Introduction

In the field of physics, the behavior of springs under oscillating conditions is a fundamental concept in simple harmonic motion (SHM). Understanding the relationship between the length of a spring and its oscillation frequency is crucial for various applications, ranging from engineering to biological systems. This article delves into the mathematical relationship between the length of a spring and its oscillation frequency, particularly in the context of a spring that is cut into two pieces with lengths in the ratio of 1:2.

Understanding Simple Harmonic Motion

Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction of the displacement. The key equation for SHM is:

F -kx

where F is the force, k is the spring constant, and x is the displacement from the equilibrium position. The displacement x is related to the angular frequency omega and frequency f by the equation:

omega 2pi f

The angular frequency omega is related to the time period T (the time for one complete oscillation) by:

omega 2pi/T

Since the force F is directly proportional to the displacement x squared (given by Hooke's law), the acceleration a is also directly proportional to the displacement squared:

F ma kx^2/2

This implies that:

a k/m * x^2

Proportionality of Frequency with Length

From the above equation, we can see that:

a -omega^2 * x

where omega^2 k/m. This shows that the acceleration a is directly proportional to the angular frequency squared, and hence the frequency f is inversely proportional to the square root of the length x of the spring. Therefore:

f^2 propto x^2

or

f propto x

This means that if the length of the spring is doubled, the frequency of oscillation is halved, and vice versa.

Case of a Spring Cut into Two Pieces

Consider a spring of length L that is cut into two pieces with lengths in the ratio of 1:2. Let the lengths of the two pieces be x and 2x, where x is the length of the shorter piece and 2x is the length of the longer piece.

Since the frequency is directly proportional to the square root of the length, the frequency f1 of the shorter piece and the frequency f2 of the longer piece are given by:

f1 k1/sqrt{x}

f2 k2/sqrt{2x}

where k1 and k2 are the spring constants of the shorter and longer pieces, respectively. However, in the context of simple harmonic motion, the spring constant k is proportional to the length of the spring.

Hence, the relationship between the spring constants and the lengths can be expressed as:

k1 k * x

k2 k * 2x

Substituting these into the frequency equations, we get:

f1 (kx)/sqrt{x} ksqrt{x}

f2 (k*2x)/sqrt{2x} ksqrt{2x}/sqrt{2} ksqrt{x}

Therefore, the frequency of the shorter piece f1 and the longer piece f2 are related by the ratio:

f1 : f2 1 : sqrt{2/2} 1 : 1

Thus, the ratio of the frequencies of the cut springs is 1:1, not 1:2 as initially thought.

Conclusion

The relationship between the length of a spring and its oscillation frequency in simple harmonic motion is a fundamental concept in physics. This analysis demonstrates that the frequency of a spring is inversely proportional to the square root of its length. When a spring is cut into two pieces, the frequency of oscillation is determined by the square root of the length of the individual pieces, leading to an unexpected result where the frequencies are equal for two pieces with lengths in the ratio of 1:2.

Keywords: Spring Oscillation, Simple Harmonic Motion, Frequency Ratio