Center of Mass (Physics)

The center of mass is the average position of the parts of an object of the system, taking the mass into account^[1]. If a force is exerted on an object the center of mass will not rotate, even if other parts do, and is the part where the object would balance^[1]. The center of mass $x_{cm}$ is, where $m_i$ is the mass of the parts and $x_i$ is the position of those parts:

$$x_{cm}=\frac{\sum m_i{x}_i}{\sum m_i}$$

If an object has uniform density the center of mass will be located at the center of the object^[1]. It is also important to consider the horizontal and vertical positions of the object^[1].

Calculating Center of Mass Examples#

Example 1#

Calculate the center of mass of the following system: a meterstick with a 20 g cylinder placed on at the 30 cm mark.

We’ll set the reference point $x=0$ to be at the 0 cm mark on the meterstick. Assuming that the meterstick has a uniform mass distribution the center of mass of the meterstick will be 0.5 m. Assuming that the cylinder has a uniform mass distribution the center of mass will be at the 0.3 m mark on the meterstick. The mass of the meterstick is measured to be 0.0857 kg. The mass of the cylinder is measured to be 0.20 kg.

Following the equation we can calculate $x_{cm}$ of the system:

$$\begin{array}{lrlr}& x_{cm}& =\frac{\left(0.0857\right)\left(0.5\right)+\left(0.20\right)\left(0.3\right)}{0.0857+0.20}& \text{<span class="tml-eqn"></span>}\\ & & \approx 0.36& \text{<span class="tml-eqn"></span>}\end{array}$$

1. A.P. Physics 1 – 2.1: Daily Video 1 ↩ ↩2 ↩3 ↩4