Parallel Axis Theorem explained with examples

📚 Quick Study Guide

• ⚛️ The Parallel Axis Theorem is used to calculate the moment of inertia of a rigid body about any axis, given the moment of inertia about a parallel axis through the center of mass.
• 📏 The formula is: $I = I_{cm} + Md^2$, where:
  • $I$ is the moment of inertia about the new axis.
  • $I_{cm}$ is the moment of inertia about the center of mass.
  • $M$ is the total mass of the object.
  • $d$ is the distance between the two parallel axes.
• 💡 The theorem simplifies calculations when the moment of inertia about the center of mass is known.
• ✍️ Remember to use consistent units (e.g., kg m² for moment of inertia, kg for mass, and m for distance).

🧪 Practice Quiz

1. Question 1: A rod of mass $M$ and length $L$ has a moment of inertia $I_{cm}$ about its center. What is its moment of inertia about an axis at one end?
   1. $I = I_{cm}$
   2. $I = I_{cm} + \frac{1}{4}ML^2$
   3. $I = I_{cm} + \frac{1}{2}ML^2$
   4. $I = I_{cm} + ML^2$
2. Question 2: A disc has a mass of 2 kg and a radius of 0.5 m. Its moment of inertia about its center is 0.25 kg m². What is the moment of inertia about an axis parallel to the central axis and 0.25 m away?
   1. 0.25 kg m²
   2. 0.375 kg m²
   3. 0.50 kg m²
   4. 0.625 kg m²
3. Question 3: A square plate has a mass $M$ and side length $a$. If $I_{cm}$ is the moment of inertia about its center, what is the moment of inertia about an axis along one edge?
   1. $I = I_{cm}$
   2. $I = I_{cm} + \frac{1}{4}Ma^2$
   3. $I = I_{cm} + \frac{1}{2}Ma^2$
   4. $I = I_{cm} + Ma^2$
4. Question 4: Which of the following statements is TRUE about the Parallel Axis Theorem?
   1. It can only be applied to symmetrical objects.
   2. It requires knowing the moment of inertia about an axis through the center of mass.
   3. It only works if the new axis is perpendicular to the original axis.
   4. It is used to find the moment of inertia about any arbitrary axis.
5. Question 5: A sphere of mass 5 kg has a moment of inertia about its center of $\frac{2}{5}MR^2$, where R = 0.2 m. What is its moment of inertia about a tangent axis?
   1. 0.04 kg m²
   2. 0.08 kg m²
   3. 0.20 kg m²
   4. 0.24 kg m²
6. Question 6: A thin hoop has a mass $M$ and radius $R$. Its moment of inertia about its center is $MR^2$. What is the moment of inertia about an axis on its edge?
   1. $\frac{1}{2}MR^2$
   2. $MR^2$
   3. $\frac{3}{2}MR^2$
   4. $2MR^2$
7. Question 7: In the formula $I = I_{cm} + Md^2$, what does 'd' represent?
   1. The diameter of the object.
   2. The distance between the two parallel axes.
   3. The density of the object.
   4. The displacement of the object.