A tunnel is dug through the centre of the Earth. Show that a body of mass 'm' when dropped from rest from one

NCERT Exemplar · Q39

m

Odisha ChseLong· 5mImportance★★★★★est

98% · 65/66 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

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). …

Start your 14-day free trial to unlock the full solution →

A body dropped through a tunnel across the Earth experiences a restoring force proportional to its displacement from the centre — exactly the condition for simple harmonic motion — with a period of about 84 minutes.

The key insight is that only the mass of the Earth inside the body's current radius pulls on it. Outside that radius, the shell of Earth above cancels its own gravity — a result of Newton's shell theorem. So as the body falls toward the centre, the gravitational force shrinks linearly with distance, not as $1/r^2$ . That linear restoring force is the hallmark of SHM.

Let's work through it.

Set up the problem.

Earth is a uniform sphere of radius $R$ and mass $M$ . A tunnel is drilled straight through the centre, from one surface to the other. A body of mass $m$ is dropped from rest at the surface ( $r = R$ ). We want to show its motion is SHM.
Gravitational force inside a uniform sphere.

For a point at distance $r$ from the centre ( $r \le R$ ), the gravitational force on $m$ is due only to the mass $M_r$ enclosed within radius $r$ . By the shell theorem, the spherical shell outside $r$ contributes zero net force.

The enclosed mass is proportional to volume:

M_r = M \cdot \frac{r^3}{R^3}

Write the force. Newton's law of gravitation gives:

F = -\frac{G M_r m}{r^2} = -\frac{G m}{r^2} \cdot M \frac{r^3}{R^3} = -\frac{G M m}{R^3} \, r

The negative sign means the force points toward the centre (restoring).

This is a linear restoring force: $F = -k r$ with $k = \frac{G M m}{R^3}$ .

F = -\frac{G M m}{R^3} \, r

Relate to SHM. For any system where $F = -k x$ , the motion is simple harmonic with angular frequency $\omega = \sqrt{k/m}$ . Here $x = r$ , so:

\omega = \sqrt{\frac{G M}{R^3}}

The equation of motion is:

$m \frac{d^2 r}{dt^2} = -\frac{G M m}{R^3} r \quad \Rightarrow \quad \frac{d^2 r}{dt^2} + \frac{G M}{R^3} r = 0$ …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ38 — One end of a V-tube containing mercury is connected to a suction pump …→Next questionQ40 — A simple pendulum of time period 1s and length l is hung from a fixed …

Q.A tunnel is dug through the centre of the Earth. Show that a body of mass 'mmm' when dropped from rest from one end of the tunnel will execute simple harmonic motion.

🎓Unlock everything free for 14 days

Q.A tunnel is dug through the centre of the Earth. Show that a body of mass ' $m$ ' when dropped from rest from one end of the tunnel will execute simple harmonic motion.

Unlock everything free for 14 days