Interactive physics simulator
Conservation of Angular Momentum
Explore how rotation speed adapts to preserve momentum. Contract dumbbell sliders on a rotating stool, toggle layout and tuck poses for an aerial diver, and watch collapsing stars spin up dynamically.
Rotational Conservation Lab
Adjust coordinates in the control panel, choose the active graph representation, and click Simulate.
Live Telemetry
- Moment of Inertia (I)
- 0.00 kg·m²
- Angular Velocity (ω)
- 0.00 rad/s
- Angular Momentum (L)
- 0.00 kg·m²/s
- Rotational Kinetic Energy (K)
- 0.0 J
- Telemetry Status
- Ready
Introduction to Rotational Conservation
The Law of Conservation of Angular Momentum is one of the primary conservation principles in physics, alongside conservation of energy and linear momentum. It states that the total angular momentum of any system remains constant in both magnitude and direction if the net external torque acting on the system is zero.
This principle explains how complex rotational tasks are performed across various scales, from an ice skater spinning faster on ice, to acrobatics in gymnastics, and down to the scale of white dwarfs or neutron stars rotating hundreds of times per second after gravitational collapse.
Mathematical Formulation
According to the rotational analogue of Newton's Second Law, net external torque is equal to the time rate of change of angular momentum:
When the net external torque is zero (τnet = 0), the change in angular momentum is zero:
For a rigid body rotating about a fixed axis, the angular momentum is the product of its moment of inertia (I) and its angular velocity (ω). Under conservation, if the system changes its shape (and thus its moment of inertia), its speed must change inversely to keep the product constant:
Where I₁ and ω₁ are the initial moment of inertia and angular velocity, and I₂ and ω₂ are the final values after the shape reconfiguration.
Solved Numerical Examples
A student sits on a rotating stool holding two heavy dumbbells. With their arms extended, the total moment of inertia of the student, stool, and dumbbells is 4.80 kg·m², and they rotate at an angular speed of 1.50 rad/s. When the student pulls the dumbbells close to their body, the moment of inertia drops to 1.60 kg·m². (a) Calculate the new angular speed of the student. (b) Find the ratio of the final rotational kinetic energy to the initial rotational kinetic energy.
View Step-by-Step Solution
- Given: Initial moment of inertia I₁ = 4.80 kg·m², initial angular velocity ω₁ = 1.50 rad/s, final moment of inertia I₂ = 1.60 kg·m².
- (a) Find New Angular Velocity (ω₂):
Since no external torques act on the stool-student system, total angular momentum (L) is conserved: L₁ = L₂ ⇒ I₁ · ω₁ = I₂ · ω₂. - Solve for ω₂: ω₂ = (I₁ · ω₁) / I₂.
- Substitute values: ω₂ = (4.80 × 1.50) / 1.60 = 7.20 / 1.60 = 4.50 rad/s.
- The rotation speed increases by a factor of 3.
- (b) Compare Rotational Kinetic Energies (KE_rot):
Initial kinetic energy: KE₁ = ½ · I₁ · ω₁² = 0.5 × 4.80 × (1.50)² = 5.40 Joules. - Final kinetic energy: KE₂ = ½ · I₂ · ω₂² = 0.5 × 1.60 × (4.50)² = 16.20 Joules.
- Ratio: KE₂ / KE₁ = 16.20 / 5.40 = 3.00.
- Note: Rotational kinetic energy increases by a factor of 3. This additional energy comes from the chemical work done by the student's arm muscles as they pull the dumbbells inward against the outward apparent centrifugal force.
- Results: (a) The final angular speed is 4.50 rad/s. (b) The ratio of final to initial kinetic energy is 3.00.
A high diver jumps off a 10-meter platform. In the straight layout position, their moment of inertia about the center of mass is 15.0 kg·m², spinning at 1.20 rad/s. They immediately curl into a tight tuck, reducing their moment of inertia to 3.00 kg·m². Calculate their rotational velocity in the tucked position.
View Step-by-Step Solution
- Given: Initial layout inertia Ilayout = 15.0 kg·m², initial spin ωlayout = 1.20 rad/s, tucked inertia Ituck = 3.00 kg·m².
- During the aerial flight, gravity acts on the diver's center of mass, producing zero external torque about their rotation axis. Thus, angular momentum is conserved: Llayout = Ltuck.
- Write the equation: Ilayout · ωlayout = Ituck · ωtuck.
- Solve for tucked angular velocity ωtuck: ωtuck = (Ilayout · ωlayout) / Ituck.
- Substitute values: ωtuck = (15.0 × 1.20) / 3.00 = 18.0 / 3.00 = 6.00 rad/s.
- Result: The diver spins at 6.00 rad/s in the tucked position (5 times faster than in layout).
A dying star collapses into a dense white dwarf. If the star originally has a radius of 7.00 × 10⁵ km and rotates once every 30.0 days, calculate its new rotation period (in hours) if it contracts uniformly to a radius of 7.00 × 10³ km, assuming its mass remains constant.
