Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Parallel Axis Theorem Consider a lamina \(S\) of mass \(m\) together with parallel lines \(L\) and \(L^{\prime}\) in the plane of \(S\), the line \(L\) passing through the center of mass of \(S\). Show that if \(I\) and \(I^{\prime}\) are the moments of inertia of \(S\) about \(L\) and \(L^{\prime}\), respectively, then \(I^{\prime}=I+d^{2} m\), where \(d\) is the distance between \(L\) and \(L^{\prime}\). Hint: Assume that \(S\) lies in the \(x y\) -plane, \(L\) is the \(y\) -axis, and \(L^{\prime}\) is the line \(x=-d\).

Step-by-Step Solution

Answer

The parallel axis theorem states: \(I' = I + d^2 m\).

Step 1: Establish Moment of Inertia about L

The moment of inertia about a line (axis) passing through the center of mass can be expressed as the integral over the lamina: \[I = \int_S r^2 \, dm,\]where \(r\) is the perpendicular distance from an element of mass \(dm\) to the axis \(L\) (here, the \(y\)-axis, so \(r = x\)). Thus, we have:\[I = \int_S x^2 \, dm.\]

Step 2: Establish Moment of Inertia about L'

When considering the line \(L'\) (given by \(x = -d\)), the distance from a mass element to \(L'\) becomes \(|x + d|\). The moment of inertia \(I'\) about \(L'\) is:\[I' = \int_S (x + d)^2 \, dm.\]

Step 3: Expand the Binomial

Expand the expression \((x + d)^2\) as follows:\[(x + d)^2 = x^2 + 2xd + d^2.\]Substitute this expansion into the integral expression for \(I'\):\[I' = \int_S (x^2 + 2xd + d^2) \, dm = \int_S x^2 \, dm + 2d \int_S x \, dm + d^2 \int_S \, dm.\]

Step 4: Use the Center of Mass Property

Since the line \(L\) passes through the center of mass of \(S\), the term \(\int x \, dm = 0\) because the average \(x\)-position is at the origin, relative to the center of mass. Therefore, the second term in the expression vanishes:\[I' = \int_S x^2 \, dm + 0 + d^2 \int_S \, dm.\]

Step 5: Simplify the Expression

The term \(\int_S \, dm\) is simply the total mass, \(m\). Thus, the expression for \(I'\) simplifies to:\[I' = I + d^2 m,\]where \(I = \int x^2 \, dm\) is the moment of inertia about \(L\). This confirms the parallel axis theorem, \(I' = I + d^2 m\).

Key Concepts

Moment of Inertia

The moment of inertia is a fundamental concept in rotational dynamics. It measures how much torque is needed for a desired angular acceleration about an axis. Essentially, it is the rotational analog of mass in linear motion. In simpler terms, it quantifies how the mass is distributed in relation to a specific axis of rotation.

For a given mass distribution, the moment of inertia depends on the geometric arrangement of mass regarding the axis. Mathematically, it is represented by:

• \( I = \int_S r^2 \, dm \)

where \( r \) is the perpendicular distance from the axis to an infinitesimal mass element \( dm \). This distance \( r \) greatly influences the moment of inertia value—the further away the mass from the axis, the greater the moment of inertia.

In the context of the Parallel Axis Theorem, the axis passing through the center of mass provides a baseline moment of inertia \( I \), crucial for calculating inertia about parallel axes.

Center of Mass

The center of mass is a point that serves as the average location of the total mass of a body or system. It is crucial for simplifying complex mass distributions into a single-point mass model. In calculations, this concept aids in determining the body's response to external forces and torques.

For symmetrical objects, the center of mass often coincides with geometric centers. But for irregular shapes, finding it requires integrating over the body's mass distribution. Importantly, when dealing with rotational dynamics, if an axis passes through this center, it simplifies the moment of inertia analysis:

• It eliminates linear motion effects because \( \int x \, dm = 0 \) for any direction perpendicular to this axis.

In our current context, the line \( L \) through the center of mass simplifies the inertia calculations and confirms the vanishing of any terms involving \( x \), aiding in reducing complex multivariable problems.

Axis of Rotation

The axis of rotation is a fundamental line around which an object rotates. Every rotational motion is defined with respect to an axis, whether linear or angular. Selecting this axis correctly simplifies dynamic calculations and offers a clearer understanding of the system's behavior.

In practical examples:

• A door rotates around its hinges, which serve as the axis.
• Earth rotates around its axis, creating day and night.

For complex systems, sometimes it’s necessary to consider parallel axes, as our exercise does.
The Parallel Axis Theorem aids in shifting moment of inertia calculations to a parallel but displaced axis, without recalculating from scratch, using \( I' = I + d^2 m \). This formula acts as a bridge, adapting inertia calculations to meet practical demands such as re-orienting or moving objects without extensive recalculations.

Mass Distribution

Mass distribution is key in understanding the hardness or ease of rotating objects. Non-uniform distribution implies that inertia will not be the same along every potential axis, making this a critical variable in design and engineering.

Think of a seesaw: adding mass to one end impacts its balance and rotation more than placing the same mass near the pivot. This is because how mass is spread affects the concentration of forces during movement or balance.

In mathematical terms, to determine the overall effect, we assess how every mass piece contributes:

• For even distribution, simplify calculations because individual component contributions symmetrical.
• Complex shapes often require calculus, integrating each smaller piece's contribution.

In our Parallel Axis context, understanding mass distribution lets us see why moments of inertia shift upon moving the axis elsewhere. It predicts how the object resists rotational alteration given the same mass lies differently against that new axis.