Suppose a beam is 10 meters long, and that there are three weights on the beam: a 10 kilogram weight 3 meters from the left end, a 5 kilogram weight 6 meters from the left end, and a 4 kilogram weight 8 meters from the left end. Where should a fulcrum be placed so that the beam balances? Let's assign a scale to the beam, from 0 at the left end to 10 at the right, so that we can denote locations on the beam simply as $x$ coordinates; the weights are at $x=3$, $x=6$, and $x=8$, as in figure 11.1.1.
Suppose to begin with that the fulcrum is placed at $x=5$. What will happen? Each weight applies a force to the beam that tends to rotate it around the fulcrum; this effect is measured by a quantity called torque, proportional to the mass times the distance from the fulcrum. Of course, weights on different sides of the fulcrum rotate the beam in opposite directions. We can distinguish this by using a signed distance in the formula for torque. So with the fulcrum at 5, the torques induced by the three weights will be proportional to $(3-5)10=-20$, $(6-5)5=5$, and $(8-5)4=12$. For the beam to balance, the sum of the torques must be zero; since the sum is $-20+5+12=-3$, the beam rotates counter-clockwise, and to get the beam to balance we need to move the fulcrum to the left. To calculate exactly where the fulcrum should be, we let $\ds \bar x$ denote the location of the fulcrum when the beam is in balance. The total torque on the beam is then $\ds (3-\bar x)10+(6-\bar x)5+(8-\bar x)4=92-19\bar x$. Since the beam balances at $\ds \bar x$ it must be that $\ds 92-19\bar x=0$ or $\ds \bar x=92/19\approx 4.84$, that is, the fulcrum should be placed at $x=92/19$ to balance the beam.
Now suppose that we have a beam with varying density—some portions of the beam contain more mass than other portions of the same size. We want to figure out where to put the fulcrum so that the beam balances.
Example 11.1.1 Suppose the beam is 10 meters long and that the density is $1+x$ kilograms per meter at location $x$ on the beam. To approximate the solution, we can think of the beam as a sequence of weights "on'' a beam. For example, we can think of the portion of the beam between $x=0$ and $x=1$ as a weight sitting at $x=0$, the portion between $x=1$ and $x=2$ as a weight sitting at $x=1$, and so on, as indicated in figure 11.1.2. We then approximate the mass of the weights by assuming that each portion of the beam has constant density. So the mass of the first weight is approximately $\ds m_0=(1+0)1=1$ kilograms, namely, $(1+0)$ kilograms per meter times 1 meter. The second weight is $\ds m_1=(1+1)1=2$ kilograms, and so on to the tenth weight with $\ds m_9=(1+9)1=10$ kilograms. So in this case the total torque is $$ (0-\bar x)m_0+(1-\bar x)m_1+\cdots+(9-\bar x)m_9= (0-\bar x)1+(1-\bar x)2+\cdots+(9-\bar x)10. $$ If we set this to zero and solve for $\ds \bar x$ we get $\ds \bar x=6$. In general, if we divide the beam into $n$ portions, the mass of weight number $i$ will be $\ds m_i=(1+x_i)(x_{i+1}-x_i)=(1+x_i)\Delta x$ and the torque induced by weight number $i$ will be $\ds (x_i-\bar x)m_i=(x_i-\bar x)(1+x_i)\Delta x$. The total torque is then $$\eqalign{ (x_0-\bar x)(1+x_0)\Delta x&+(x_1-\bar x)(1+x_1)\Delta x+\cdots+ (x_{n-1}-\bar x)(1+x_{n-1})\Delta x\cr &=\sum_{i=0}^{n-1} x_i(1+x_i)\Delta x- \sum_{i=0}^{n-1}\bar x(1+x_i)\Delta x\cr &=\sum_{i=0}^{n-1} x_i(1+x_i)\Delta x- \bar x\sum_{i=0}^{n-1}(1+x_i)\Delta x.\cr }$$ If we set this equal to zero and solve for $\ds \bar x$ we get an approximation to the balance point of the beam: $$\eqalign{ 0&=\sum_{i=0}^{n-1} x_i(1+x_i)\Delta x- \bar x\sum_{i=0}^{n-1}(1+x_i)\Delta x\cr \bar x\sum_{i=0}^{n-1}(1+x_i)\Delta x&=\sum_{i=0}^{n-1} x_i(1+x_i)\Delta x\cr \bar x&={\ds\sum_{i=0}^{n-1} x_i(1+x_i)\Delta x\over \ds\sum_{i=0}^{n-1}(1+x_i)\Delta x}.\cr }$$ The denominator of this fraction has a very familiar interpretation. Consider one term of the sum in the denominator: $\ds (1+x_i)\Delta x$. This is the density near $\ds x_i$ times a short length, $\Delta x$, which in other words is approximately the mass of the beam between $\ds x_i$ and $\ds x_{i+1}$. When we add these up we get approximately the mass of the beam.
