1. What is the Center of Mass Formula?

The center of mass is the average location of the mass of a system. In calculus, it's calculated by integrating the position multiplied by the mass distribution and dividing by the total mass.

2. How Does the Calculator Work?

The calculator uses the center of mass formula:

Where:

• \( x_{cm} \) — Center of mass position (meters)
• \( M \) — Total mass of the system (kg)
• \( \int x \, dm \) — Integral of position multiplied by mass distribution (kg·m)

Explanation: The formula gives the weighted average of all mass elements' positions in a system.

3. Importance of Center of Mass Calculation

Details: The center of mass is crucial in physics and engineering for analyzing motion, stability, and balance of objects and systems.

4. Using the Calculator

Tips: Enter the total mass in kilograms and the value of the integral ∫ x dm in kg·m. Both values must be valid (mass > 0).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between center of mass and center of gravity?
A: They coincide when gravity is uniform, but differ in non-uniform gravitational fields.

Q2: How is this formula extended to 2D or 3D systems?
A: Similar formulas exist for y and z coordinates: \( y_{cm} = \frac{1}{M} \int y \, dm \), etc.

Q3: What if the mass distribution isn't continuous?
A: For discrete masses, the integral becomes a sum: \( x_{cm} = \frac{1}{M} \sum x_i m_i \).

Q4: Why is center of mass important in physics?
A: The system moves as if all mass were concentrated at this point, simplifying motion analysis.

Q5: How does symmetry affect center of mass?
A: For symmetric objects with uniform density, the center of mass lies on the symmetry axes/planes.