To determine the frequency of oscillation for the block connected to two springs, we start by considering the system's behavior in terms of its effective spring constant. This approach parallels how capacitors behave in series in electrical circuits, where the total capacitance is modified by the individual capacitances. Let's dive into the details of how we arrive at the frequency expression for this mechanical oscillation system.

The Mass-Spring System

When a block is attached to two springs, the overall spring constant (k_eff) of the system can be derived from the individual spring constants of each spring. We'll denote the spring constants of the two springs as k1 and k2.

Finding the Effective Spring Constant

For springs in series, the effective spring constant is given by the formula:

• 1/k_eff = 1/k1 + 1/k2

Rearranging this equation gives:

• k_eff = (k1 * k2) / (k1 + k2)

This equation shows that the overall spring constant for the two springs connected in series is less than the spring constant of either spring alone, which is a key characteristic of series systems.

Understanding Oscillation Frequency

The frequency of oscillation for a mass-spring system is determined by the formula:

• f = (1/2π) * √(k/m)

Where:

• f is the frequency of oscillation,
• k is the effective spring constant, and
• m is the mass of the block.

Combining the Concepts

Now, substituting our expression for k_eff into the frequency formula gives us:

• f = (1/2π) * √((k1 * k2) / (k1 + k2) * (1/m))

Relating to Individual Frequencies

Next, let's express the individual frequencies f1 and f2 for each spring when used alone:

• f1 = (1/2π) * √(k1/m)
• f2 = (1/2π) * √(k2/m)

Now, we can relate the effective spring constant back to these individual frequencies:

• k1 = (2πf1)² * m
• k2 = (2πf2)² * m

Final Expression for Frequency

By substituting these expressions back into our original effective spring constant equation, we find that the overall frequency of the oscillating block is represented as:

• f_eff = (1/2π) * √((k1 * k2) / (k1 + k2) * (1/m)) = (1/2π) * √((√(k1/m) * √(k2/m))^2)

This results in:

• f_eff = (1/2π) * √(f1^2 * f2^2) = (1/2π) * √(f1 * f2)

This shows that the frequency of oscillation of the combined system is the geometric mean of the individual frequencies f1 and f2. This relationship illustrates how the system's dynamics change when multiple springs are involved, similar to how capacitors behave in a series circuit. The effective frequency reflects the contributions of both springs, leading to a unique oscillatory behavior of the mass.