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Angular Momentum Calculator

Calculate angular momentum, moment of inertia, and rotational kinetic energy with interactive visualization

Parameters

kgⓘ
Mass of the rotating object
mⓘ
Radius of rotation
rad/sⓘ
Angular velocity (positive for counterclockwise)
kg·m²ⓘ
Moment of inertia (leave 0 for automatic calculation)
Rotational Motion Analysis
Enter values to calculate angular momentum and rotational energy

Calculated Values

Angular Momentum:
2.5000;kg⋅m2/s2.5000;kg·m²/s
Rotational Kinetic Energy:
12.5000;J12.5000;J
Moment of Inertia:
0.2500;kg⋅m20.2500;kg·m²
Linear Velocity:
5.0000;m/s5.0000;m/s

Examples

Example 1: Spinning Disk

A 2kg disk with 0.5m radius spinning at 10 rad/s.

  • Angular Momentum: 2.52.5
  • Rotational Kinetic Energy: 12.512.5
  • Moment of Inertia: 0.250.25
  • Linear Velocity: 55

Example 2: Figure Skater

A 60kg skater with 0.2m radius spinning at 5 rad/s.

  • Angular Momentum: 66
  • Rotational Kinetic Energy: 1515
  • Moment of Inertia: 1.21.2
  • Linear Velocity: 11

Example 3: Earth's Rotation

Earth (5.97×10²⁴ kg, 6.37×10⁶ m) rotating at 7.27×10⁻⁵ rad/s.

  • Angular Momentum: 7.07e+337.07e+33
  • Rotational Kinetic Energy: 2.57e+292.57e+29
  • Moment of Inertia: 9.72e+379.72e+37
  • Linear Velocity: 463463

Angular Momentum and Rotational Motion

Angular momentum is the rotational equivalent of linear momentum. It is a conserved quantity in isolated systems and plays a crucial role in understanding rotational motion, from spinning tops to planetary orbits.

The angular momentum of a point particle is L = r × p = mvr, where r is the position vector, p is linear momentum, m is mass, and v is velocity. For extended objects, L = Iω, where I is moment of inertia and ω is angular velocity.

Moment of inertia (I) is the rotational equivalent of mass. It depends on the mass distribution relative to the axis of rotation. For common shapes: disk (I = ½mr²), sphere (I = ⅖mr²), rod about center (I = ¹/₁₂ml²).

Rotational kinetic energy is KE = ½Iω², analogous to translational kinetic energy KE = ½mv². This energy is stored in the rotational motion of the object.

Angular momentum is conserved when no external torque acts on the system. This conservation principle explains phenomena like figure skaters spinning faster when they pull their arms in.

Key Concepts

  • Angular Momentum: L = Iω (rotational equivalent of linear momentum)
  • Moment of Inertia: I = Σmr² (rotational equivalent of mass)
  • Rotational Kinetic Energy: KE = ½Iω² (energy of rotation)
  • Conservation of Angular Momentum: L = constant (no external torque)
  • Torque: τ = Iα (rotational equivalent of force)
  • Angular Velocity: ω = v/r (rate of rotation)

Real-World Applications

  • Gyroscopes: Stability and navigation systems
  • Planetary Motion: Conservation of angular momentum in orbits
  • Sports: Spinning objects and angular momentum transfer
  • Quantum Physics: Electron spin and atomic structure
  • Engineering: Rotating machinery and flywheels

Physics Equations

Angular Momentum:
L=IωL = I\omega
Moment of Inertia (Disk):
I=12mr2I = \frac{1}{2}mr^2
Rotational Kinetic Energy:
KE=12Iω2KE = \frac{1}{2}I\omega^2
Linear Velocity:
v=rωv = r\omega
Torque:
τ=Iα\tau = I\alpha

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Moment of Inertia

Determine the moment of inertia:

Equation:

I=12mr2 (for disk)I = \frac{1}{2}mr^2 \text{ (for disk)}

Calculation:

I=0.25 kg\cdotpm2I = 0.25 \text{ kg·m}^2

Explanation:

Moment of inertia depends on mass distribution relative to the axis of rotation.

2

Step 2: Calculate Angular Momentum

Use the angular momentum formula:

Equation:

L=IωL = I\omega

Calculation:

L=0.250×10=2.500 kg\cdotpm2/sL = 0.250 \times 10 = 2.500 \text{ kg·m}^2\text{/s}

Explanation:

Angular momentum is the product of moment of inertia and angular velocity.

3

Step 3: Calculate Rotational Kinetic Energy

Find the rotational kinetic energy:

Equation:

KE=12Iω2KE = \frac{1}{2}I\omega^2

Calculation:

KE=12×0.250×(10)2=12.500 JKE = \frac{1}{2} \times 0.250 \times (10)^2 = 12.500 \text{ J}

Explanation:

Rotational kinetic energy is the energy associated with rotational motion.

4

Step 4: Calculate Linear Velocity

Find the linear velocity at the edge:

Equation:

v=rωv = r\omega

Calculation:

v=0.5×10=5.000 m/sv = 0.5 \times 10 = 5.000 \text{ m/s}

Explanation:

Linear velocity is the tangential speed at the edge of the rotating object.

5

Step 5: Conservation Check

Verify angular momentum conservation:

Equation:

L=constant (no external torque)L = \text{constant} \text{ (no external torque)}

Calculation:

L=2.500 kg\cdotpm2/s (conserved)L = 2.500 \text{ kg·m}^2\text{/s} \text{ (conserved)}

Explanation:

In the absence of external torque, angular momentum remains constant.

Frequently Asked Questions (FAQ)

What is angular momentum?

Angular momentum is the rotational equivalent of linear momentum. It measures the amount of rotational motion and is conserved in isolated systems when no external torque acts.

How do I calculate angular momentum?

For a point particle: L = mvr. For extended objects: L = Iω, where I is moment of inertia and ω is angular velocity. The direction follows the right-hand rule.

What is moment of inertia?

Moment of inertia is the rotational equivalent of mass. It depends on how mass is distributed relative to the axis of rotation. Larger moment of inertia means more resistance to angular acceleration.

Why is angular momentum conserved?

Angular momentum is conserved when no external torque acts on the system. This conservation explains phenomena like figure skaters spinning faster when they pull their arms in.

What is rotational kinetic energy?

Rotational kinetic energy is the energy associated with rotational motion: KE = ½Iω². It's analogous to translational kinetic energy KE = ½mv².

How does moment of inertia change with shape?

Moment of inertia depends on mass distribution. For the same mass, objects with mass farther from the axis have larger moment of inertia. Common formulas: disk (I = ½mr²), sphere (I = ⅖mr²).

Practice MCQs

  1. Angular momentum is conserved when:
  2. For a disk, moment of inertia is:
  3. If angular velocity doubles, rotational kinetic energy:
  4. Angular momentum has units of:
  5. A figure skater spins faster when: