spring constant is same as both are in parallel.

Keff = k1 + k2  

Figure c has spring constant

Keff = k1k2 / (k1 + k2 )

a)

From the above calcualtion, the decrease order of spring constant

B = D >A > C

b)

Time period is inversely proportional to effective spring constant so the order is

C > A > B = D

c)

Energy is directly proportional to the effective spring constant

B = D >A > C

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