Question

When a mass \(\mathrm{m}\) is suspended from the free end of a massless spring having force constant \(\mathrm{k}\), its oscillates with frequency \(\mathrm{f}\). Now if the spring is divided into two equal parts and a mass \(2 \mathrm{~m}\) is suspended from the end of anyone of them, it will oscillate with a frequency equal to......... (A) \(\mathrm{f}\) (B) \(2 \mathrm{f}\) (C) \((\mathrm{f} / \sqrt{2})\) (D) \(\sqrt{2 f}\)

Short answer

The frequency of oscillation when a mass $2\mathrm{m}$ is suspended from the end of one of the divided springs is equal to the initial frequency, $f$. So, the answer is (A) \(\mathrm{f}\).

Step-by-step solution

Expression for frequency of oscillation for mass m

We can first express the frequency of oscillation for a mass m suspended from a spring with force constant k using the formula: \(f = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}\)

Force constant of the divided spring

When the spring is divided into two equal parts, its force constant changes. The force constant for each part of the spring when it is divided is given by: \(k' = 2k\) The new force constant, \(k'\), is twice the original force constant.

Expression for frequency of oscillation for mass 2m and divided spring

Now we need to find the frequency of oscillation when a mass 2m is suspended from the end of one of the divided springs having force constant \(k'\). We can use the same formula for frequency, with the new values for mass and force constant: \(f' = \dfrac{1}{2\pi}\sqrt{\dfrac{k'}{2m}}\)

Substitute the new force constant

We can now substitute the new force constant, \(k' = 2k\), into the expression for frequency: \(f' = \dfrac{1}{2\pi}\sqrt{\dfrac{2k}{(2m)}}\)

Simplify and compare with the original frequency

Simplify the expression for the new frequency: \(f' = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}\) Comparing this expression with the original frequency expression, we can see that they are equal: \(f' = f\) Therefore, the correct answer is: (A) \(\mathrm{f}\)

Key concepts

Spring Force Constant

The spring force constant, often denoted as "k," is a measure of the stiffness of a spring. It tells us how much force is needed to compress or stretch the spring by a unit length. Higher values of "k" indicate a stiffer spring that requires more force to stretch or compress.
When a spring is divided into two equal parts, each part behaves as if it has a higher stiffness compared to the whole spring. This is because the number of coils is reduced, effectively increasing the resistance of the spring to deformation. In such cases, the new spring's force constant doubles, thus:

• Original force constant for the full spring: \( k \)
• Force constant for each half: \( k' = 2k \)

Understanding this change in the force constant is vital for analyzing changes in frequency or motion related to the spring.

Frequency of Oscillation

The frequency of oscillation in a spring-mass system indicates how often the system completes a cycle of motion in a given time frame, typically measured in Hertz (Hz). For a mass-suspended oscillating spring, the frequency can be determined using the formula:
\[ f = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}} \]
where "f" is the frequency, "k" is the spring force constant, and "m" is the mass. The formula tells us that frequency depends on both the stiffness of the spring and the mass attached.
When modifications occur—like changing the spring length or the suspended mass—the frequency alters. In mathematical terms, any increase in stiffness (such as doubling the spring force constant) can potentially affect the cycling rate. In the example given, when a mass of \( 2m \) is suspended from a segment with a doubled force constant, the new frequency remains:

• \( f' = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}} \)

This demonstrates that, despite changes in spring characteristics, the frequency in this problem scenario does not change.

Mass-Spring System

The mass-spring system is a simple mechanical model that describes how mass attached to a spring can move back and forth, displaying what is known as Simple Harmonic Motion (SHM). Understanding this system is critical in physics as it provides insights into oscillatory movements, similar to those in pendulums or waves.
The key components of this system include:

• Mass ("m"): The object that is attached to the spring.
• Spring Force Constant ("k"): This constant tells how stiff the spring is.

The system operates by compressing and stretching as the spring tries to return to its equilibrium, or rest position, after being displaced. For an instance where the spring is split and a new mass is introduced, as seen in the exercise, the characteristics like the force constant adapt accordingly.
The relationship between these elements dictates the system's natural frequency and period of oscillation, which are determined using factors like the weight of the suspended mass and the spring's constant. Through calculating and comparing values, we can predict system behaviors, as demonstrated by the given solution where frequency remained unchanged despite modifications.