Question

Using the parallel axis theorem, what is the moment of inertia of the rod of mass m about the axis shown below? (Use the following as necessary: m and L.)

Answer 1

To find the moment of inertia of the rod of mass m about the given axis using the parallel-axis theorem, we can use the formula I = (mL^2)/12 + m(2L/3)^2 = (7mL^2)/36.

Step-by-step explanation:

To find the moment of inertia of the rod of mass m about the axis shown, we can use the parallel-axis theorem. The parallel-axis theorem states that the moment of inertia of a compound object is equal to the sum of the moment of inertia of its components plus the product of their masses and the square of the distance between their respective axes of rotation.

In this case, the rod is a thin rod with mass m and length L. The moment of inertia of a thin rod about an axis through its centre perpendicular to its length is given by the formula I = (mL^2)/12. To find the moment of inertia about the axis shown (L/3), we need to calculate the distance between the two axes of rotation. This distance is 2L/3. Using the parallel-axis theorem, the moment of inertia about the axis shown is I = (mL^2)/12 + m(2L/3)^2 = (7mL^2)/36.

Answer 2

The moment of inertia of the rod about the given axis can be calculated using the parallel axis theorem. It is (7/36)mL².

Step-by-step explanation:

The moment of inertia of the rod of mass m about the axis shown can be calculated using the parallel axis theorem. The moment of inertia of a rod about an axis through its center perpendicular to its length is given by the formula I = (1/12)mL², where m is the mass of the rod and L is the length of the rod. However, in this case, the axis of rotation is shifted to a distance of L/3 from the center, so we need to use the parallel axis theorem. The parallel axis theorem states that the moment of inertia about an axis parallel to and a distance d from an axis through the center of mass is given by Iparallel-axis = Icenter of mass + md².

In this case, the moment of inertia about the given axis is the sum of the moment of inertia about the center of mass (which is (1/12)mL²) and the product of the mass m and the square of the distance d (which is mL/3)² = (1/9)mL². Therefore, the moment of inertia of the rod about the given axis is (1/12)mL² + (1/9)mL² = (7/36)mL².