Kinetic Energy and Moment of Inertia of Rotating Bodies. How to Use Parallel Axis Theorem

Introduction to Kinetic Energy of Rotating Bodies

Calculating the kinetic energy of an object moving in a straight line is straightforward using the formula KE = 1/2 mv², where every particle of the body moves at the same velocity (v). However, when dealing with rotating bodies, the linear velocity of each point varies depending on its distance from the axis of rotation. This difference in velocities complicates the calculation using the same formula. In this lesson, we will explore how to calculate the kinetic energy of rotating bodies.

Understanding Rotational Kinetic Energy

Step-by-Step Calculation

To understand the kinetic energy of rotating bodies, let's consider a rotating blade of a helicopter. We divide the blade into very small masses, each with its own speed based on its distance from the axis of rotation. By calculating the kinetic energy of each small mass and summing them, we can find the total kinetic energy of the blade.

1. Identify Small Masses and Velocities: Assume the blade is divided into particles m₁, m₂, m₃, ..., mₙ, each with respective velocities v₁, v₂, v₃, ..., vₙ. The kinetic energy of each particle is given by 1/2 mᵢ vᵢ².
2. Summing Kinetic Energies: The total kinetic energy (K) of the blade is the sum of the kinetic energies of all the particles:

• • K = 1/2 m₁ v₁² + 1/2 m₂ v₂² + 1/2 m₃ v₃² + ... + 1/2 mₙ vₙ²
• In shorthand notation, this can be written as:
  • K = Σ (1/2 mᵢ vᵢ²)

Finding Velocities of Particles

To find the velocities of the particles, we use the formula v = ωr, where ω is the angular velocity and r is the distance from the axis of rotation.

1. Substitute Velocities: By substituting vᵢ = ωrᵢ into the kinetic energy equation, we get:

• • K = Σ (1/2 mᵢ (ωrᵢ)²)

1. Simplify the Expression: This simplifies to:

• • K = 1/2 (Σ mᵢ rᵢ²) ω²

Moment of Inertia

The term Σ mᵢ rᵢ² represents the distribution of mass around the axis of rotation and is known as the moment of inertia (I). It is a measure of how the mass of the body is distributed relative to the axis of rotation.

1. Moment of Inertia Definition: The moment of inertia (I) is given by:

• • I = Σ mᵢ rᵢ²

1. Kinetic Energy in Terms of I: The kinetic energy of the rotating body can now be written as:

• • K = 1/2 I ω²

Example: Rotating Rod

To illustrate the concept, let's calculate the moment of inertia for a rod of mass M and length L about its central axis.

1. Integrate Over the Rod's Length: Since the rod has many particles at different distances from the axis, we use integration to sum their contributions:

• • I = ∫ (r² dm)

1. Relate Mass to Length: If the rod has uniform mass distribution, the mass per unit length is M/L. For a small segment of length dx, the mass is (M/L) dx.
2. Integral Expression: The moment of inertia for the rod is:

• • I = (M/L) ∫ (x² dx) from -L/2 to L/2

1. Solve the Integral: Solving this integral gives:

• • I = (1/12) M L²

Parallel Axis Theorem

If we need to find the moment of inertia about an axis parallel to the central axis but passing through one end of the rod, we use the parallel axis theorem:

1. Parallel Axis Theorem: The theorem states:

• • I = I_central + Mh²

1. where I_central is the moment of inertia about the central axis and h is the distance between the two axes.
2. Apply the Theorem: For the rod, h = L/2, so:

• • I = (1/12) M L² + M (L/2)² = (1/12) M L² + 1/4 M L² = 1/3 M L²

Conclusion

The kinetic energy of a rotating body is a function of its moment of inertia and angular velocity. The moment of inertia depends not only on the mass of the body but also on how that mass is distributed relative to the axis of rotation. Understanding these principles is crucial for solving problems involving rotational motion in physics