Calculate the moment of inertia of each of the following uniform objects about the axes indicated. Consult Table 9.2 as needed. (a) A thin 2.50-kg rod of length 75.0 cm, about an axis perpendicular to it and passing through (i) one end and (ii) its center, and (iii) about an axis parallel to the rod and passing through it.

Verified step by step guidance

1. Identify the formula for the moment of inertia of a thin rod about different axes. For a rod of mass \( m \) and length \( L \), the moment of inertia about an axis perpendicular to the rod and passing through one end is \( I = \frac{1}{3}mL^2 \).
2. For the moment of inertia about an axis perpendicular to the rod and passing through its center, use the formula \( I = \frac{1}{12}mL^2 \).
3. For the moment of inertia about an axis parallel to the rod and passing through it, the moment of inertia is zero because all points on the rod are at the axis of rotation, so \( I = 0 \).
4. Substitute the given values into the formulas. For the rod with mass \( m = 2.50 \) kg and length \( L = 0.75 \) m, calculate \( I \) for each case using the respective formulas.
5. Ensure units are consistent and verify calculations by checking each step for accuracy, ensuring that the correct formula is applied for each axis of rotation.

Key Concepts

Moment of Inertia

Moment of inertia is a measure of an object's resistance to rotational motion about an axis. It depends on the object's mass distribution relative to the axis of rotation. For a rod, the moment of inertia varies based on whether the axis is through the center, one end, or parallel to the rod.

Parallel Axis Theorem

The parallel axis theorem allows calculation of the moment of inertia about any axis parallel to an axis through the center of mass. It states that the moment of inertia about the new axis equals the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the axes.

Uniform Rod Moment of Inertia Formulas

For a uniform rod, specific formulas are used to calculate the moment of inertia depending on the axis location. For an axis through the center, the formula is (1/12)ML², and for an axis through one end, it is (1/3)ML², where M is mass and L is length. These formulas are derived from integrating mass distribution along the rod.