Solved Examples: Finding the Center of Mass of Objects

📚 Quick Study Guide

• ⚖️ The center of mass (COM) is the unique point where the weighted relative position of the distributed mass sums to zero. It's the average position of all the parts of the system, weighted by their masses.
• 📐 For a system of particles, the COM is given by: $ \vec{R} = \frac{\sum_{i} m_i \vec{r_i}}{\sum_{i} m_i} $, where $m_i$ are the masses and $ \vec{r_i} $ are their positions.
• 🧱 For continuous objects, the summation becomes an integral: $ \vec{R} = \frac{\int \vec{r} dm}{\int dm} = \frac{1}{M} \int \vec{r} dm $, where M is the total mass of the object.
• 💡 Symmetry can simplify calculations. If an object has an axis of symmetry, the COM lies on that axis.
• 🧪 The COM of a composite object can be found by treating each part as a point mass located at its own COM.

Practice Quiz

1. Two objects of masses 3 kg and 5 kg are located at positions (2, -1) m and (-3, 2) m, respectively. What is the x-coordinate of the center of mass of this system?

   1. -0.875 m
   2. 0.875 m
   3. -0.5 m
   4. 0.5 m

2. A uniform rod of length L and mass M is placed along the x-axis from x = 0 to x = L. Where is its center of mass located?

   1. L
   2. L/4
   3. L/2
   4. 3L/4

3. A system consists of three particles: 1 kg at (0, 0), 2 kg at (1, 1), and 3 kg at (2, -2). What is the y-coordinate of the center of mass?

   1. -1/2
   2. 1/2
   3. -1
   4. 1

4. A square of side 'a' has a mass 'm' uniformly distributed over it. What is the x-coordinate of its center of mass if one corner is at the origin (0,0) and the sides are along x and y axis?

   1. a
   2. a/4
   3. a/2
   4. 3a/4

5. A circular disc of radius R has a hole of radius R/2 cut out of it, with the center of the hole at R/2 from the center of the disc. Where is the center of mass of the remaining part?

   1. R/6
   2. R/2
   3. R/3
   4. R/4

6. Two masses, $m_1$ and $m_2$, are connected by a massless rod of length L. If $m_1 = 2m_2$, how far is the center of mass from $m_1$?

   1. L/3
   2. 2L/3
   3. L/4
   4. 3L/4

7. A composite figure consists of a square (side a, mass m) and an equilateral triangle (side a, mass m) attached to one side of the square. What is the y-coordinate of the center of mass, if the base of the square lies on the x-axis and its left corner is at the origin?

   1. (a + \frac{\sqrt{3}}{2}a)/2
   2. (a + \frac{\sqrt{3}}{4}a)/2
   3. (a + \frac{\sqrt{3}}{2}a)/4
   4. (a + \frac{\sqrt{3}}{4}a)/4