1- Suppose a bicycle wheel is rotated about an axis through its rim and parallel to its axle. Is its moment of inertia about this axis greater than, less than, or equal to its moment of inertia about its axle?
a. less than
b. greater than
c. equal to
2- Suppose a bicycle wheel is rotated about an axis through its rim and parallel to its axle. Choose the best explanation from among the following:
a. The mass and shape of the wheel remain the same.
b. Mass is farther from the axis when the wheel is rotated about the rim.
c. The moment of inertia is greatest when an object is rotated about its center.
choose the correct one
a. option B
b. option B
1- The moment of inertia about the axis through the rim and parallel to the axle is greater than the moment of inertia about the axle. Therefore, the correct answer is b. greater than.
2- When a bicycle wheel is rotated about an axis through its rim and parallel to its axle, the mass is farther from the axis compared to when it is rotated about the axle. Therefore, the correct answer is b. Mass is farther from the axis when the wheel is rotated about the rim.
1- To determine whether the moment of inertia of a bicycle wheel about an axis through its rim and parallel to its axle is greater than, less than, or equal to its moment of inertia about its axle, we need to consider the distribution of mass in the wheel.
The moment of inertia of an object depends on both its mass and how that mass is distributed around the axis of rotation. In the case of a bicycle wheel, the majority of its mass is concentrated near the rim, while the axle has very little mass.
The moment of inertia is given by the formula: I = Σmr^2, where I is the moment of inertia, m is the mass of each small element, and r is the distance of each small element from the axis of rotation.
Since the mass is mainly concentrated near the rim, the distance from the axis of rotation will be greater for the rim than for the axle. Therefore, the moment of inertia about the axis through the rim will be larger because of the larger distance between the axis and the mass elements.
Hence, the moment of inertia about the rim axis is greater than the moment of inertia about the axle axis. Therefore, the answer is option (b) greater than.
2- To determine the best explanation for why the moment of inertia is greatest when an object is rotated about its center, we need to understand the concept of moment of inertia and how it relates to the distribution of mass.
The moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on both the mass of the object and how that mass is distributed around the axis of rotation. The moment of inertia is given by the formula: I = Σmr^2, where I is the moment of inertia, m is the mass of each small element, and r is the distance of each small element from the axis of rotation.
When an object is rotated about its center, the distribution of mass is symmetrical. This means that for every mass element on one side of the center, there is an equivalent mass element on the opposite side at the same distance from the axis of rotation. This symmetrical distribution of mass leads to a cancellation of the moment of inertia contributions, resulting in the lowest total moment of inertia.
On the other hand, if an object is rotated about a different axis, such as the rim of the bicycle wheel, the distribution of mass becomes asymmetrical. This means that the distances of the mass elements from the axis are different, and there is no cancellation of the contributions to the moment of inertia. As a result, the moment of inertia is increased compared to rotation about the center.
Therefore, the best explanation for why the moment of inertia is greatest when an object is rotated about its center is option (c) - The moment of inertia is greatest when an object is rotated about its center due to the cancellation of moment of inertia contributions resulting from the symmetrical distribution of mass.