Center of Mass and Moments

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In this section we consider centers of mass and moments.

The center of mass for an object can be thought as the point about which the entire mass of the object is equally distributed.

If the object has a uniform density \(\rho,\) then the center of mass is also the geometric center of the object called the centroid.

Center of Mass and Moment of a Thin Rod

Suppose that we have a thin rod lying on the \(x-\)axis between \(x = a\) and \(x = b.\) At a point \(x,\) the rod has mass density (mass per unit length) \(\rho \left( x \right).\)

The center of mass of the rod is given by

\[\bar x = \frac{{{M_0}}}{m} = \frac{{\int\limits_a^b {x\rho \left( x \right)dx} }}{{\int\limits_a^b {\rho \left( x \right)dx} }}.\]

The integral in the numerator \({{M_0} = \int\limits_a^b {x\rho \left( x \right)dx} }\) is called the moment (or the first moment) of the one-dimensional object around zero.

The integral in the denominator \({m = \int\limits_a^b {\rho \left( x \right)dx} }\) gives the total mass of the rod.

Center of Mass and Moments of a Planar Lamina

In general, the center of mass and moments of a lamina can be determined using double integrals. However, in certain special cases when the density only depends on one coordinate, the calculations can be performed using single integrals.

Case 1. Density Depends on the \(x-\)Coordinate

Consider a thin planar lamina bounded by the curves \(y = f\left( x \right)\) and \(y = g\left( x \right)\) on the interval \(\left[ {a,b} \right].\)

If the density only depends on the \(x-\)coordinate, the moments of the lamina about the \(x-\) and \(y-\)axes are

\[{M_x} = \frac{1}{2}\int\limits_a^b {\rho \left( x \right)\left[ {{f^2}\left( x \right) - {g^2}\left( x \right)} \right]dx} ,\]

\[{M_y} = \int\limits_a^b {x\rho \left( x \right)\left[ {f\left( x \right) - g\left( x \right)} \right]dx} .\]

The center of mass \(\left( {\bar x,\bar y} \right)\) is given by

\[\bar x = \frac{{{M_y}}}{m},\;\; \bar y = \frac{{{M_x}}}{m},\]

where \(m = \int\limits_a^b {\rho \left( x \right)\left[ {f\left( x \right) - g\left( x \right)} \right]dx} \) is the mass of the lamina.

If the density \(\rho\) is constant, these formulas are simplified. In this case the centroid of the lamina is determined by formulas

\[\bar x = \frac{{{M_y}}}{m} = \frac{{\int\limits_a^b {x\left[ {f\left( x \right) - g\left( x \right)} \right]dx} }}{{\int\limits_a^b {\left[ {f\left( x \right) - g\left( x \right)} \right]dx} }},\]

\[\bar y = \frac{{{M_x}}}{m} = \frac{{\frac{1}{2}\int\limits_a^b {\left[ {{f^2}\left( x \right) - {g^2}\left( x \right)} \right]dx} }}{{\int\limits_a^b {\left[ {f\left( x \right) - g\left( x \right)} \right]dx} }}.\]

Case 2. Density Depends on the \(y-\)Coordinate

Similarly, we can consider a region enclosed by two curves \(x = f\left( y \right),\) \(x = g\left( y \right)\) and two horizontal lines \(y = c,\) \(y = d.\)

Assuming the density of the region only depends on the \(y-\)coordinate, the moments of the lamina about the \(x-\) and \(y-\)axes are expressed by the integrals

\[{M_x} = \int\limits_c^d {y\rho \left( y \right)\left[ {f\left( y \right) - g\left( y \right)} \right]dy} ,\]

\[{M_y} = \frac{1}{2}\int\limits_c^d {\rho \left( y \right)\left[ {{f^2}\left( y \right) - {g^2}\left( y \right)} \right]dy} .\]

The center of mass is calculated by the formulas

\[\bar x = \frac{{{M_y}}}{m},\;\; \bar y = \frac{{{M_x}}}{m},\]

where the mass of the lamina is now determined by the integral

\[m = \int\limits_c^d {\rho \left( y \right)\left[ {f\left( y \right) - g\left( y \right)} \right]dy} .\]

In case of constant density, the centroid has the following coordinates

\[\bar x = \frac{{{M_y}}}{m} = \frac{{\frac{1}{2}\int\limits_c^d {\left[ {{f^2}\left( y \right) - {g^2}\left( y \right)} \right]dy} }}{{\int\limits_c^d {\left[ {f\left( y \right) - g\left( y \right)} \right]dy} }},\]

\[\bar y = \frac{{{M_x}}}{m} = \frac{{\int\limits_c^d {y\left[ {f\left( y \right) - g\left( y \right)} \right]dy} }}{{\int\limits_c^d {\left[ {f\left( y \right) - g\left( y \right)} \right]dy} }}.\]

