A uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 4.00 kg, while the balls each have mass 0.300 kg and can be treated as point masses. Find the moment of inertia of this combination about an axis parallel to the bar through both balls;

검증된 단계별 안내

Identify the formula for the moment of inertia of the system about the given axis. Since the axis is parallel to the bar and passes through both balls, the moment of inertia will only depend on the two point masses (the balls). The bar itself does not contribute to the moment of inertia because it is parallel to the axis of rotation.

The moment of inertia for a point mass is given by the formula:

I = m r^{2}

, where

m

is the mass of the point and

r

is the perpendicular distance from the axis of rotation.

In this case, each ball has a mass of

0.300

kg, and the distance of each ball from the axis of rotation is

0

m because the axis passes through both balls.

Substitute the values into the formula for each ball. For the first ball:

I

₁ = 0.300 × 02. Similarly, for the second ball:

I

₂ = 0.300 × 02.

Add the contributions of both balls to find the total moment of inertia:

I

_total = I₁ + I₂. Since both terms are zero, the total moment of inertia is zero.

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주요 개념

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Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion about a specific axis. It depends on the mass distribution relative to the axis of rotation. For point masses, it is calculated as the product of the mass and the square of the distance from the axis of rotation. In this case, the moment of inertia will include contributions from both the bar and the balls at its ends.

Parallel Axis Theorem

The parallel axis theorem allows us to calculate the moment of inertia of a body about any axis parallel to an axis through its center of mass. It states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass axis plus the product of the mass and the square of the distance between the two axes. This theorem is essential for determining the moment of inertia of the bar and the balls about the specified axis.

Center of Mass

The center of mass is the point at which the mass of a system is concentrated and can be considered to act for translational motion. For composite systems, like the bar with balls, the center of mass can be found by considering the masses and their positions. Understanding the center of mass is crucial for applying the parallel axis theorem and calculating the moment of inertia accurately.