1. What is an Angular Momentum Calculator?

Definition: This calculator determines the angular momentum of an object, either by using mass, velocity, and radius for an object rotating around a central point, or by using moment of inertia and angular velocity for an object rotating around its own axis.

Purpose: It is used in physics to analyze rotational motion, such as in orbiting planets or spinning objects.

2. How Does the Calculator Work?

The calculator supports two methods to calculate angular momentum:

1. For an Object Rotating Around a Central Point: \[ L = m v r \] 2. For an Object Rotating Around Its Own Axis: \[ L = I \omega \] Where:

• \( L \): Angular momentum (kg·m²/s)
• \( m \): Mass (mg, g, dag, kg, lb, st, Earth)
• \( v \): Velocity (m/s, km/h, ft/s, mph, km/s, mi/s)
• \( r \): Radius (mm, cm, m, km, in, ft, yd, mi)
• \( I \): Moment of inertia (kg·m², lb·ft², lbf·ft·s²)
• \( \omega \): Angular velocity (rpm, rad/s, Hz)

Unit Conversions:

• Mass (\( m \)): mg (1 mg = 0.000001 kg), g (1 g = 0.001 kg), dag (1 dag = 0.01 kg), kg, lb (1 lb = 0.45359237 kg), st (1 st = 6.35029318 kg), Earth (1 Earth = 5.972 × 10²⁴ kg)
• Velocity (\( v \)): m/s, km/h (1 km/h = \( \frac{1000}{3600} \) m/s ≈ 0.277777778 m/s), ft/s (1 ft/s = 0.3048 m/s), mph (1 mph = 0.44704 m/s), km/s (1 km/s = 1000 m/s), mi/s (1 mi/s = 1609.344 m/s)
• Radius (\( r \)): mm (1 mm = 0.001 m), cm (1 cm = 0.01 m), m, km (1 km = 1000 m), in (1 in = 0.0254 m), ft (1 ft = 0.3048 m), yd (1 yd = 0.9144 m), mi (1 mi = 1609.344 m)
• Moment of Inertia (\( I \)): kg·m², lb·ft² (1 lb·ft² = 0.0421401100938048 kg·m²), lbf·ft·s² (1 lbf·ft·s² = 1.3558179483314004 kg·m²)
• Angular Velocity (\( \omega \)): rpm (1 rpm = \( \frac{2\pi}{60} \) rad/s ≈ 0.104719755 rad/s), rad/s, Hz (1 Hz = \( 2\pi \) rad/s ≈ 6.283185307 rad/s)
• Angular Momentum (\( L \)): kg·m²/s

Steps:

• Select the calculation mode (Rotating Around a Central Point or Rotating Around Its Own Axis).
• Enter the required values and select their units (mass, velocity, and radius, or moment of inertia and angular velocity).
• Convert all inputs to base units (kg, m/s, m, kg·m², rad/s).
• Calculate the angular momentum using the selected formula.
• If the result is less than 0.001, display it in scientific notation; otherwise, display it with 3 decimal places.

3. Importance of Angular Momentum Calculation

Calculating angular momentum is crucial for:

• Physics Education: Understanding rotational dynamics.
• Astronomy: Analyzing the motion of planets and stars.
• Engineering: Designing rotating machinery like turbines and flywheels.

4. Using the Calculator

Examples:

• Example 1 (Rotating Around a Central Point): For \( m = 1 \, \text{kg} \), \( v = 2 \, \text{m/s} \), \( r = 3 \, \text{m} \):
  • Angular Momentum: \( L = 1 \times 2 \times 3 = 6.000 \, \text{kg·m²/s} \)
• Example 2 (Rotating Around a Central Point with Different Units): For \( m = 500 \, \text{g} \), \( v = 10 \, \text{km/h} \), \( r = 50 \, \text{cm} \):
  • Convert: \( m = 500 \times 0.001 = 0.5 \, \text{kg} \), \( v = 10 \times \frac{1000}{3600} = 2.778 \, \text{m/s} \), \( r = 50 \times 0.01 = 0.5 \, \text{m} \)
  • Angular Momentum: \( L = 0.5 \times 2.778 \times 0.5 = 0.694 \, \text{kg·m²/s} \)
• Example 3 (Rotating Around Its Own Axis): For \( I = 2 \, \text{kg·m²} \), \( \omega = 5 \, \text{rad/s} \):
  • Angular Momentum: \( L = 2 \times 5 = 10.000 \, \text{kg·m²/s} \)
• Example 4 (Rotating Around Its Own Axis with Different Units): For \( I = 10 \, \text{lb·ft²} \), \( \omega = 60 \, \text{rpm} \):
  • Convert: \( I = 10 \times 0.0421401100938048 = 0.421 \, \text{kg·m²} \), \( \omega = 60 \times \frac{2\pi}{60} = 6.283 \, \text{rad/s} \)
  • Angular Momentum: \( L = 0.421 \times 6.283 = 2.645 \, \text{kg·m²/s} \)
• Example 5 (Small Value with Scientific Notation): For \( m = 1 \, \text{mg} \), \( v = 0.1 \, \text{m/s} \), \( r = 0.01 \, \text{m} \):
  • Convert: \( m = 1 \times 0.000001 = 0.000001 \, \text{kg} \)
  • Angular Momentum: \( L = 0.000001 \times 0.1 \times 0.01 = 0.000000001 \, \text{kg·m²/s} \)
  • Since 0.000000001 < 0.001, display as \( 1.000 \times 10^{-9} \, \text{kg·m²/s} \)

5. Frequently Asked Questions (FAQ)

Q: What is angular momentum?
A: Angular momentum is a measure of the rotational motion of an object, defined as the product of mass, velocity, and radius for an object rotating around a central point, or the product of moment of inertia and angular velocity for an object rotating around its own axis.

Q: How is angular momentum conserved?
A: Angular momentum is conserved in a closed system where no external torques act, according to the law of conservation of angular momentum.

Q: Can angular momentum be negative?
A: Yes, a negative angular momentum indicates rotation in the opposite direction (e.g., clockwise if positive is counterclockwise).

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