Which of the following has a greater moment of inertia?
(A) long, fat leg (C) short, fat leg
(B) long, thin leg (D) short, thin leg
The correct answer and explanation is :
The correct answer is (A) long, fat leg.
Explanation:
The moment of inertia is a measure of an object’s resistance to changes in its rotational motion about an axis. The formula for the moment of inertia ( I ) for a rotating object is:
[
I = \sum m_i r_i^2
]
Where:
- ( m_i ) is the mass of each infinitesimal element of the object.
- ( r_i ) is the distance of each mass element from the axis of rotation.
When considering the shape and mass distribution of a leg, the main factors that determine the moment of inertia are:
- Length of the leg: The moment of inertia increases with the square of the distance from the axis of rotation. A longer leg has mass elements that are farther from the axis, increasing the overall moment of inertia.
- Thickness of the leg: A fat leg has more mass distributed farther from the axis compared to a thin leg. More mass farther from the axis increases the moment of inertia significantly.
Now, let’s break down each option:
- Long, fat leg: This leg has both a large length and a significant mass distributed over a wider radius. Since both the length and the thickness increase the moment of inertia, this leg will have the largest moment of inertia.
- Short, fat leg: While this leg is thick, it is short. The mass is distributed farther from the axis, but the shorter length reduces the moment of inertia compared to the long leg.
- Long, thin leg: This leg is long, so its mass elements are spread farther from the axis, but being thin means the mass is concentrated more toward the center, which reduces its moment of inertia compared to a fat leg.
- Short, thin leg: This leg has the least length and the least thickness. Both of these factors reduce the moment of inertia.
Therefore, the long, fat leg (A) has the greatest moment of inertia because both its length and thickness work together to maximize the distribution of mass further from the axis of rotation.
Visual comparison of the legs to highlight the differences in mass distribution and rotational inertia.
Here is the visual representation comparing the different legs with varying characteristics. As shown, the long, fat leg has the most mass distributed farther from the axis of rotation, leading to the greatest moment of inertia. This makes it harder to rotate compared to the other legs.