Question

The following notation is used in the problems: \(M=\) mass, \(\bar{x}, \bar{y}, \bar{z}=\) coordinates of center of mass (or centroid if the density is constant), \(I=\) moment of inertia (about axis stated), \(I_{x}, I_{y}, I_{z}=\) moments of inertia about \(x, y, z\) axes, \(I_{m}=\) moment of inertia (about axis stated) through the center of mass. Note: It is customary to give answers for \(I, I_{m}, I_{x},\) etc., as multiples of \(M\) (for example, \(I=\frac{1}{3} M l^{2}\) ). A chain in the shape \(y=x^{2}\) between \(x=-1\) and \(x=1\) has density \(|x| .\) Find (a) \(M\), (b) \(\bar{x}, \bar{y}\).

Short answer

M = \int_{-1}^{1} |x| \, \sqrt{1+(2x)^{2}} \ dx, \; \bar{x}=0, \; \bar{y} = \frac{2}{M} \int_{0}^{1} x^{3} \, \sqrt{1+4x^{2}} \ dx.

Step-by-step solution

- Determine the mass (M)

To find the mass of the chain, integrate the density over the length of the chain. The density function is given by \(|x|\). The mass \(M\) is calculated as: \[ M = \int_{-1}^{1} |x| \, \sqrt{1+(2x)^{2}} \, dx \]\ To simplify the integral, note that \|x| = x\ for \x \geq 0\ and \|x| = -x\ for \x < 0\. Thus, the integral can be split: \[ M = 2 \int_{0}^{1} x \, \sqrt{1+4x^{2}} \, dx \] Calculate this integral to find the value of M.

- Calculate the x-coordinate of the center of mass (\bar{x})

The x-coordinate of the center of mass is calculated using: \[\bar{x} = \frac{1}{M} \int_{-1}^{1} x |x| \, \sqrt{1+(2x)^{2}} \, dx \] Since the density function \|x|\ is an even function and the domain is symmetric around the y-axis, the product \x |x|\ is odd. Hence, \[\bar{x} = 0 \]

- Calculate the y-coordinate of the center of mass (\bar{y})

The y-coordinate of the center of mass is calculated by finding: \[\bar{y} = \frac{1}{M} \int_{-1}^{1} y |x| \, \sqrt{1+(2x)^{2}} \, dx \] Since the given equation of the chain is \y=x^{2}\, substitute y as \x^{2}\. The integral becomes: \[ \bar{y} = \frac{1}{M} \int_{-1}^{1} x^{2} |x| \, \sqrt{1+4x^{2}} \, dx \] This integral simplifies to: \[ \bar{y} = \frac{2}{M} \int_{0}^{1} x^{3} \, \sqrt{1+4x^{2}} \ dx \] Use substitution \u = 1+4x^{2}\ to solve the integral and find \bar{y}\.

Key concepts

Mass Calculation

To find the mass of a chain shaped like the curve \(y = x^2\) with endpoints between \(x = -1\) and \(x = 1\), we have to integrate the density function along the length of the chain. Integrating density helps us accumulate the small bits of mass over the length.

The density function given is \(|x|\). Thus, the integral for mass (\(M\)) is:

\[ M = \int_{-1}^{1} |x| \, \sqrt{1+ (2x)^2} \, dx \]

To simplify, note that \(|x| = x\) for \(x \ge 0\) and \(|x| = -x\) for \(x < 0\). Hence, we split the integral:

\[ M = 2 \int_{0}^{1} x \, \sqrt{1 + 4x^2} \, dx \]

Compute this integral to find \(M\). The result tells us the total mass accumulated along the chain’s length.

Moment of Inertia

The moment of inertia (\(I\)) quantifies the distribution of the chain's mass relative to a rotational axis, indicating resistance to angular acceleration. Moments of inertia for different axes are denoted differently (e.g., \(I_x, I_y, I_z\)). Moments are crucial for understanding rotational dynamics.

The formula for moment of inertia about an axis involves integrating the mass elements multiplied by the square of their distance from the axis.

For our exercise involving the chain, if we calculated the moments about different axes, it would involve expressions like:

\[I_x = \int_{-1}^{1} y^2|x| \sqrt{1 + (2x)^2} \, dx\]
Using such moments helps in deeper analyses of the chain's rotational properties.

Density Function

Density function represents how mass is distributed over a region or along a curve. Here, the density function is given by \(|x|\), which signifies how concentrated the mass is at any point \(x\).

Understanding how to interpret and use density functions is vital for computations like mass calculation. When density varies, as in this case, the integral approach effectively accumulates small mass bits into the total mass.

In other words, density function \(|x|\) here tells us the mass density depends on distance from the origin, increasing linearly with \(x\), whether positive or negative.

Symmetry in Integrals

Symmetry is a powerful concept simplifying many integrals. Recognizing symmetry can often halve the work.

For example, our density function \(|x|\) exhibits symmetry about the y-axis.
Given symmetry, an integral from \(-a\) to \(a\) with an odd integrand (a function where \(f(-x) = -f(x)\)) will evaluate to zero.

In our case for \(\bar{x}\):

\[ \bar{x} = \frac{1}{M} \int_{-1}^{1} x|x| \sqrt{1+(2x)^2} \, dx = 0 \]

Here the integrand is odd, hence integration over symmetric bounds yields zero.

Substitution in Integrals

Substitution is a technique to simplify integrals by transforming variables. It helps convert a complex integral into a more manageable form.

In the given problem, we use substitution to find \(\bar{y}\).

Starting with:
\[\bar{y} = \frac{1}{M} \int_{-1}^{1} x^2 |x| \sqrt{1+4x^2} \, dx\]

We substitute \(u = 1 + 4x^2\), then find \(du = 8x \, dx\), simplifying the integral. This helps us in evaluating it more easily.

Such techniques are especially useful handling complex integrands involving composites and products.