How to Find the Centre of Mass by Integration?

Finding the centre of mass by integration involves using calculus to determine the average position of mass within an object or system. The method varies depending on whether you're dealing with a discrete system of point masses, a one-dimensional object (like a rod), a two-dimensional object (like a lamina), or a three-dimensional object.

1. Centre of Mass for Discrete Systems

If you have a system of point masses, the centre of mass coordinates (x_cm, y_cm, z_cm) are given by:

• x_cm = (∑m_ix_i) / M
• y_cm = (∑m_iy_i) / M
• z_cm = (∑m_iz_i) / M

Where:

• m_i is the mass of the i-th particle.
• x_i, y_i, z_i are the coordinates of the i-th particle.
• M is the total mass of the system (M = ∑m_i).

2. Centre of Mass for Continuous Objects (Integration)

When dealing with continuous objects (rods, laminas, solids), you need to use integration because the mass is distributed continuously.

2.1. One-Dimensional Object (Rod)

For a rod of length L, lying along the x-axis, with a linear mass density λ(x) (mass per unit length), the x-coordinate of the centre of mass is:

x_cm = (∫₀^L x * λ(x) dx) / (∫₀^L λ(x) dx)

Steps:

1. Identify the linear mass density λ(x). This describes how the mass is distributed along the rod. It could be constant (uniform density) or a function of x (non-uniform density).
2. Set up the integral. The numerator involves integrating the product of position (x) and linear density (λ(x)) over the length of the rod. The denominator is the integral of the linear density, which gives the total mass (M) of the rod.
3. Integrate. Evaluate both integrals.
4. Calculate. Divide the result of the numerator by the result of the denominator to find x_cm.

Example: Consider a rod of length L with a density λ(x) = kx, where k is a constant.

• x_cm = (∫₀^L x * (kx) dx) / (∫₀^L kx dx)
• x_cm = (k ∫₀^L x² dx) / (k ∫₀^L x dx)
• x_cm = (k [x³/3]₀^L) / (k [x²/2]₀^L)
• x_cm = (kL³/3) / (kL²/2)
• x_cm = (2/3)L

2.2. Two-Dimensional Object (Lamina)

For a lamina (a thin, flat object), you'll need to calculate both x_cm and y_cm. Let σ(x, y) be the surface mass density (mass per unit area).

• x_cm = (∫∫ x * σ(x, y) dA) / (∫∫ σ(x, y) dA)
• y_cm = (∫∫ y * σ(x, y) dA) / (∫∫ σ(x, y) dA)

Where:

• dA is the differential area element (e.g., dx dy in Cartesian coordinates).
• The double integrals are taken over the entire area of the lamina.
• The denominator in each case is the total mass (M) of the lamina.

Steps:

1. Identify the surface mass density σ(x, y). This tells you how the mass is distributed across the surface.
2. Define the area element dA. Choose appropriate coordinates (Cartesian, polar, etc.) and express dA accordingly.
3. Set up the double integrals. Define the limits of integration based on the shape of the lamina.
4. Integrate. Evaluate the integrals for both x_cm and y_cm.
5. Calculate. Divide the results of the numerator by the results of the denominator to find the x and y coordinates of the centre of mass.

2.3. Three-Dimensional Object (Solid)

For a three-dimensional object, you extend the concept to include a z-coordinate and use a volume mass density ρ(x, y, z) (mass per unit volume).

• x_cm = (∭ x * ρ(x, y, z) dV) / (∭ ρ(x, y, z) dV)
• y_cm = (∭ y * ρ(x, y, z) dV) / (∭ ρ(x, y, z) dV)
• z_cm = (∭ z * ρ(x, y, z) dV) / (∭ ρ(x, y, z) dV)

Where:

• dV is the differential volume element (e.g., dx dy dz in Cartesian coordinates).
• The triple integrals are taken over the entire volume of the solid.
• The denominator is the total mass (M) of the solid.

Steps:

1. Identify the volume mass density ρ(x, y, z).
2. Define the volume element dV. Choose an appropriate coordinate system (Cartesian, cylindrical, spherical) and express dV.
3. Set up the triple integrals. Define the limits of integration based on the shape of the solid.
4. Integrate. Evaluate the integrals for x_cm, y_cm, and z_cm.
5. Calculate. Divide the results of the numerator by the denominator for each coordinate.

3. Key Considerations

• Symmetry: If the object has symmetry, the centre of mass will lie along the axis of symmetry. This can simplify the calculations.
• Coordinate System: Choosing the right coordinate system can significantly ease the integration process.
• Density Function: The accuracy of the centre of mass calculation depends on the accuracy of the density function.

By using integration and understanding the mass distribution of an object, you can accurately determine its centre of mass.