IB MYP 4-5 Physics- Centre of mass- Study Notes - New Syllabus
IB MYP 4-5 Physics-Centre of mass- Study Notes
Key Concepts
- Centre of mass
Centre of Mass
Centre of Mass
The centre of mass (COM) of a body is the point where the entire mass of the body can be considered to be concentrated for the purposes of translational motion analysis. It is also the point where the net gravitational force acts on the body.
- The position of the centre of mass depends on the shape and mass distribution of the body.
- For uniform, symmetrical bodies, the centre of mass lies at the geometric centre.
- For irregular or non-uniform bodies, the centre of mass is found by calculation or experiment.
- In free fall or projectile motion, the centre of mass follows a parabolic path.
- If the pivot point is below the centre of mass, the object becomes unstable.
Key Note:
- The centre of mass may lie outside the physical body (e.g., in a ring).
- For composite objects, the COM can be found by taking the weighted average of the positions of each part.
- It affects stability: lower COM → more stable; higher COM → less stable.
Centre of Mass for Different Bodies
| Object | Position of Centre of Mass |
|---|---|
| Uniform Rod | At the midpoint of the length |
| Solid Sphere | At the geometric centre |
| Solid Cylinder | At the geometric centre of the axis |
| Ring | At the geometric centre (hollow region) |
| Irregular Body | Found experimentally or by calculation |
Example:
A uniform rod of length \( 2 \, \text{m} \) is supported at its midpoint. If a \( 10 \, \text{kg} \) mass is hung \( 0.5 \, \text{m} \) from one end, find the position of the centre of mass from the left end of the rod.
▶️ Answer/Explanation
Let the rod’s mass be negligible. Taking the left end as reference:
\(\text{COM} = \dfrac{(0 \times 0) + (10 \times 0.5)}{10} = 0.5 \, \text{m}\)
The centre of mass lies \( 0.5 \, \text{m} \) from the left end, towards the load.
Final Answer: \(\boxed{0.5 \, \text{m}}\) from left end
Example
Two point masses are placed on a thin rod (x measured from the left end): \(m_1=3\,\text{kg}\) at \(x_1=0.20\,\text{m}\) and \(m_2=5\,\text{kg}\) at \(x_2=0.80\,\text{m}\). Find the centre of mass.
▶️ Answer/Explanation
Use weighted average:
\( x_{\text{COM}} = \dfrac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \)
\( x_{\text{COM}} = \dfrac{(3)(0.20) + (5)(0.80)}{3+5} = \dfrac{0.60 + 4.00}{8} = \dfrac{4.60}{8} = 0.575\,\text{m} \)
Final Answer: \( \boxed{0.575\,\text{m from the left end}} \)
Example
A uniform rod of length \(2.0\,\text{m}\) and mass \(4.0\,\text{kg}\) has a small mass \(2.0\,\text{kg}\) attached at \(0.50\,\text{m}\) from the left end. Find the combined centre of mass measured from the left end.
▶️ Answer/Explanation
Rod’s COM is at its midpoint: \(x_{\text{rod}}=1.0\,\text{m}\). Use composite formula:
\( x_{\text{COM}} = \dfrac{m_{\text{rod}} x_{\text{rod}} + m_{\text{mass}} x_{\text{mass}}}{m_{\text{rod}} + m_{\text{mass}}} \)
\( x_{\text{COM}} = \dfrac{(4.0)(1.0) + (2.0)(0.50)}{4.0 + 2.0} = \dfrac{4.0 + 1.0}{6.0} = \dfrac{5.0}{6.0} \approx 0.833\,\text{m} \)
Final Answer: \( \boxed{0.833\,\text{m from the left end}} \)
Example
A uniform ring (thin circular hoop) has its centre of mass at the geometric centre which lies in empty space inside the ring. What does it mean physically that the COM lies outside the material? Why is this important?
▶️ Answer/Explanation
The centre of mass is a point representing the average position of mass distribution it need not lie inside solid material. For a ring, all mass is distributed around a circle, so the average position is at the centre (empty space).
Physically this means:
- If you apply a force at the ring’s centre (through the COM), the ring will translate without causing rotation.
- When analyzing motion (e.g., free fall, projectile motion), you can treat the whole ring as if its mass were concentrated at that centre point its COM follows the same trajectory as a point particle of the same mass would.
- Stability and balance depend on the COM location relative to support points — even if COM is outside the material, the stability rules still apply.
Final Note: COM is a mathematical/physical point (can be in empty space) used to simplify translational dynamics.