Motion of an oscillating liquid column in a U-tube is (a) periodic but not simple harmonic. (b) non-periodic. (c) simple harmonic and time period is independent of the density of the liquid. (d) simple harmonic and time-period is directly proportional to the density of the liquid.

> [!TLDR] > The restoring force on a displaced liquid column is proportional to displacement, making the motion simple harmonic. The time period depends on column length and gravity but cancels out density. The answer is **(C)**. ## Why the liquid oscillates: the restoring mechanism When you displace liquid in a U-tube from equilibrium, gravity creates an imbalance. If one arm rises by height $h$ above the equilibrium level, the other arm falls by the same amount (conservation of volume in an incompressible liquid). This creates a height difference of $2h$ between the two arms, and the excess weight of the taller column pulls the system back toward equilibrium. The key insight: this restoring force is directly proportional to the displacement. That proportionality is the signature of simple harmonic motion. ## Step-by-step analysis 1. **Set up the displacement.** Suppose the liquid column has total length $L$ and cross-sectional area $A$. At equilibrium, the liquid sits at the same height in both arms. Displace it so one side rises by $y$ and the other falls by $y$; the height difference is now $2y$. 2. **Find the restoring force.** The extra column of liquid on the higher side has height $2y$, volume $A \cdot 2y$, and mass $\rho \cdot A \cdot 2y$, where $\rho$ is the liquid's density. Gravity pulls this excess mass downward with force $$ F = -\rho A (2y) g = -2\rho A g \, y. $$ The negative sign indicates the force opposes the displacement (restoring). 3. **Apply Newton's second law.** The entire liquid column (mass $m = \rho A L$) accelerates as one unit: $$ m \frac{d^2 y}{dt^2} = -2\rho A g \, y. $$ Substitute $m$: $$ \rho A L \frac{d^2 y}{dt^2} = -2\rho A g \, y. $$ 4. **Simplify to standard SHM form.** Cancel $\rho A$ from both sides: $$ L \frac{d^2 y}{dt^2} = -2g \, y \quad \Rightarrow \quad \frac{d^2 y}{dt^2} = -\frac{2g}{L} \, y. $$ This is the equation of simple harmonic motion with angular frequency $\om

Motion of an oscillating liquid column in a U-tube is (a) periodic but not simple harmonic. (b) non-periodic.

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Q. Motion of an oscillating liquid column in a U-tube is

(a) periodic but not simple harmonic.

(b) non-periodic.

(d) simple harmonic and time-period is directly proportional to the density of the liquid.

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Concept understanding — Simple Harmonic Motion

Simple Harmonic Motion: The Natural Rhythm of Things

Imagine a ball placed at the bottom of a perfectly smooth, U-shaped bowl. If you give it a gentle push, what happens? It rolls up one side, slows down, stops for an instant, then rolls back down, past the bottom, up the other side, stops, and returns. Left alone, it keeps doing this — back and forth, back and forth — in a steady, repeating rhythm.

That rhythm is the heart of Simple Harmonic Motion (SHM). It's the most fundamental kind of oscillatory (back-and-forth) motion in physics.

The Intuition: A Restoring Force That Fights Displacement

The key idea is this: the further you push the object from its resting (equilibrium) position, the stronger the force that tries to pull it back.

In the bowl, when the ball is at the bottom (equilibrium), gravity pulls straight down, and the bowl pushes straight up — no sideways force. But when you push the ball up the side, gravity now has a component that pulls it down the slope. The higher up the side you push it, the steeper the slope, and the stronger that pull-back force becomes.

This is a restoring force — it always points toward equilibrium. And crucially, in SHM, this restoring force is directly proportional to the displacement from equilibrium. Double the displacement, double the restoring force.

F = -k x

The Precise Statement

Simple Harmonic Motion is the motion of an object where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.

That's it. That single condition — $F = -kx$ — is the entire definition. Everything else (the sine waves, the formulas for period and frequency) follows mathematically from this one law.

What Does This Motion Look Like?

If you track the ball's position over time, you get a beautiful, smooth wave — a sine wave (or cosine wave). It's the same shape as the shadow of a spinning wheel cast on a wall.

The motion has three key descriptors:

Amplitude ( $A$ ): The maximum displacement from equilibrium. How far you initially pushed the ball up the side of the bowl.
Period ( $T$ ): The time it takes to complete one full back-and-forth cycle (e.g., from the leftmost point, back to the leftmost point).
Frequency ( $f$ ): How many cycles happen per second. $f = 1/T$ .

Note

A remarkable fact: for a given system (fixed $k$ and fixed mass $m$ ), the period and frequency do not depend on the amplitude. A big push and a tiny push take exactly the same time to complete one cycle. This is called isochronism — and it's why pendulums were used to keep time in clocks.

The Mathematical Description (Derived from $F = -kx$ )

Using Newton's second law ( $F = ma$ ) and the definition of acceleration ( $a = \frac{d^2x}{dt^2}$ ), the condition $F = -kx$ becomes:

m \frac{d^2x}{dt^2} = -k x

This is a differential equation. Its solution — the position as a function of time — is:

x(t) = A \cos(\omega t + \phi)

Where:

$\omega = \sqrt{\frac{k}{m}}$ is the angular frequency (radians per second). It tells you how fast the oscillation is.
$\phi$ is the phase constant (determines where in the cycle you start measuring time). …

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The restoring force on a displaced liquid column is proportional to displacement, making the motion simple harmonic. The time period depends on column length and gravity but cancels out density. The answer is (C).

Why the liquid oscillates: the restoring mechanism

When you displace liquid in a U-tube from equilibrium, gravity creates an imbalance. If one arm rises by height $h$ above the equilibrium level, the other arm falls by the same amount (conservation of volume in an incompressible liquid). This creates a height difference of $2h$ between the two arms, and the excess weight of the taller column pulls the system back toward equilibrium.

The key insight: this restoring force is directly proportional to the displacement. That proportionality is the signature of simple harmonic motion.

Step-by-step analysis

Set up the displacement. Suppose the liquid column has total length $L$ and cross-sectional area $A$ . At equilibrium, the liquid sits at the same height in both arms. Displace it so one side rises by $y$ and the other falls by $y$ ; the height difference is now $2y$ .
Find the restoring force. The extra column of liquid on the higher side has height $2y$ , volume $A \cdot 2y$ , and mass $\rho \cdot A \cdot 2y$ , where $\rho$ is the liquid's density. Gravity pulls this excess mass downward with force

F = -\rho A (2y) g = -2\rho A g \, y.

The negative sign indicates the force opposes the displacement (restoring).

Apply Newton's second law. The entire liquid column (mass $m = \rho A L$ ) accelerates as one unit:

m \frac{d^2 y}{dt^2} = -2\rho A g \, y.

Substitute $m$ :

\rho A L \frac{d^2 y}{dt^2} = -2\rho A g \, y.

Simplify to standard SHM form. Cancel $\rho A$ from both sides:

L \frac{d^2 y}{dt^2} = -2g \, y \quad \Rightarrow \quad \frac{d^2 y}{dt^2} = -\frac{2g}{L} \, y.

This is the equation of simple harmonic motion with angular frequency $\omega^2 = \frac{2g}{L}$ .

Extract the time period. Since $\omega = \sqrt{\frac{2g}{L}}$ , the period is $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{L}{2g}} = \pi \sqrt{\frac{2L}{g}}.$ …

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Q.Motion of an oscillating liquid column in a U-tube is

Why the liquid oscillates: the restoring mechanism

Step-by-step analysis

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