Centroid of a Planar Lamina in Polar Coordinates

Suppose that a polar region is bounded by two radius vectors \(\theta = \alpha,\) \(\theta = \beta\) and a continuous curve \(r = f\left( \theta \right).\)

The centroid \(\left( {\bar x,\bar y} \right)\) of the polar region is given by the formulas

\[\bar x = \frac{2}{3}\frac{{\int\limits_\alpha ^\beta {{r^3}\left( \theta \right)\cos \theta d\theta } }}{{\int\limits_\alpha ^\beta {{r^2}\left( \theta \right)d\theta } }},\;\; \bar y = \frac{2}{3}\frac{{\int\limits_\alpha ^\beta {{r^3}\left( \theta \right)\sin \theta d\theta } }}{{\int\limits_\alpha ^\beta {{r^2}\left( \theta \right)d\theta } }}.\]

Centroid of a Triangle

The centroid of a triangle is the point of intersection of the medians of the triangle.

The centroid \(G\left( {\bar x,\bar y} \right)\) of a triangle with vertices \(\left( {{x_1},{y_1}} \right),\) \(\left( {{x_2},{y_2}} \right),\) \(\left( {{x_1},{y_1}} \right)\) has the coordinates

\[\bar x = \frac{{{x_1} + {x_2} + {x_3}}}{3},\;\; \bar y = \frac{{{y_1} + {y_2} + {y_3}}}{3}.\]

Solved Problems

Click or tap a problem to see the solution.

Example 1

A rod has a length of \(40\,\text{cm}.\) The rod's density changes linearly along its length from \(20\frac{\text{g}}{\text{cm}}\) to \(60\frac{\text{g}}{\text{cm}}.\) Find the center of mass of the rod.

Example 2

Find the centroid of a semicircular region of radius \(1\) centered at the origin.

Example 3

Determine the center of mass of the lamina of uniform density bounded by the sine curve \[y = \sin x\] and the \(x-\)axis on the interval \(\left[ {0,\pi } \right].\)

Example 4

A square with side length \(a\) lies in the first quadrant with one vertex at the origin. Its sides are parallel to the coordinate axes. Find the center of mass of the square if the density distribution is given by the linear function \[\rho \left( x \right) = kx.\]

Example 1.

Solution.

Let's first derive the density distribution function. The equation of a straight line passing through the points \(\rho \left( 0 \right) = 20,\) \(\rho \left( 40 \right) = 60\) is given by

\[\frac{{\rho - \rho \left( 0 \right)}}{{\rho \left( {40} \right) - \rho \left( 0 \right)}} = \frac{{x - 0}}{{40 - 0}},\;\; \Rightarrow \frac{{\rho - 20}}{{60 - 20}} = \frac{x}{{40}},\;\; \Rightarrow \rho = x + 20,\]

where \(\rho\) is measured in \(\frac{\text{g}}{\text{cm}},\) and \(x\) is measured in \(\text{cm}.\)

Calculate the mass \(m\) and the first moment \({M_0}\) of the rod:

\[m = \int\limits_a^b {\rho \left( x \right)dx} = \int\limits_0^{40} {\left( {x + 20} \right)dx} = \left. {\left( {\frac{{{x^2}}}{2} + 20x} \right)} \right|_0^{40} = 800 + 800 = 1600\,\text{g}.\]

\[{M_0} = \int\limits_a^b {x\rho \left( x \right)dx} = \int\limits_0^{40} {x\left( {x + 20} \right)dx} = \int\limits_0^{40} {\left( {{x^2} + 20x} \right)dx} = \left. {\left( {\frac{{{x^3}}}{3} + 10{x^2}} \right)} \right|_0^{40} = \frac{{64000}}{3} + 16000 \approx 37333\,\text{g} \cdot \text{cm}.\]

Hence, the center of mass \(G\left( {\bar x} \right)\) is located at the point

\[\bar x = \frac{{{M_0}}}{m} = \frac{{37333}}{{1600}} \approx 23.3\,\text{cm}\]

Example 2.

Find the centroid of a semicircular region of radius \(1\) centered at the origin.

Solution.

