Class 11 >> Physics >> motion-of-system-of-particles-and-rigid-body >> moment-of-inertia-and-theorems >> state-perpendicular-axis-theorem
Skip to Practice

state-perpendicular-axis-theorem

🚀 The Perpendicular Axis Theorem is a fundamental principle in the study of rigid body dynamics, particularly in calculating the moment of inertia of a planar object. This theorem states that for a flat, rigid body lying in the XY-plane, the moment of inertia about an axis perpendicular to the plane (the Z-axis) is equal to the sum of the moments of inertia about two perpendicular axes (X and Y) that lie in the plane. Mathematically, this can be expressed as: \[ I_z = I_x + I_y \] where \( I_z \) is the moment of inertia about the Z-axis, and \( I_x \) and \( I_y \) are the moments of inertia about the X and Y axes, respectively. This theorem is particularly useful for simplifying the calculation of moments of inertia for complex shapes by breaking them down into simpler components.

Theory Explanation

Understanding Moment of Inertia

Moment of inertia is a measure of an object's resistance to rotational motion about an axis. It depends on the mass distribution of the object relative to the axis of rotation. The further the mass is from the axis, the greater the moment of inertia.

\[ I = \sum m_i r_i^2 \]
Applying the Perpendicular Axis Theorem

To apply the Perpendicular Axis Theorem, identify the two axes in the plane of the object and calculate their respective moments of inertia. Then, sum these values to find the moment of inertia about the perpendicular axis.

\[ I_z = I_x + I_y \]
Example Calculation

For a rectangular plate, calculate the moments of inertia about the X and Y axes using the standard formulas, then apply the Perpendicular Axis Theorem to find the moment of inertia about the Z-axis.

\[ I_x = \frac{1}{12} m (b^2), I_y = \frac{1}{12} m (h^2) \]

Key Points

  • 🎯 The Perpendicular Axis Theorem applies only to planar objects.
  • 🎯 Moment of inertia depends on the mass distribution relative to the axis of rotation.
  • 🎯 The theorem simplifies calculations by allowing the use of two axes in the plane.

🛠 Simulation is being generated. Please check back in a few moments.

Examples:💡

Calculate the moment of inertia of a rectangular plate of mass 10 kg, width 2 m, and height 1 m about an axis perpendicular to the plate.

Solution:

Step 1: Calculate the moment of inertia about the X-axis: \( I_x = \frac{1}{12} m (b^2) = \frac{1}{12} \times 10 \times (2^2) = \frac{40}{12} = 3.33 \, kg \, m^2 \).

Step 2: Calculate the moment of inertia about the Y-axis: \( I_y = \frac{1}{12} m (h^2) = \frac{1}{12} \times 10 \times (1^2) = \frac{10}{12} = 0.83 \, kg \, m^2 \).

Step 3: Apply the Perpendicular Axis Theorem: \( I_z = I_x + I_y = 3.33 + 0.83 = 4.16 \, kg \, m^2 \).

Find the moment of inertia of a circular disk of mass 5 kg and radius 0.5 m about an axis perpendicular to its plane.

Solution:

Step 1: Calculate the moment of inertia about the X-axis: \( I_x = \frac{1}{4} m r^2 = \frac{1}{4} \times 5 \times (0.5^2) = 0.3125 \, kg \, m^2 \).

Step 2: Since the disk is symmetric, \( I_y = I_x = 0.3125 \, kg \, m^2 \).

Step 3: Apply the Perpendicular Axis Theorem: \( I_z = I_x + I_y = 0.3125 + 0.3125 = 0.625 \, kg \, m^2 \).

Common Mistakes

  • Mistake: Confusing the axes when applying the theorem, leading to incorrect calculations of \( I_x \) and \( I_y \).

    Correction: Always clearly label the axes and ensure you are using the correct dimensions for each axis.

  • Mistake: Forgetting that the Perpendicular Axis Theorem only applies to planar objects.

    Correction: Remember to check if the object is planar before applying the theorem.

  • Mistake: Not considering the mass distribution when calculating moments of inertia.

    Correction: Ensure to use the correct formulas for the moment of inertia based on the shape and mass distribution.

← Back Start Practice