How to Apply Parallel and Perpendicular Axis Theorems in Numerical Problems
The Parallel and Perpendicular Axis Theorems are important concepts in rotational mechanics. These theorems help students calculate the moment of inertia of various bodies about different axes with ease. They are especially useful when direct computation is complex or not practical. Understanding these theorems allows you to solve many numerical and conceptual problems in rotational dynamics confidently.
Parallel Axis Theorem
The Parallel Axis Theorem states: The moment of inertia (M.I.) of a body about any axis is equal to the sum of its moment of inertia about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two axes.
Mathematically, it is expressed as:
Here, I is the moment of inertia about the required axis,
Icm is the moment of inertia about the axis through the centre of mass,
M is the mass of the body,
d is the perpendicular distance between the axes.
Perpendicular Axis Theorem
The Perpendicular Axis Theorem applies only to planar (2D) bodies or laminas. It states: The moment of inertia of a lamina about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes lying in the plane and intersecting at the point where the perpendicular axis passes through the body.
Iz is the moment of inertia about the axis perpendicular to the plane,
Ix and Iy are moments of inertia about the two axes in the plane.
| Theorem | Formula | Applicability |
|---|---|---|
| Parallel Axis Theorem | I = Icm + Md2 | Any body, any shape |
| Perpendicular Axis Theorem | Iz = Ix + Iy | Planar (2D) bodies only |
Step-by-Step Approach to Problem Solving
- Identify the type of object (planar/3D) and the location of the axis of rotation.
- Check if the standard moment of inertia (about CM) is known. If not, refer to standard values.
- If the axis is parallel to the CM axis:
Use the parallel axis theorem.
Calculate Icm.
Find the perpendicular distance (d), and evaluate I = Icm + Md2. - If the axis is perpendicular to the lamina (planar object):
Use the perpendicular axis theorem.
Compute Ix and Iy.
Add to get Iz = Ix + Iy. - Substitute all known values with correct units (kg·m2).
- Simplify the calculation and double-check placement of axes and body type.
Example Problems
| Example | Application | Solution Steps |
|---|---|---|
| Find the moment of inertia of a disc of mass M and radius R about an axis normal to the disc and passing through a point on its edge. | Apply both Perpendicular Axis and Parallel Axis Theorems. |
1. Moment of inertia about diameter: I = MR2/4 2. By symmetry, Ix = Iy 3. Iz = 2 × (MR2/4) = MR2/2 4. About edge: I' = Iz + MR2 = 3MR2/2 |
| State the conditions for applying each theorem. | Theorems and their mathematical statements. |
Parallel Axis: Any shape, for axes parallel to that passing through CM. Perpendicular Axis: Only planar bodies, summing axes at intersection. |
Key Formulas for Quick Reference
| Physical Quantity | Expression |
|---|---|
| Moment of inertia about x-axis (point mass) | Ix = m x2 |
| Moment of inertia about y-axis (point mass) | Iy = m y2 |
| Moment of inertia about z-axis | Iz = m (x2 + y2) |
| Perpendicular Axis Theorem for lamina | Iz = Ix + Iy |
| Parallel Axis Theorem (general) | I = Icm + Md2 |
Applications and Tips
- Use the Parallel Axis Theorem to shift axes in any body, regardless of shape.
- Perpendicular Axis Theorem is only for planar shapes (like a disc, ring, or plate).
- Standard M.I. values (e.g., disc, rod) often refer to axes through the centre of mass—apply the theorems when your axis is different.
- Check units at each step; always use kg·m2 for moment of inertia.
Practice Questions
- Find the moment of inertia of a square plate of mass M and side a about an axis passing through its corner and perpendicular to its plane, using both theorems.
- Given a circular ring of radius r, calculate its radius of gyration about a line perpendicular to its plane and passing through one of its particles.
- A uniform cylinder of mass M and radius R is rolled over a step by force F. What is the minimum value of F required?
Further Learning and Vedantu Resources
- Review the detailed explanation and more solved examples on the Parallel and Perpendicular Axis Theorems page.
- Practice comprehensive Physics questions for mastering moment of inertia and rotational dynamics.
FAQs on Parallel and Perpendicular Axis Theorems Explained for Physics Exams
1. What is parallel axis theorem statement?
Parallel Axis Theorem states that the moment of inertia (I) of a body about any axis parallel to an axis passing through its center of mass (CM) is equal to the sum of the moment of inertia about the CM and the product of the mass (M) and the square of the distance (d) between the two axes.