It is obvious that the centroid \(G\left( {\bar x,\bar y} \right)\) lies on the \(y-\)axis, so \(\bar x = 0.\) To find the \(\bar y-\)coordinate, we need to compute the first moment \({M_x}.\) Assuming \(\rho = 1,\) we get

\[{M_x} = \frac{1}{2}\int\limits_a^b {{f^2}\left( x \right)dx} = \frac{1}{2}\int\limits_{ - 1}^1 {\left( {1 - {x^2}} \right)dx} = \frac{1}{2}\left. {\left( {x - \frac{{{x^3}}}{3}} \right)} \right|_{ - 1}^1 = \frac{1}{2} \cdot \frac{4}{3} = \frac{2}{3}\]

Since \(\rho = 1,\) the mass \(m\) of the lamina is numerically equal to the area \(A\) of the semicircle:

\[m = A = \frac{{\pi {R^2}}}{2} = \frac{\pi }{2}.\]

Then

\[\bar y = \frac{{{M_x}}}{m} = \frac{{\frac{2}{3}}}{{\frac{\pi }{2}}} = \frac{4}{{3\pi }}.\]

Hence, the centroid is located at the point

\[G\left( {\bar x,\bar y} \right) = G\left( {0,\frac{4}{{3\pi }}} \right).\]

Example 3.

Determine the center of mass of the lamina of uniform density bounded by the sine curve \[y = \sin x\] and the \(x-\)axis on the interval \(\left[ {0,\pi } \right].\)

Solution.

We first find the mass of the lamina:

\[m = \rho \int\limits_a^b {f\left( x \right)dx} = \rho \int\limits_0^\pi {\sin xdx} = \rho \left. {\left( { - \cos x} \right)} \right|_0^\pi = 2\rho ,\]

where \(\rho\) denotes the density of the plate.

Calculate the moments \({M_x}\) and \({M_y}:\)

\[{M_x} = \frac{\rho }{2}\int\limits_a^b {\left[ {{f^2}\left( x \right) - {g^2}\left( x \right)} \right]dx} = \frac{\rho }{2}\int\limits_0^\pi {{{\sin }^2}xdx} = \frac{\rho }{4}\int\limits_0^\pi {\left( {1 - \cos 2x} \right)dx} = \frac{\rho }{4}\left. {\left( {x - \frac{{\sin 2x}}{2}} \right)} \right|_0^\pi = \frac{{\pi \rho }}{4};\]

\[{M_y} = \rho \int\limits_a^b {x\left[ {f\left( x \right) - g\left( x \right)} \right]dx} = \rho \int\limits_0^\pi {x\sin xdx} = \left[ {\begin{array}{*{20}{l}} {du = \sin xdx}\\ {v = x}\\ {u = - \cos x}\\ {dv = 1} \end{array}} \right] = \rho \left[ {\left. {\left( { - x\cos x} \right)} \right|_0^\pi + \int\limits_0^\pi {\cos xdx} } \right] = \rho \left. {\left[ { - x\cos x + \sin x} \right]} \right|_0^\pi = \pi \rho .\]

Hence, the coordinates of the center of mass are

\[\bar x = \frac{{{M_y}}}{m} = \frac{{\pi \cancel{\rho} }}{{2 \cancel{\rho} }} = \frac{\pi }{2};\]

\[\bar y = \frac{{{M_x}}}{m} = \frac{{\frac{{\pi \bcancel{\rho} }}{4}}}{{2 \bcancel{\rho} }} = \frac{\pi }{8}.\]

Example 4.

Solution.

Calculate the first moments of the square region about the coordinate axes. Here \(f\left( x \right) = a,\) \(g\left( x \right) = 0,\) so we have

\[{M_x} = \frac{1}{2}\int\limits_0^a {\rho \left( x \right){f^2}\left( x \right)dx} = \frac{{k{a^2}}}{2}\int\limits_0^a {xdx} = \left. {\frac{{k{a^2}{x^2}}}{4}} \right|_0^a = \frac{{k{a^4}}}{4};\]

\[{M_y} = \int\limits_0^a {x\rho \left( x \right)f\left( x \right)dx} = ka\int\limits_0^a {{x^2}dx} = \left. {\frac{{ka{x^3}}}{3}} \right|_0^a = \frac{{k{a^4}}}{3}.\]

The mass of the square region is given by

\[m = \int\limits_0^a {\rho \left( x \right)f\left( x \right)dx} = ka\int\limits_0^a {xdx} = \left. {\frac{{ka{x^2}}}{2}} \right|_0^a = \frac{{k{a^3}}}{2}.\]

Hence, the center of mass is at the point \(G\left( {\bar x,\bar y} \right)\) with coordinates

\[\bar x = \frac{{{M_y}}}{m} = \frac{{\frac{{k{a^4}}}{3}}}{{\frac{{k{a^3}}}{2}}} = \frac{{2a}}{3},\;\; \bar y = \frac{{{M_x}}}{m} = \frac{{\frac{{k{a^4}}}{4}}}{{\frac{{k{a^3}}}{2}}} = \frac{a}{2}.\]