Now each of the sums in the fraction has the right form to turn into an integral, which in turn gives us the exact value of $\ds \bar x$: $$ \bar x={\ds\int_0^{10} x(1+x)\,dx\over\ds\int_{0}^{10}(1+x)\,dx}. $$ The numerator of this fraction is called the moment of the system around zero: $$\int_0^{10} x(1+x)\,dx=\int_0^{10} x+x^2\,dx={1150\over 3},$$ and the denominator is the mass of the beam: $$\int_0^{10} (1+x)\,dx=60,$$ and the balance point, officially called the center of mass, is $$\bar x={1150\over 3}{1\over 60} = {115\over 18}\approx 6.39.$$ $\square$
It should be apparent that there was nothing special about the density function $\sigma(x)=1+x$ or the length of the beam, or even that the left end of the beam is at the origin. In general, if the density of the beam is $\sigma(x)$ and the beam covers the interval $[a,b]$, the moment of the beam around zero is $$M_0=\int_a^b x\sigma(x)\,dx$$ and the total mass of the beam is $$M=\int_a^b \sigma(x)\,dx$$ and the center of mass is at $$\bar x={M_0\over M}.$$
Example 11.1.2 Suppose a beam lies on the $x$-axis between 20 and 30, and has density function $\sigma(x)=x-19$. Find the center of mass. This is the same as the previous example except that the beam has been moved. Note that the density at the left end is $20-19=1$ and at the right end is $30-19=11$, as before. Hence the center of mass must be at approximately $20+6.39=26.39$. Let's see how the calculation works out. $$\eqalign{ M_0&=\int_{20}^{30} x(x-19)\,dx=\int_{20}^{30} x^2-19x\,dx= \left.{x^3\over3}-{19x^2\over2}\right|_{20}^{30}={4750\over3}\cr M&=\int_{20}^{30} x-19\,dx=\left.{x^2\over2}-19x\right|_{20}^{30}=60\cr {M_0\over M}&={4750\over3}{1\over60}={475\over18}\approx 26.39.\cr }$$ $\square$
Example 11.1.3 Suppose a flat plate of uniform density has the shape contained by $\ds y=x^2$, $y=1$, and $x=0$, in the first quadrant. Find the center of mass. (Since the density is constant, the center of mass depends only on the shape of the plate, not the density, or in other words, this is a purely geometric quantity. In such a case the center of mass is called the centroid.)
