To analyze the oscillation of the block connected to two springs with force constants k1 and k2, we need to consider how these springs behave when they are combined. The key to understanding the system lies in recognizing how the springs affect the motion of the block and how we can derive the overall frequency of oscillation.

Understanding the System

In this scenario, the block experiences forces from both springs simultaneously. When you pull the block, both springs will stretch, and the total restoring force acting on the block will be the sum of the forces exerted by both springs. According to Hooke's Law, the force exerted by a spring is proportional to the displacement from its equilibrium position, given by:

F = -kx

Here, F is the force, k is the spring constant, and x is the displacement. For our system, if we denote the displacements of the block from its equilibrium position as x, the forces from the two springs can be expressed as:

• Force from spring 1: F1 = -k1 * x
• Force from spring 2: F2 = -k2 * x

Calculating the Total Restoring Force

The total restoring force acting on the block when displaced by x can be derived from both springs:

F_total = F1 + F2 = -k1 * x - k2 * x = -(k1 + k2)x

Finding the Effective Spring Constant

From the expression for the total restoring force, we can identify an effective spring constant (k_eff) for the system:

k_eff = k1 + k2

Connecting Spring Constants to Frequency

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

f = (1/2π) * √(k/m)

Where k is the spring constant, and m is the mass of the block. For our combined spring system:

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

Relating to Individual Spring Frequencies

Now, let’s consider the individual frequencies of the block if it were attached solely to each spring:

• Frequency with spring 1: f1 = (1/2π) * √(k1/m)
• Frequency with spring 2: f2 = (1/2π) * √(k2/m)

To express k1 and k2 in terms of the individual frequencies, we rearrange these formulas:

• k1 = (2πf1)² * m
• k2 = (2πf2)² * m

Final Frequency Expression

Substituting these expressions back into the effective spring constant gives us:

k_eff = (2πf1)² * m + (2πf2)² * m

Now, factoring out the mass m:

k_eff = m * ((2πf1)² + (2πf2)²)

Finally, plugging this back into our frequency formula for the effective system:

f_eff = (1/2π) * √((k1 + k2)/m) = (1/2π) * √((m * ((2πf1)² + (2πf2)²))/m) = (1/2π) * √(((2πf1)² + (2πf2)²))

Thus, we derive that the frequency of oscillation of the block is determined by the individual frequencies of the two springs, confirming that:

f_eff = √(f1² + f2²)

Conclusion

This result illustrates how the combined effect of two springs can be analyzed through their individual frequencies, akin to capacitors in parallel where the total capacitance is the sum of the individual capacitances. Understanding these relationships is fundamental in mechanics and helps in predicting the behavior of more complex systems.