Understanding the Centre of Mass of a Uniform Rod
To demonstrate that the centre of mass of a uniform rod lies at its center, we need to analyze the distribution of mass along its length.
Definition of Centre of Mass
- The centre of mass of an object is the point where its mass is evenly distributed. For symmetrical objects, this point often lies at the geometric center.
Properties of a Uniform Rod
- A uniform rod is characterized by:
- Equal mass distribution throughout its length.
- Length denoted as "l" and mass as "m".
Mathematical Analysis
1. Choice of Coordinates:
- Place the rod along the x-axis, ranging from 0 to l.
2. Mass Element:
- Consider an infinitesimal mass element, dm, at a position x along the rod. Since the rod is uniform, dm can be expressed as:
- dm = (m/l) dx
- Here, (m/l) is the mass per unit length.
3. Calculating the Centre of Mass (x_cm):
- The formula for the centre of mass is given by:
- x_cm = (1/M) ∫ x dm
- Substituting for dm:
- x_cm = (1/m) ∫ (x (m/l)) dx from 0 to l
4. Solving the Integral:
- This evaluates to:
- x_cm = (1/m) * (m/l) * [x^2/2] from 0 to l
- x_cm = (1/l) * (l^2/2) = l/2
Conclusion
- Thus, the centre of mass (x_cm) of a uniform rod is located at l/2, which is precisely the midpoint of the rod. This confirms that for a uniform rod, the centre of mass lies at its center.