Problem 21 - Thomas Calculus

Question

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.

Step-by-Step Solution

Answer

a. The first moment about any plane through the COM is zero. b. The parallel axis theorem is proven as \(I_{L} = I_{L_{cm}} + Mh^{2}\).

Step 1: Understanding the Center of Mass

The center of mass (COM) of a body is a point where the mass of the body is considered to be concentrated. When positioned at the origin, the coordinates of the COM are \( (0, 0, 0) \). At this point, the first moment (the integral of the mass distribution) along any plane through the COM is zero because there is no net torque about the COM.

Step 2: Express the First Moment about the yz-plane

The first moment of a body about the \(yz\)-plane is expressed as \( M_{yz} = \int_{D} x \, dm \), where \(x\) is the distance of mass element \(dm\) from the \(yz\)-plane. Since the COM is at the origin and using the relationship \( \overline{x}= M_{yz} / M \), where \( M \) is the total mass, \( \overline{x} = 0 \) implies \( M_{yz} = 0 \).

Step 3: Understanding the Parallel Axis Theorem

The Parallel Axis Theorem states that the moment of inertia about any axis parallel to an axis through the COM is the sum of the inertia through the COM axis and the product of the total mass and the square of the distance between the axes.

Step 4: Set up the Integral for Moment of Inertia

Given the line \(L\) is perpendicular to the \(xy\)-plane at \((h, 0, 0)\) and \(L_{cm}\) along the \(z\)-axis, the moment of inertia \(I_{L}\) is defined as \(I_{L} = \int_{D} |\mathbf{v} - h \mathbf{i}|^{2} \, dm\), where \(\mathbf{v}\) is the position vector of a mass element.

Step 5: Expand the Integrand

The expression \(|\mathbf{v} - h \mathbf{i}|^{2} = (x - h)^{2} + y^{2} + z^{2}\). Thus, the integrand becomes \[ \int_{D} ((x-h)^{2} + y^{2} + z^{2}) \, dm \].

Step 6: Solve the Integral

The expanded integral becomes \[ \int_{D} ((x-h)^{2} + y^{2} + z^{2}) \, dm = \int_{D} (x^{2} + y^{2} + z^{2}) \, dm - 2h \int_{D} x \, dm + h^{2} \int_{D} \, dm \].

Step 7: Relate to Moment about the COM

The term \( \int_{D} x \, dm \) is zero by the placement of the center of mass at the origin. Thus, \(I_{L} = I_{L_{cm}} + Mh^{2}\), where \( I_{L_{cm}} = \int_{D} (y^{2} + z^{2}) \, dm \) is the moment of inertia through the axis passing the COM.

Key Concepts

Statistical Mechanics

Statistical mechanics is a branch of physics that connects the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties we observe. This field allows us to understand phenomena like temperature, pressure, and volume from a particle-level perspective.
By analyzing the probabilistic behavior of particles, statistical mechanics provides averaged quantities that make up classical thermodynamics.

• Through concepts like entropy, it bridges the gap between microstates (specific configurations of particles) and macrostates (observable states).
• Applications include understanding gases, liquids, and solids as well as explaining phenomena in astrophysics and material science.

It relies on the law of large numbers, where large ensembles of particles result in predictable behaviors despite individual randomness.

Moment of Inertia

The moment of inertia is a fundamental concept in mechanics, describing how mass is distributed relative to an axis. It determines the torque needed for a desired angular acceleration around that axis.
Mathematically, it’s represented by the integral over the mass distribution as: \[ I = \int r^2 \, dm \]
Here, \( r \) is the perpendicular distance of a mass element \( dm \) from the axis of rotation.

• This property changes with the shape of an object and the location of the axis relative to the object.
• Higher moments of inertia mean it's harder to change the object's rotational state.

In engineering and physical sciences, calculating the moment of inertia is crucial for designing rotating machinery and predicting rotational movements.

Center of Mass

The center of mass (COM) is the point at which the entire mass of an object or system can be considered to be concentrated. It's key in understanding motion dynamics.
For an object with a continuous mass distribution, the COM coordinates can be found using:

• \( \overline{x} = \frac{1}{M} \int x \, dm \)
• \( \overline{y} = \frac{1}{M} \int y \, dm \)
• \( \overline{z} = \frac{1}{M} \int z \, dm \)

Here, \( M \) is the total mass. When COM is at the origin in a coordinate system, any object can be balanced on a point, as it acts as the weighted average of its mass distribution. Finding the COM simplifies solving for gravitational and inertial forces acting on the body.

Parallel Axis Theorem

The Parallel Axis Theorem is a principle in mechanics used to determine the moment of inertia of a body about any axis parallel to an axis through its center of mass.
It states that:
\[ I = I_{cm} + Md^2 \]
where \( I \) is the moment of inertia about the new axis, \( I_{cm} \) is the moment of inertia about an axis through the center of mass, \( M \) is the total mass of the body, and \( d \) is the perpendicular distance between the two axes.

• This theorem simplifies calculations in engineering, allowing designers to easily calculate moments of inertia for bodies in motion under parallel translational or rotational conditions.
• It also explains why a body becomes harder to rotate as its mass is distributed further from the axis of rotation.

This is particularly useful in biomechanical models, robotics, and spacecraft dynamics, where objects often rotate about multiple axes.