Problem 51 - University Physics with Modern Physics

Question

A thin, rectangular sheet of metal has mass \(M\) and sides of length \(a\) and \(b\). Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the sheet.

Step-by-Step Solution

Answer

The moment of inertia is \( \frac{1}{3}M(a^2 + b^2) \).

Step 1: Identify Known Variables

We have a rectangular sheet with mass \( M \), length \( a \), and width \( b \). We are tasked with finding the moment of inertia through the corner of the sheet and perpendicular to its plane.

Step 2: Determine Moment of Inertia through Center

The moment of inertia of a rectangular sheet with respect to an axis passing through its center and perpendicular to its plane is given by the formula: \( I_c = \frac{1}{12}M(a^2 + b^2) \).

Step 3: Use the Parallel-Axis Theorem

The parallel-axis theorem states: \( I = I_c + Md^2 \), where \( I \) is the moment of inertia about the new axis, and \( d \) is the distance between the center of mass and the new axis.

Step 4: Calculate Distance from Center to Corner

The center of the rectangle is at \( (\frac{a}{2}, \frac{b}{2}) \), and the corner is at (0,0). So the distance \( d \) is the diagonal from the center to the corner: \( d = \sqrt{(\frac{a}{2})^2 + (\frac{b}{2})^2} = \frac{\sqrt{a^2 + b^2}}{2} \).

Step 5: Compute Moment of Inertia about Corner

Use the parallel-axis theorem to compute: \[ I = \frac{1}{12}M(a^2 + b^2) + M\left(\frac{\sqrt{a^2 + b^2}}{2}\right)^2 = \frac{1}{12}M(a^2 + b^2) + \frac{M(a^2 + b^2)}{4} \]Combine the terms: \[ I = \frac{1}{12}M(a^2 + b^2) + \frac{3}{12}M(a^2 + b^2) = \frac{1}{3}M(a^2 + b^2). \]

Step 6: Conclusion

The moment of inertia of the metal sheet around an axis through its corner, perpendicular to its plane, is given by: \( I = \frac{1}{3}M(a^2 + b^2) \).

Key Concepts

Parallel-Axis Theorem

The parallel-axis theorem is an important tool in calculating the moment of inertia for a body about an axis parallel to one through its center of mass.
This theorem is particularly useful when we need to find the moment of inertia for an axis not passing through the center of mass.

• The formula for the parallel-axis theorem is: \( I = I_c + Md^2 \), where:
• \( I \) is the moment of inertia about the new axis,
• \( I_c \) is the moment of inertia about a parallel axis through the center of mass, and
• \( d \) is the distance between the center of mass and the new axis.

In our original exercise, we used this theorem to shift the axis from the center of the rectangular sheet to a corner. By accurately calculating \( d \), we precisely determined the new moment of inertia.

Rectangular Sheet

A rectangular sheet, like the one in our original exercise, is a flat surface with two distinct dimensions: length and width.
When calculating physical properties like the moment of inertia, considering its shape is crucial. The moment of inertia depends heavily on the distribution of mass around the axis of rotation.

• For our exercise, the sheet has a mass \( M \), a length \( a \), and a width \( b \).
• The moment of inertia calculations hinge on these dimensions as they dictate how mass is spread out in relation to the axis.

Thus, understanding the properties of the sheet will help us in adjusting the calculation using tools like the parallel-axis theorem.

Center of Mass

The center of mass is a point at which the total mass of a body can be considered to be concentrated for the purposes of motion and inertia calculations.
In the case of a uniformly dense object, like our rectangular sheet, this point is located geometrically in the center.

• For a rectangle, the center of mass is at the coordinate \((\frac{a}{2}, \frac{b}{2})\) if the origin is at one corner.
• Knowing the center of mass allows us to find the distance \( d \) in the parallel-axis theorem.

By identifying this point, one can effectively use it to simplify the moment of inertia calculations by applying it to mathematically manipulate the axis to the desired position.

Axis of Rotation

The axis of rotation is the line about which a body rotates. Understanding its placement is key when calculating the moment of inertia.
For our rectangular sheet, the axis of rotation was initially through its center and was adjusted to pass through a corner.

• When through the center, it is perpendicular to the plane of the sheet.
• The altered axis resulted in recalculating the moment of inertia using the parallel-axis theorem.

The exercise required shifting the axis of rotation, demonstrating how vital its positioning is for accurate computation.
By moving it to the corner, the calculations were adjusted to reflect the new distribution of the mass around this point.