Question

Rotational Inertia The rotational inertia of an object varies directly with the square of the perpendicular distance from the object to the axis of rotation. If the rotational inertia is \(0.4 \mathrm{~kg} \cdot \mathrm{m}^{2}\) when the perpendicular distance is \(0.6 \mathrm{~m},\) what is the rotational inertia of the object if the perpendicular distance is \(1.5 \mathrm{~m} ?\)

Short answer

The new rotational inertia is \( 2.5 \text{ kg} \bullet \text{m}^2 \).

Step-by-step solution

Understand the Problem

The problem involves finding the new rotational inertia when the perpendicular distance from the axis of rotation changes. The rotational inertia varies directly with the square of the distance.

Identify Given Values

The given rotational inertia is \( I_1 = 0.4 \text{ kg} \bullet \text{m}^2 \) and the given distance is \( d_1 = 0.6 \text{ m} \). The new distance is \( d_2 = 1.5 \text{ m} \).

Write the Proportional Relationship

Since rotational inertia varies directly with the square of the distance: \( I \propto d^2 \). This can be expressed as \( I_1 / d_1^2 = I_2 / d_2^2 \).

Solve for the New Rotational Inertia

Rearrange the proportional relationship to solve for the new rotational inertia: \( I_2 = I_1 \times (d_2^2 / d_1^2) \). Substitute the known values: \( I_2 = 0.4 \times (1.5^2 / 0.6^2) \).

Calculate the New Rotational Inertia

First, calculate \( 1.5^2 \ = 2.25 \) and \( 0.6^2 \ = 0.36 \). Then compute the ratio: \( 2.25 / 0.36 \ = 6.25 \). Finally, compute: \( 0.4 \times 6.25 = 2.5 \text{ kg} \bullet \text{m}^2 \).

Key concepts

Direct Variation

Direct variation is a relationship between two variables where if one variable changes, the other variable changes in a directly proportional manner. This means if one variable doubles, the other doubles as well.
In the case of rotational inertia, this relationship can be represented as: \( I \propto d^2 \).

This tells us that the rotational inertia increases when the square of the perpendicular distance increases.
So, if you increase the perpendicular distance, the rotational inertia will increase according to the square of that distance.

Perpendicular Distance

In the context of rotational inertia, the perpendicular distance is the straight-line distance from the object to the axis of rotation. This distance determines how much the object resists rotating. The farther the object is from the axis, the greater the rotational inertia.

For example, in the problem given, we compare two different distances, 0.6 meters and 1.5 meters, to see how the rotational inertia changes: \( I_1 = 0.4 \text{ kg} \bullet \text{m}^2 \), and \( d_1 = 0.6 \text{ m} \). The new distance is \( d_2 = 1.5 \text{ m} \).
The new rotational inertia will be larger because the second distance is longer.

Axis of Rotation

The axis of rotation is an imaginary line about which an object rotates. For any rotating object, each point moves in a circle around the axis. This concept is crucial for understanding rotational inertia and how objects behave when they spin.

In our problem, an object's rotational inertia depends on its distance from this axis. The farther away an object’s mass is from the axis, the more it contributes to the rotational inertia.

• For example, a figure skater pulls their arms in to reduce their rotational inertia to spin faster.
• In our exercise, changing the distance from the axis affects how much the object resists rotating.

Proportional Relationships

Proportional relationships are relationships between two quantities where they increase or decrease at the same rate. To solve our problem, we used a proportional relationship between rotational inertia and the square of the perpendicular distance.

The formula we used was:
\( \frac{I_1}{d_1^2} = \frac{I_2}{d_2^2} \)
The given values allowed us to solve for the new rotational inertia. Breaking it down:

• First, we calculated \( d_2^2 \) and \( d_1^2 \), which are 2.25 and 0.36, respectively.
• Then, we formed a ratio which is 6.25, indicating how much the rotational inertia increased.
• Finally, multiplying this ratio by the initial rotational inertia \( 0.4 \text{ kg} \bullet \text{m}^2 \), we found the new rotational inertia to be \( 2.5 \text{ kg} \bullet \text{m}^2 \).

Understanding how proportional relationships work helps solve similar problems effortlessly.