Q.a. Prove Kepler's law of areas as a consequence of conservation of angular momentum principle.
OR
b. Show that the time period of a satellite depends only on the orbital radius.
Nagaland NbseNagaland Board of School Education (Class XI) 2025Subjective· 3mImportance★★★★★
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
L=constantifτext=0
Where:
L is the angular momentum of the system
τext 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 v at position r from a reference point:
L=r×mv
For a rigid body rotating about a fixed axis:
L=Iω
Where:
I = moment of inertia (how mass is distributed relative to the axis)
ω = 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:
τext=dtdL
If τext=0, then dtdL=0, so L is constant.
Since L=Iω, if I changes (you pull arms in), ω must change in the opposite way to keep L the same:
I1ω1=I2ω2
Smaller I → larger ω (spin faster). Larger I → smaller ω (spin slower).
Real-World Examples
Situation
What happens
Why
Ice skater pulling arms in
Spins faster
I decreases, ω 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.
Kepler's second law (equal areas in equal times) follows directly from the fact that the gravitational force on a planet is central (always along the Sun–planet line), which conserves the planet's angular momentum about the Sun.
Consider a planet of mass m at position vector r from the Sun, moving with velocity v, so its linear momentum is p=mv and its angular momentum about the Sun is L=r×p.
Step 1 — angular momentum is conserved: The gravitational force F on the planet is always directed along r (attractive, central force). The torque about the Sun is:
τ=r×F=0 (since F is parallel/anti-parallel to r)
Since τ=dL/dt=0, L is constant in both magnitude and direction throughout the orbit (this is also why planetary orbits are planar).
Step 2 — relate L to the areal velocity: In a small time dt, the planet moves through dr=vdt. The area swept out by the position vector r in this time is that of the triangle with sides r and dr:
Q.The motion of a particle under the central force is always confined to a plane. This is a consequence of
(a) conservation of linear momentum
(b) conservation of angular momentum
(c) conservation of energy
(d) none of these
›Reveal solutionSolution
Motion under a central force stays in a plane because angular momentum is conserved. Answer (B).
A central force always points toward (or away from) a fixed centre, so its torque about that centre is zero (r and F are parallel/anti-parallel, r x F = 0).
Kepler's second law (the radius vector sweeps equal areas in equal times) is a direct geometric consequence of angular momentum being conserved in a planet's orbit around the Sun.
Gravitational force on a planet from the Sun always acts along the line joining them (a central force), so it produces zero torque about the Sun. With zero net torque, the planet's angular momentum L = m v r sin(theta) about the Sun stays constant throughout the orbit.
Q.A particle undergoes uniform circular motion. The angular momentum of the particle remains conserved about:
(a) any point inside the circle
(b) the centre point of the circle
(c) any point outside the circle
(d) the point on the circumference of the circle
›Reveal solutionSolution
Angular momentum in uniform circular motion is conserved only about the centre of the circle, because the centripetal force passes through that point and produces zero torque there.
For a particle moving in a circle of radius r with constant speed, the only force acting is the centripetal force F, which is always directed radially inward, straight at the centre of the circle.
Torque about any point O is tau = r' x F, where r' is the position vector from O to the particle. If O is the centre of the circle, r' is exactly along the same (or opposite) line as F, so r' x F = 0 at every instant -- zero torque, hence angular momentum about the centre stays constant (both in magnitude, m v r, and direction, perpendicular to the plane of motion).
Q.The motion of planets in the solar system is an example of conservation of
(a) mass
(b) linear momentum
(c) angular momentum
(d) energy.
›Reveal solutionSolution
The gravitational force of the Sun on a planet always acts along the line joining the Sun and the planet (a central force), so it produces zero torque about the Sun.
Since τ=dtdL=r×F, and F is always anti-parallel to r (radial), τ=0, so the angular momentum L of the planet about the Sun st …
Q.The motion of planets in the solar system is an example of the conservation of
(a) mass
(b) linear momentum
(c) angular momentum
(d) energy.
›Reveal solutionSolution
Planetary motion is a direct illustration of conservation of angular momentum, since gravity is a central force.
The Sun's gravitational pull on a planet always points from the planet towards the Sun, i.e. along the position vector r of the planet (measured from the Sun). Torque is τ=r×F; since F is anti-parallel to r, this cross product is zero, so the torque on the planet about the Sun is always zero. Because τ=dL/dt=0, the angular momentum L of the planet about …