What is the direction of areal velocity of the earth around the sun?

NCERT Exemplar · Q20

Q. What is the direction of areal velocity of the earth around the sun?

Punjab PsebShort· 2mImportance★★★★★est

73% · 49/67 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 — Conservation of Angular Momentum

Conservation of Angular Momentum

The Intuition First

Imagine you're sitting on a spinning office chair with your arms stretched out. Someone gives you a gentle push, and you start rotating slowly. Now pull your arms in tight against your chest. What happens? You spin faster. Push your arms back out — you slow down again.

Nothing external pushed you to go faster or slower. The change came from inside — from how you arranged your mass relative to the axis of rotation.

That's the core idea: Angular momentum is a quantity that stays constant for a rotating system unless an external torque acts on it. When you pulled your arms in, you didn't change your angular momentum — you changed your distribution of mass, and your rotation speed had to adjust to keep the total constant.

The Precise Statement

\vec{L} = \text{constant} \quad \text{if} \quad \vec{\tau}_{\text{ext}} = 0

Where:

$\vec{L}$ is the angular momentum of the system
$\vec{\tau}_{\text{ext}}$ is the net external torque acting on the system

In words: The total angular momentum of an isolated system (no external torque) remains constant in both magnitude and direction.

What Is Angular Momentum?

For a point mass $m$ moving with velocity $\vec{v}$ at position $\vec{r}$ from a reference point:

\vec{L} = \vec{r} \times m\vec{v}

For a rigid body rotating about a fixed axis:

L = I\omega

Where:

$I$ = moment of inertia (how mass is distributed relative to the axis)
$\omega$ = angular velocity (how fast it spins)

Note

Moment of inertia $I$ depends on where the mass is, not just how much. Mass far from the axis gives larger $I$ ; mass close to the axis gives smaller $I$ .

Why It Works: The Physics

Newton's second law for rotation says:

\tau_{\text{ext}} = \frac{dL}{dt}

If $\tau_{\text{ext}} = 0$ , then $\frac{dL}{dt} = 0$ , so $L$ is constant.

Since $L = I\omega$ , if $I$ changes (you pull arms in), $\omega$ must change in the opposite way to keep $L$ the same:

I_1 \omega_1 = I_2 \omega_2

Smaller $I$ → larger $\omega$ (spin faster). Larger $I$ → smaller $\omega$ (spin slower).

Real-World Examples

Situation	What happens	Why
Ice skater pulling arms in	Spins faster	$I$ decreases, $\omega$ increases to keep $L$ constant
Diver tucking into a ball	Rotates faster in midair	Same principle — no external torque during flight
Cat falling upside-down	Twists body to land on feet	Changes $I$ of different body parts to rotate without external torque
Planet orbiting the Sun	Speeds up when closer, slows down when farther	Gravitational force is central (torque = 0), so $L$ is constant

Watch out

Angular momentum is a vector. Its direction matters too. If no external torque acts, the axis of rotation stays fixed in space. This is why a spinning gyroscope or a bicycle wheel resists being tilted.

Common Mistake to Avoid …

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

Areal velocity is a vector whose direction is given by the angular velocity vector of the orbiting body — for Earth around the Sun, that direction is perpendicular to the orbital plane, pointing out of the page (or northward, along the axis of the orbit).

The idea of areal velocity comes from Kepler’s second law: a line joining a planet to the Sun sweeps out equal areas in equal times. That sweeping motion has a direction — just like the area swept has an orientation. In physics, any quantity that involves rotation or sweeping has a vector associated with it, and its direction is given by the right-hand rule.

Areal velocity is defined as $\frac{d\vec{A}}{dt}$ , the rate of change of the area vector. The area vector itself is perpendicular to the plane of the orbit, by convention pointing in the direction of the angular velocity $\vec{\omega}$ . So the direction of areal velocity is the same as the direction of $\vec{\omega}$ .

Connect areal velocity to angular momentum. For a planet of mass $m$ orbiting the Sun, the angular momentum $\vec{L} = m (\vec{r} \times \vec{v})$ is conserved. The areal velocity is $\frac{d\vec{A}}{dt} = \frac{1}{2} \vec{r} \times \vec{v}$ . This is directly proportional to the specific angular momentum $\vec{L}/m$ . So the direction of $\frac{d\vec{A}}{dt}$ is exactly the direction of $\vec{L}$ .
Find the direction of $\vec{L}$ for Earth’s orbit. Earth orbits the Sun in a counterclockwise direction when viewed from above the North Pole (i.e., looking down from the north). Using the right-hand rule: curl the fingers of your right hand in the direction of Earth’s motion (counterclockwise), and your thumb points upward — out of the orbital plane, toward the north celestial pole. …

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

←Previous questionQ19 — Draw areal velocity versus time graph for mars.→Next questionQ21 — How is the gravitational force between two point masses affected when …

Q.What is the direction of areal velocity of the earth around the sun?

🎓Unlock everything free for 14 days

Unlock everything free for 14 days