Mathematically: I = Ic + Md2
• Ic = Moment of inertia about center of mass axis
• M = Mass of the body
• d = Distance between the two parallel axes
2. What is the correct formula for the theorem of perpendicular axis?
The Perpendicular Axis Theorem formula is: Iz = Ix + Iy
• Iz = Moment of inertia about axis perpendicular to the plane of the body
• Ix and Iy = Moments of inertia about two perpendicular axes in the plane, intersecting at the same point
• Applicable only to planar lamina (flat bodies).
3. Where is parallel axis theorem used?
Parallel Axis Theorem is used to calculate the moment of inertia of a rigid body about any axis that does not pass through its center of mass.
Common applications include:
• Calculating M.I. of rods, discs, spheres about axes at their ends or tangents
• Engineering and rotational dynamics
• Solving JEE/NEET rotational motion problems
4. How do you use the parallel and perpendicular axis theorem?
To use the parallel axis theorem:
1. Find the moment of inertia about the center of mass (Ic)
2. Measure the perpendicular distance (d) between this axis and the required parallel axis
3. Apply: I = Ic + Md2
To use the perpendicular axis theorem:
1. Find the M.I. about two perpendicular axes (Ix, Iy) lying in the plane
2. Add them: Iz = Ix + Iy (for planar bodies)
5. State and prove the theorem of parallel axes about moment of inertia.
Statement: The moment of inertia of a body about any axis is equal to the sum of its moment of inertia about a parallel axis through its center of mass and the product of its mass and square of the distance between the two axes.
Proof Outline:
• Let O be the CM, and axis at distance d from O
• M.I. about CM = Ic = Σmixi2
• M.I. about axis parallel and distance d away: I = Σmi(xi + d)2 = Σmixi2 + 2dΣmixi + Md2
• Since Σmixi (about CM) = 0, I = Ic + Md2
6. Prove the theorem of perpendicular axes about the moment of inertia.
Perpendicular Axis Theorem Proof:
• For a planar lamina: Iz is the M.I. about axis perpendicular to plane at O, Ix and Iy about axes in plane at O.
• For each mass element, r2 = x2 + y2
• Iz = Σmi(xi2 + yi2) = Σmixi2 + Σmiyi2 = Ix + Iy
• Hence, Iz = Ix + Iy
7. State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.
Parallel Axis Theorem:
• Applicable to any rigid body
• Used when axis is parallel to axis through center of mass
• Expression: I = Ic + Md2
Perpendicular Axis Theorem:
• Applicable only for planar bodies (laminae)
• Axis must be perpendicular to the lamina's plane
• Expression: Iz = Ix + Iy
8. Give an example using the parallel axis theorem to find moment of inertia.
Example: Find the moment of inertia of a disc of mass M and radius R about a tangent to its edge, perpendicular to its plane.
• M.I. through center and perpendicular axis: Ic = (1/2)MR2
• Distance from center to tangent = R
• Apply Parallel Axis Theorem: I = Ic + MR2 = (1/2)MR2 + MR2 = (3/2)MR2
9. Why is the perpendicular axis theorem not applicable to a solid sphere?
Perpendicular Axis Theorem is only valid for planar laminae (flat 2D objects), not for solid or 3D bodies such as spheres.
• Reason: The theorem requires all mass to be distributed in a single plane.
• A solid sphere has mass throughout its volume, so the sum of M.I. about x and y axes in a plane does not equal the M.I. about the perpendicular z axis.
• Always use standard M.I. formulas for 3D bodies.
10. What are common mistakes while applying parallel and perpendicular axis theorems?
Common mistakes to avoid:
• Applying perpendicular axis theorem to non-planar or 3D bodies
• Using the wrong distance d in parallel axis theorem (must be perpendicular distance between axes)
• Mixing up formulae for Ic and the required axis
• Ignoring unit conversions (ensure all values are in SI units: kg·m2)
• Assuming PeAT is valid for cylinders or spheres (it is not)
11. What is the physical meaning of moment of inertia?
Moment of inertia (M.I.) measures how much a body resists rotational acceleration about a specific axis.
• It depends on both the mass and its distribution relative to the axis.
• Higher M.I. means more torque is needed for the same angular acceleration.
• M.I. is crucial in rotational motion, just as mass is in linear motion.
12. How do you choose between parallel and perpendicular axis theorem in a problem?
Choose the theorem based on problem requirements:
• Use parallel axis theorem when you know M.I. about the center of mass and need M.I. about any parallel axis.
• Use perpendicular axis theorem only with planar bodies when finding M.I. about an axis perpendicular to the plane by knowing M.I. about two perpendicular in-plane axes.
• Read the question carefully to identify the type of body and axes involved.