View Step-by-Step Solution
- Given: Initial radius R₁ = 7.00 × 10⁵ km, initial period T₁ = 30.0 days, final radius R₂ = 7.00 × 10³ km.
- For a uniform sphere, the moment of inertia is I = ⅖ · M · R². Since mass (M) is constant, moment of inertia scales as R²: I ∝ R².
- Angular momentum conservation: I₁ · ω₁ = I₂ · ω₂.
- Since angular speed ω = 2π / T, we can rewrite the conservation equation as: I₁ / T₁ = I₂ / T₂ ⇒ R₁² / T₁ = R₂² / T₂.
- Solve for final rotation period T₂: T₂ = T₁ · (R₂ / R₁)².
- Calculate the radius ratio: R₂ / R₁ = 7.00 × 10³ / 7.00 × 10⁵ = 1 / 100 = 0.01.
- Substitute values: T₂ = 30.0 days × (0.01)² = 30.0 × 10⁻⁴ days = 0.0030 days.
- Convert period to hours: T₂ = 0.0030 days × 24 hours/day = 0.072 hours (or 4.32 minutes).
- Result: The collapsed white dwarf star rotates once every 0.072 hours (approx. 4.32 minutes).
Conceptual Practice
State the Law of Conservation of Angular Momentum and specify its mathematical criteria.
Show Explanation
The Law of Conservation of Angular Momentum states that the total angular momentum of a system remains constant in magnitude and direction if the net external torque acting on the system is zero. Mathematically:
τnet = 0 ⇒ dL/dt = 0 ⇒ L = constant. For a rigid system: I₁ω₁ = I₂ω₂.
When a spinning acrobat pulls their arms in, their rotational speed increases. Does their rotational kinetic energy change? Explain the source of any difference.
Show Explanation
Yes, their rotational kinetic energy increases. Since angular momentum is conserved ($L = Iomega = ext{const}$), rotational kinetic energy is $K = rac{1}{2}Iomega^2 = rac{L^2}{2I}$. When they pull their arms in, moment of inertia ($I$) decreases, which increases $K$. The source of this additional kinetic energy is the positive work done by the acrobat's muscles pulling their limbs inward against the centrifugal forces.
Why is a helicopter built with a small tail rotor? Explain using conservation of angular momentum.
Show Explanation
When a helicopter engine spins its main horizontal rotor blades, it exerts a torque on them. By Newton's third law (or conservation of angular momentum), the helicopter cabin experiences an equal and opposite torque, causing it to spin in the opposite direction. The small vertical tail rotor blows air sideways, creating a countering torque that keeps the cabin stable and prevents it from spinning out of control.
How does conservation of angular momentum explain why a cat always lands on its feet (the cat-turning reflex)?
Show Explanation
A falling cat starts with zero total angular momentum ($L = 0$). By twisting its upper body in one direction and its lower body in the opposite direction, the cat can rotate its alignment in mid-air while keeping its total $L$ exactly zero. By curling and extending legs dynamically, it changes the relative moments of inertia of its front and back halves, allowing it to complete a 180-degree rotation and land safely.
Explain how tidal friction affects the Earth-Moon system's angular momentum over long time scales.
Show Explanation
Tidal friction between Earth's oceans and the rotating crust slows down the Earth's rotation rate, decreasing its spin angular momentum. Since there are no external torques on the Earth-Moon system, the total angular momentum must remain constant. Consequently, the orbital angular momentum of the Moon must increase, which causes the Moon to slowly spiral outward to a wider orbit (approx. 3.8 cm per year).
Frequently Asked Questions
What is the conservation of angular momentum?
It is a fundamental physics principle stating that the total rotational momentum of a system remains constant unless acted upon by an external twisting force (torque).
What happens to the moment of inertia when a spinner curls up?
Curling up brings mass closer to the center of rotation, which decreases the moment of inertia (I). To keep L constant, the angular velocity (ω) increases.
Does kinetic energy remain constant when angular momentum is conserved?
No. While momentum is conserved, rotational kinetic energy (KE = L²/2I) increases when I decreases because internal work is performed to pull the mass inward.
How is angular momentum conserved in planetary orbits?
A central gravitational force acts directly along the radial vector, producing zero torque on the planet. Thus, the planet's orbital angular momentum is conserved, making it speed up at perihelion.
What is the relationship between torque and angular momentum?
Net torque is the rate of change of angular momentum (τ = dL/dt). If net torque is zero, angular momentum is constant (conserved).
Why do gyroscopes resist changing their orientation?
A spinning gyroscope has a large angular momentum vector. According to conservation rules, the vector resists changes in both magnitude and direction, maintaining orientation stability.
Can a system change its rotation speed without internal work?
No. Changing the mass distribution (radius) requires radial forces. Doing work to pull mass inward increases kinetic energy; letting mass slide outward does negative work, slowing spin down.