Question

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you have two springs. one has a greater spring constant than the other. you also have two objects, one with a greater mass than the other. which object should be attached to which spring, so that the resulting spring - object system has the greatest possible period of oscillation?
the object with the smaller mass should be attached to the spring with the greater spring constant.
the object with the greater mass should be attached to the spring with the smaller spring constant.
the object with the greater mass should be attached to the spring with the greater spring constant.
the object with the smaller mass should be attached to the spring with the smaller spring constant.

Explanation:

Step1: Recall the formula for the period of a spring - mass system

The period \(T\) of a spring - mass system is given by \(T = 2\pi\sqrt{\frac{m}{k}}\), where \(m\) is the mass of the object and \(k\) is the spring constant.

Step2: Analyze the effect of \(m\) and \(k\) on \(T\)

From the formula \(T = 2\pi\sqrt{\frac{m}{k}}\), we can see that \(T\) is directly proportional to \(\sqrt{m}\) (as \(m\) increases, \(T\) increases) and inversely proportional to \(\sqrt{k}\) (as \(k\) increases, \(T\) decreases).
To maximize \(T\), we need to maximize \(m\) and minimize \(k\).

Answer:

The object with the greater mass should be attached to the spring with the smaller spring constant.