The Parallel Axis Theorem Let Lcm be a line through the center of mass of a body of mass m and l

step 1 So part A wants us to prove that the first moment of a body in space about any plane through the

The Parallel Axis Theorem Let Lcm be a line through the...

The Parallel Axis Theorem Let $L_{\mathrm{cm}}$ be a line through the center of mass of a body of mass $m$ and let $L$ be a parallel line $h$ units away from $L_{\mathrm{cm}} .$ The Parallel Axis Theorem says the moments of inertia $I_{\mathrm{cm}}$ and $I_{L}$ of the body about $L_{\mathrm{cm}}$ and $L$ satisfy the equation $$I_{L}=I_{\mathrm{c.m.}}+m h^{2}$$ As in the two-dimensional case, the theorem gives a quick way to calculate one moment when the other moment and the mass are known.\ Proof of the Parallel Axis Theorem a. Show that the first moment of a body in space about any plane through the body's center of mass is zero. (Hint: Place the body's center of mass at the origin and let the plane be the $y z$ -plane. What does the formula $\overline{x}=M_{y z} / M$ then tell you? b. To prove the Parallel Axis Theorem, place the body with its center of mass at the origin, with the line $L_{c . m .}$ along the $z$ -axis and the line $L$ perpendicular to the $x y$ -plane at the point $(h, 0,0)$ . Let $D$ be the region of space occupied by the body. Then, in the notation of the figure, $$I_{L}=\iiint_{D}|\mathbf{v}-h \mathbf{i}|^{2} d m$$ Expand the integrand in this integral and complete the proof.

The Parallel Axis Theorem Let $L_{\mathrm{cm}}$ be a line through the center of mass of a body of mass $m$ and let $L$ be a parallel line $h$ units away from $L_{\mathrm{cm}} .$ The Parallel Axis Theorem says the moments of inertia $I_{\mathrm{cm}}$ and $I_{L}$ of the body about $L_{\mathrm{cm}}$ and $L$ satisfy the equation
$$I_{L}=I_{\mathrm{c.m.}}+m h^{2}$$
As in the two-dimensional case, the theorem gives a quick way to calculate one moment when the other moment and the mass are known.\
Proof of the Parallel Axis Theorem
a. Show that the first moment of a body in space about any
plane through the body's center of mass is zero. (Hint: Place
the body's center of mass at the origin and let the plane be
the $y z$ -plane. What does the formula $\overline{x}=M_{y z} / M$ then tell
you?
b. To prove the Parallel Axis Theorem, place the body with its
center of mass at the origin, with the line $L_{c . m .}$ along the $z$ -axis
and the line $L$ perpendicular to the $x y$ -plane at the point $(h, 0,0)$ .
Let $D$ be the region of space occupied by the body. Then, in the
notation of the figure,
$$I_{L}=\iiint_{D}|\mathbf{v}-h \mathbf{i}|^{2} d m$$
Expand the integrand in this integral and complete the proof.

Key Concepts

Parallel Axis Theorem

The Parallel Axis Theorem provides a relationship between the moment of inertia of a body about any axis and its moment of inertia about a parallel axis that passes through its center of mass. Specifically, it states that the moment of inertia about the shifted axis is equal to the moment of inertia about the center of mass axis plus the product of the total mass and the square of the distance between the two axes. This theorem simplifies many rotational dynamics problems by relating them back to the center of mass properties.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to rotational motion about a specific axis, analogous to mass in linear motion. It is defined as the sum or integral of the products of the mass elements and the squares of their distances from the axis. It plays a crucial role in determining how an object will respond to applied torques and is a central concept in rotational dynamics.

Center of Mass

The center of mass of a body is the weighted average position of its mass distribution and serves as the balance point where the total moment of the mass is zero. In problems involving rotational dynamics and moment calculations, locating the center of mass allows for simplifying assumptions, such as setting the first moment about that point to zero, which is essential for applying the Parallel Axis Theorem.

First Moment of Mass Distribution

The first moment of the mass distribution measures the distribution of mass relative to a specific reference point or plane. When calculated about the center of mass, this moment equals zero, indicating that the positive and negative contributions balance out. This property is exploited in proofs involving the Parallel Axis Theorem to simplify integrals and relate moments about different axes.

Coordinate Shifts and Integral Expansion

Employing coordinate shifts allows one to position the center of mass at the origin, which simplifies the mathematical treatment of the problem. This technique, along with expanding integrals (by breaking down expressions like the square of distances into separate terms), is fundamental in the derivation of the Parallel Axis Theorem. It also illustrates how transformations in coordinate systems can streamline the integration process over a body’s volume.

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