Question

Let a wire be given as a curve \(C\) in space. Let its density (mass per unit length) be \(\delta=\delta(x, y, z)\), where \((x, y, z)\) is a variable point in \(C\). Justify the following formulas: a) length of wire \(=\int_{C} d s=L\); b) mass of wire \(=\int_{C} \delta d s=M\); c) center of mass of the wire is \((\bar{x}, \bar{y}, \bar{z})\), where $$ M \bar{x}=\int_{C} x \delta d s, \quad M \bar{y}=\int_{C} y \delta d s, \quad M \bar{z}=\int_{C} z \delta d s ; $$ d) moment of inertia of the wire about the \(z\) axis is $$ I_{z}=\int_{C}\left(x^{2}+y^{2}\right) \delta d s . $$

Short answer

In this exercise, we first find the length of the wire by integrating the small segment ds over the whole curve, which gives us L = ∫ds. Then, we find the mass of the wire by integrating the mass density function δ times the small segment ds over the whole curve, which gives us M = ∫δds. To find the center of mass, we find the coordinates (x̄, ȳ, ż) by integrating xδds, yδds, and zδds over the curve, respectively. Finally, we compute the moment of inertia Iₓ about the z-axis by integrating (x²+y²)δds over the curve.

Step-by-step solution

Length of Wire

If we consider a small segment (d\(s\)) of wire \(C\), the total length of the wire \(L\) can be found by integrating this small segment over the whole curve. This is given by $$ L = \int_{C} ds. $$

Mass of Wire

Now, let's consider the mass of the wire. To find the mass of the small segment d\(s\), we multiply it by the mass density function, \(\delta(x, y, z)\). $$ dm = \delta ds $$ To find the total mass \(M\) of the wire, we integrate over the entire curve \(C\). $$ M = \int_{C} \delta ds $$

Center of Mass

To find the center of mass of the wire, we first need to compute the coordinates of the center of mass, denoted by \((\bar{x}, \bar{y}, \bar{z})\). $$ M \bar{x} = \int_{C} x \delta ds,\\ M \bar{y} = \int_{C} y \delta ds,\\ M \bar{z} = \int_{C} z \delta ds $$ Here, \(x, y, z\) represent the coordinates of a point in the curve \(C\), and \(\delta(x, y, z)\) represents the mass per unit length at that particular point.

Moment of Inertia

Finally, we need to compute the moment of inertia \(I_{z}\) of the wire about the \(z\)-axis. The infinitesimal moment of inertia of a small segment d\(s\) is given by $$ dI_{z}=\left(x^{2}+y^{2}\right) \delta ds $$ Integrating this over the entire curve \(C\), we get the moment of inertia \(I_{z}\), $$ I_{z} = \int_{C}\left(x^{2}+y^{2}\right) \delta ds $$

Key concepts

Center of Mass Calculation

Understanding the center of mass (COM) is essential when analyzing the balance and motion of objects. For a wire shaped like a curve in space, calculating the COM can seem tricky, but it boils down to finding a point that represents the average position of all the mass in the wire.

The center of mass \( (\bar{x}, \bar{y}, \bar{z}) \) is a point where we could balance the wire if it were possible. To calculate the coordinates of the COM, we apply the curve integral to each spatial dimension—integrating the position weighted by the mass distribution along the curve.

For instance, to find \( \bar{x} \) we integrate the x-coordinate multiplied by the mass per unit length \( \delta(x, y, z) \) along the curve \( C \) and then divide by the total mass of the wire \( M \) to get the average. This gives us \( M\bar{x} = \int_{C} x \delta ds \) and similarly for \( \bar{y} \) and \( \bar{z} \).

Physics and engineering applications use COM calculations for understanding how objects will react under various forces, such as gravity or tension, which can be essential in building stable structures or understanding body dynamics in biomechanics.

Moment of Inertia Computation

The moment of inertia (MOI) is a crucial concept that measures an object's resistance to changes in its rotation. For a wire following a spatial curve \( C \) with a variable mass per unit length \( \delta(x, y, z) \) and concerning an axis, in this case, the \( z \) axis, we calculate MOI using an integral.

The differential segment of MOI is given by \( dI_{z}=(x^{2}+y^{2}) \delta ds \), where \( x^{2}+y^{2} \) represents the perpendicular distance from the axis of rotation to the mass element squared, and it's essential as it reflects how the distribution of mass affects rotational inertia.

By integrating this expression over the whole curve \( C \) with \( I_{z} = \int_{C}(x^{2}+y^{2}) \delta ds \), we find the total MOI of the wire. This computation is widely used in mechanical design and analysis, as it affects how forces result in rotational motion, stability, and strain within materials.

Mass per Unit Length

The concept of mass per unit length \( \delta \) is integral to various calculations concerning linear objects like wires or rods. It describes how much mass is found along a particular segment of the curve and can often vary along its length.

In our exercise, \( \delta(x, y, z) \) can change at every point on the wire, described by the curve \( C \) in space. This variable density function requires that we integrate over the wire's length to find mass-related properties, as we do not assume a constant density.

For continuous bodies with complex shapes and non-uniform mass density, the mass per unit length must be described by a function of position, such as \( \delta(x, y, z) \) rather than a constant. This allows for precise calculations and realistic modeling of physical objects in disciplines like materials science and structural engineering.

Curve Integral

A curve integral is a powerful mathematical tool used to calculate various properties like length, mass, and center of mass for objects defined along a curve in space. Essentially, it sums up the contributions of infinitesimally small segments along the curve.

In our context, whether we're integrating \( ds \) to find the length of the wire, \( \delta ds \) for the mass, or \( x\delta ds \) for the center of mass computation, we're applying the curve integral concept to account for every minute portion of the wire comprehensively.

It's analogous to adding up infinitely small contributions along the curve to compute a finite, meaningful property for the entire wire. Learning to set up and solve curve integrals is foundational in fields that include physics, engineering, and advanced calculus.