Center of Mass of a Semi-Circular Rod: A Worked Example

📚 Quick Study Guide

• 📏 The center of mass (COM) is the unique point where the weighted relative position of the distributed mass sums to zero.
• 📐 For a uniform semi-circular rod, the COM lies on the axis of symmetry.
• ➗ The formula for the y-coordinate of the COM ($y_{cm}$) of a semi-circular rod of radius $R$ is: $y_{cm} = \frac{2R}{\pi}$.
• ➕ The x-coordinate of the COM ($x_{cm}$) is 0 due to symmetry along the y-axis.
• 💡 Remember to use radians when integrating over angles.

🧪 Practice Quiz

1. What is the primary characteristic that simplifies the center of mass calculation for a uniform semi-circular rod?

   1. Symmetry about the x-axis
   2. Uniform mass distribution
   3. Constant radius
   4. Symmetry about the y-axis

2. The center of mass of an object is the point:

   1. Where the entire mass of the object is concentrated.
   2. Located at the geometric center of the object.
   3. Where the gravitational force acts.
   4. That remains stationary when the object rotates.

3. If a semi-circular rod has a radius $R$, what is the formula for its y-coordinate of the center of mass ($y_{cm}$)?

   1. $y_{cm} = \frac{R}{\pi}$
   2. $y_{cm} = \frac{4R}{3\pi}$
   3. $y_{cm} = \frac{2R}{\pi}$
   4. $y_{cm} = R$

4. Why is the x-coordinate of the center of mass ($x_{cm}$) equal to zero for a uniform semi-circular rod?

   1. Because the rod is symmetric about the x-axis.
   2. Because the rod is symmetric about the y-axis.
   3. Because the rod is placed on the origin.
   4. Because the total mass is zero.

5. What is the angular range used for integrating over the semi-circular rod in radians?

   1. $0$ to $\pi/2$
   2. $0$ to $2\pi$
   3. $-\pi/2$ to $\pi/2$
   4. $0$ to $\pi$

6. Which of the following statements is true regarding the center of mass of any object?

   1. It must lie within the physical boundaries of the object.
   2. It depends on the gravitational field.
   3. It represents the average position of the mass of the object.
   4. It is always at the geometric center.

7. What happens to the y-coordinate of the center of mass if the radius of the semi-circular rod is doubled?

   1. It remains the same.
   2. It is halved.
   3. It is doubled.
   4. It is quadrupled.