Parallel Axis Theorem Example

Summary

In this video, the concept of the parallel-axis theorem is explored as it applies to rotational inertia. The narrator walks through the derivation of the rotational inertia of a uniform, long, thin rod about its end, using the parallel-axis theorem to simplify the process. By adding the rotational inertia about the center of mass and the additional mass times the square of the distance from the center to the end, the final result is easily obtained: one-third of the total mass times the length squared. The video highlights how the parallel-axis theorem provides a much simpler solution compared to earlier, more complex methods.

Takeaways

• 😀 The parallel-axis theorem relates the moment of inertia of an object about an axis parallel to the center of mass to its moment of inertia about the center of mass.
• 😀 The rotational inertia of an object about an axis parallel to its center of mass is the sum of the inertia about the center of mass and the product of the object's mass and the square of the distance between the axes.
• 😀 The moment of inertia for a uniform, long, thin rod about an axis through its center of mass is 1/12 * mass * length^2.
• 😀 The moment of inertia of the same rod about one end is 1/3 * mass * length^2.
• 😀 Bo is asked to derive the rotational inertia of a uniform rod about its end using the parallel-axis theorem.
• 😀 The rotational inertia about the end of the rod is the sum of the rotational inertia about the center of mass and the mass of the rod multiplied by the square of the distance from the center of mass to the end of the rod.
• 😀 The distance between the center of mass and the end of a uniform rod is half its length.
• 😀 Using the parallel-axis theorem, the rotational inertia of the rod about its end equals (1/12 * mass * length^2) + (mass * (length/2)^2).
• 😀 Simplifying the equation results in factoring out mass * length^2 and getting a common denominator of 12, ultimately leading to the rotational inertia of 1/3 * mass * length^2.
• 😀 The process demonstrates that the parallel-axis theorem yields the same result as a more complicated integral method previously used.
• 😀 The final conclusion reaffirms that the physics works, and the derived result is correct.

Q & A

What is the parallel-axis theorem?

The parallel-axis theorem allows us to calculate the rotational inertia (moment of inertia) of an object about an axis parallel to an axis through the center of mass by adding the moment of inertia about the center of mass and the product of the object's total mass and the square of the distance between the two axes.

How is the rotational inertia of a uniform, long, thin rod derived about an axis through its center of mass?

The rotational inertia about the center of mass of a uniform, long, thin rod is given by (1/12) * M * L^2, where M is the total mass and L is the length of the rod.

What is the formula for the rotational inertia of a uniform rod about an axis through one end?

The rotational inertia about one end of a uniform, long, thin rod is (1/3) * M * L^2, where M is the total mass and L is the length of the rod.

How does the parallel-axis theorem simplify the calculation of the rotational inertia of a rod about its end?

The parallel-axis theorem simplifies the calculation by stating that the rotational inertia about the end of the rod is the sum of the rotational inertia about the center of mass and the mass times the square of the distance between the center of mass and the end, which is half the length of the rod.

What is the distance between the center of mass and the end of a uniform, long, thin rod?

The distance between the center of mass and the end of the rod is half of the rod's length, or L/2.

What does the equation (1/12) * M * L^2 + M * (L/2)^2 simplify to?

The equation simplifies to (1/3) * M * L^2, which is the rotational inertia of a uniform, long, thin rod about its end.

Why is factoring out M * L^2 helpful in the calculation?

Factoring out M * L^2 helps simplify the equation and allows you to combine terms more easily, leading to the final result.

What is the significance of the common denominator of 12 in the calculation?

The common denominator of 12 allows the terms to be added more easily. It helps in simplifying the equation by making the numerators compatible for addition.

How does the parallel-axis theorem relate to integral equations in this context?

The parallel-axis theorem provides a simpler way to calculate the rotational inertia, giving the same result as the more complicated integral approach that was used earlier in the lesson.

What conclusion does the script reach about the parallel-axis theorem?

The script concludes that the parallel-axis theorem is a simpler method for calculating rotational inertia and that it produces the same result as the more complex integral method, confirming the validity of the theorem.