Centres of mass: position, applications to uniform bodies and composite bodies

Centres of Mass: Position, Applications to Uniform Bodies and Composite Bodies

1. What is the Centre of Mass?

The centre of mass (CoM) is the point at which the total mass of a body can be considered to be concentrated. If you could balance the body on a single point, that point would be its CoM. Think of it as the “balance point” of a seesaw – the spot where the seesaw stays level. 🧭

2. Calculating the Centre of Mass for Uniform Bodies

A uniform body has the same density everywhere. For such bodies, the CoM is simply the geometric centre. The formulas are easy to remember:

Example: A uniform wooden stick 2 m long. The CoM is at $x = \frac{2\,\text{m}}{2} = 1\,\text{m}$ from either end. If you balance it on your finger at 1 m, it stays level. 🎯

3. Composite Bodies – The “Two‑Mass” System

When a body is made of several parts with different masses, the CoM is found by weighting each part’s position by its mass. The general formula is:

$$(x_{\text{cm}}, y_{\text{cm}}, z_{\text{cm}}) = \frac{1}{M}\sum_{i=1}^{n} m_i (x_i, y_i, z_i)$$

where $M = \sum m_i$ is the total mass. Think of it like a seesaw with several children of different weights sitting at different distances from the fulcrum. The heavier child pulls the CoM closer to them. 👦👧

Step‑by‑Step Example

1. Identify each part’s mass $m_i$ and its position $(x_i, y_i, z_i)$.
2. Compute the total mass $M = \sum m_i$.
3. Calculate the weighted sum $S = \sum m_i (x_i, y_i, z_i)$.
4. Divide by $M$ to get the CoM coordinates.

Concrete Example: Two blocks on a frictionless table:

• Block A: mass 3 kg, positioned at $x = 1$ m.
• Block B: mass 5 kg, positioned at $x = 4$ m.

• $M = 3 + 5 = 8$ kg.
• $S = 3(1) + 5(4) = 3 + 20 = 23$ kg·m.
• $x_{\text{cm}} = \frac{23}{8} \approx 2.875$ m.

4. Practical Applications

• Engineering: Designing bridges and aircraft – the CoM must be within safe limits to avoid tipping.
• Sports: A gymnast’s CoM determines how easily they can flip or balance.
• Everyday Life: When carrying a bag, placing the heavier side closer to your body keeps your balance.

5. Quick Quiz

Answer the following to test your understanding. Write your answers in the margin or on a sheet of paper. 📚

1. A uniform triangular plate with vertices at (0,0), (4,0), and (0,3). Where is its CoM?
2. Three masses: $m_1 = 2$ kg at $(2,0)$, $m_2 = 3$ kg at $(5,0)$, $m_3 = 1$ kg at $(0,4)$. Find the CoM.
3. Explain why the CoM of a uniform sphere is at its centre.

Answers (for teachers only):

1. $x = \frac{4}{3}$, $y = \frac{1}{3}$ (centroid of a right triangle).
2. $M = 6$ kg, $x_{\text{cm}} = \frac{2(2)+3(5)+1(0)}{6} = \frac{16}{6} \approx 2.67$, $y_{\text{cm}} = \frac{2(0)+3(0)+1(4)}{6} = \frac{4}{6} \approx 0.67$.
3. All points inside a uniform sphere are equally distant from the centre, so the average position is the centre.

6. Take‑Away Summary

• The CoM is the “balance point” of a body.
• For uniform bodies, the CoM is the geometric centre.
• For composite bodies, use weighted averages of each part’s position.
• Understanding CoM helps in engineering, sports, and everyday balance.