A uniform cylinder of length L and mass M having cross-sectional area A is suspended, with its lengt

Q: Why does the buoyant force increase when the cylinder is pushed down?

When the cylinder is pushed down, more of its volume becomes submerged in the liquid. According to Archimedes' principle, the buoyant force is equal to the weight of the displaced liquid. Since the submerged volume increases by $A \cdot x$, the buoyant force increases by $A x \rho g$.

Q: What would happen if the cylinder was completely submerged in the liquid?

If the cylinder was completely submerged, pushing it down further would not change the submerged volume. Therefore, the extra buoyant force would be zero, and the restoring force would only come from the spring. The frequency of oscillation would simply be $f = \frac{1}{2\pi}\sqrt{\frac{k}{M}}$.

Q: Does the density of the cylinder affect the frequency of oscillation?

Yes, the mass $M$ of the cylinder is directly related to its density and volume ($M = \text{density} \times \text{volume}$). Since $M$ appears in the denominator of the frequency formula, a denser cylinder of the same dimensions would have a larger mass, resulting in a lower frequency of oscillation.

Q: Why is there a negative sign in the restoring force equation?

The negative sign indicates that the restoring force is always directed opposite to the displacement. When the cylinder is displaced downwards ($x > 0$), the restoring force acts upwards ($F < 0$), trying to bring it back to the equilibrium position.

Q: Can we use this setup to measure the density of an unknown liquid?

Absolutely! By measuring the frequency of oscillation of the cylinder in air (or vacuum) and then in the unknown liquid, we can easily calculate the density $\rho$ of the liquid using the formula, provided we know the spring constant $k$, mass $M$, and cross-sectional area $A$.

Question

A uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half-submerged in a liquid of density $\rho$ at equilibrium position. When the cylinder is given a small downward push and released it starts oscillating vertically with a small amplitude. If the force constant of the spring is $k$, the frequency of oscillation of the cylinder is

Options:

(A) $\frac{1}{2\pi}\left(\frac{k - A\rho g}{M}\right)^{1/2}$

(B) $\frac{1}{2\pi}\left(\frac{k + A\rho g}{M}\right)^{1/2}$

(C) $\frac{1}{2\pi}\left(\frac{k + \rho g L^2}{M}\right)^{1/2}$

(D) $\frac{1}{2\pi}\left(\frac{k + A\rho g}{A\rho g}\right)^{1/2}$

Sigma Percentile · Accepted Answer

### The Symphony of Springs and Fluids

Imagine a heavy metal cylinder suspended from a spring, bobbing up and down in a calm pool of water. This is not just a classic physics problem; it is a beautiful dance between two of nature's most fundamental restoring mechanisms: **elasticity** and **buoyancy**.

In this journey, we will explore how these two distinct physical phenomena join forces to create a faster, more energetic simple harmonic motion.

Let us dive deep into the mechanics of this system and discover the elegant mathematics that governs its oscillations.

### The Baseline: Equilibrium State

Before we disturb our system, we must understand its state of perfect peace—the **equilibrium position**.

At this position, the downward gravitational pull on the cylinder is exactly balanced by two upward forces.

First, we have the upward spring force, which is proportional to the initial extension of the spring.

Second, we have the buoyant force, which is equal to the weight of the liquid displaced by the half-submerged cylinder.

Mathematically, we can write this balance as:

$$Mg = kx_0 + ho A \left(\frac{L}{2}ight) g$$

Here, $x_0$ is the initial stretch of the spring, and $\frac{L}{2}$ is the submerged depth of the cylinder.

### The Disturbance: Introducing Displacement

Now, let us gently push the cylinder downwards by a small distance $x$ and release it.

This displacement disturbs the delicate balance of forces.

As the cylinder moves downwards, the spring stretches by an additional distance $x$, which increases the upward elastic force by $kx$.

Simultaneously, a greater volume of the cylinder enters the liquid, displacing more fluid and increasing the upward buoyant force.

This extra submerged volume is simply the cross-sectional area $A$ multiplied by the downward displacement $x$.

Therefore, the extra buoyant force is:

$$\Delta F_b = A x ho g$$

### The Combined Restoring Force

Both of these additional forces act in the upward direction, directly opposing our downward displacement.

This is the definition of a **restoring force**—it always points towards the equilibrium position.

We can write the net restoring force $F$ as the sum of these two extra forces:

$$F = -\left(kx + A x ho gight)$$

Factoring out the displacement $x$, we get:

$$F = -\left(k + A ho gight)x$$

Notice how beautifully the spring constant $k$ and the fluid term $A ho g$ add up!

It is as if the liquid acts as an **effective spring** of force constant $k_{	ext{eff}} = A ho g$.

### The Equation of Motion

Using Newton's second law, we can equate this net restoring force to the mass times acceleration of the cylinder:

$$M a = -\left(k + A ho gight)x$$

Solving for acceleration $a$, we get:

$$a = -\left(\frac{k + A ho g}{M}ight)x$$

This equation is in the classic form of simple harmonic motion, $a = -\omega^2 x$, where $\omega$ is the angular frequency.

Therefore, we can identify:

$$\omega^2 = \frac{k + A ho g}{M}$$

$$\omega = \sqrt{\frac{k + A ho g}{M}}$$

### Finding the Frequency

Finally, we can find the linear frequency of oscillation $f$ using the relation $f = \frac{\omega}{2\pi}$:

$$f = \frac{1}{2\pi}\sqrt{\frac{k + A ho g}{M}}$$

This matches option (b) perfectly!

This result shows that the presence of the liquid increases the frequency of oscillation, making the cylinder bob up and down faster than it would in empty space.

Select Answer:

Visualized Solution

Visual Anchor (Orient)

Equilibrium Analysis

Logic Bridge (Tool)

Raw Setup (Buoyancy Increase)

Raw Setup (Spring Force Increase)

Net Restoring Force

Equation of Motion

Standard SHM Comparison

Frequency of Oscillation

Final Answer & Verification

The Way Forward

The Sigma Insight: Force and Energy Method in SHM

The Symphony of Springs and Fluids

The Baseline

The Disturbance

The Combined Restoring Force

The Equation of Motion

Finding the Frequency

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