This is a two dimensional problem, but it can be solved as if it were two one dimensional problems: we need to find the $x$ and $y$ coordinates of the center of mass, $\ds \bar x$ and $\ds \bar y$, and fortunately we can do these independently. Imagine looking at the plate edge on, from below the $x$-axis. The plate will appear to be a beam, and the mass of a short section of the "beam'', say between $\ds x_i$ and $\ds x_{i+1}$, is the mass of a strip of the plate between $\ds x_i$ and $\ds x_{i+1}$. See figure 11.1.3 showing the plate from above and as it appears edge on. Since the plate has uniform density we may as well assume that $\sigma=1$. Then the mass of the plate between $\ds x_i$ and $\ds x_{i+1}$ is approximately $\ds m_i=\sigma(1-x_i^2)\Delta x=(1-x_i^2)\Delta x$. Now we can compute the moment around the $y$-axis: $$M_y=\int_0^1 x(1-x^2)\,dx={1\over4}$$ and the total mass $$M=\int_0^1 (1-x^2)\,dx={2\over3}$$ and finally $$\bar x = {1\over4}{3\over2}={3\over8}.$$ Next we do the same thing to find $\ds \bar y$. The mass of the plate between $\ds y_i$ and $\ds y_{i+1}$ is approximately $\ds n_i=\sqrt{y}\Delta y$, so $$M_x=\int_0^1 y\sqrt{y}\,dy={2\over5}$$ and $$\bar y={2\over5}{3\over2}={3\over5},$$ since the total mass $M$ is the same. The center of mass is shown in figure 11.1.3. $\square$
Example 11.1.4 Find the center of mass of a thin, uniform plate whose shape is the region between $y=\cos x$ and the $x$-axis between $x=-\pi/2$ and $x=\pi/2$. It is clear that $\ds \bar x=0$, but for practice let's compute it anyway. We will need the total mass, so we compute it first: $$ M=\int_{-\pi/2}^{\pi/2} \cos x\,dx=\sin x\Big|_{-\pi/2}^{\pi/2}=2. $$ The moment around the $y$-axis is $$ M_y=\int_{-\pi/2}^{\pi/2} x\cos x\,dx= \cos x+x\sin x\Big|_{-\pi/2}^{\pi/2}=0 $$ and the moment around the $x$-axis is $$ M_x=\int_{0}^{1} y\cdot2\arccos y\,dy= \left.y^2\arccos y-{y\sqrt{1-y^2}\over2}+{\arcsin y\over 2} \right|_{0}^{1}={\pi\over4}. $$ Thus $$\bar x={0\over2},\quad \bar y={\pi\over8}\approx 0.393.$$ $\square$
Exercises 11.1
You can use Sage to help check your work.
Ex 11.1.1 A beam 10 meters long has density $\ds \sigma(x)=x^2$ at distance $x$ from the left end of the beam. Find the center of mass $\ds \bar x$. (answer)
Ex 11.1.2 A beam 10 meters long has density $\sigma(x)=\sin(\pi x/10)$ at distance $x$ from the left end of the beam. Find the center of mass $\ds \bar x$. (answer)
Ex 11.1.3 A beam 4 meters long has density $\ds \sigma(x)=x^3$ at distance $x$ from the left end of the beam. Find the center of mass $\ds \bar x$. (answer)
Ex 11.1.4 Verify that $\ds\int 2x\arccos x\,dx= x^2\arccos x-{x\sqrt{1-x^2}\over2}+{\arcsin x\over 2}+C$.
Ex 11.1.5 A thin plate lies in the region between $\ds y=x^2$ and the $x$-axis between $x=1$ and $x=2$. Find the centroid. (answer)
Ex 11.1.6 A thin plate fills the upper half of the unit circle $\ds x^2+y^2=1$. Find the centroid. (answer)
Ex 11.1.7 A thin plate lies in the region contained by $y=x$ and $\ds y=x^2$. Find the centroid. (answer)
Ex 11.1.8 A thin plate lies in the region contained by $\ds y=4-x^2$ and the $x$-axis. Find the centroid. (answer)
Ex 11.1.9 A thin plate lies in the region contained by $\ds y=x^{1/3}$ and the $x$-axis between $x=0$ and $x=1$. Find the centroid. (answer)
Ex 11.1.10 A thin plate lies in the region contained by $\ds \sqrt{x}+\sqrt{y}=1$ and the axes in the first quadrant. Find the centroid. (answer)
Ex 11.1.11 A thin plate lies in the region between the circle $\ds x^2+y^2=4$ and the circle $\ds x^2+y^2=1$, above the $x$-axis. Find the centroid. (answer)
Ex 11.1.12 A thin plate lies in the region between the circle $\ds x^2+y^2=4$ and the circle $\ds x^2+y^2=1$ in the first quadrant. Find the centroid. (answer)
Ex 11.1.13 A thin plate lies in the region between the circle $\ds x^2+y^2=25$ and the circle $\ds x^2+y^2=16$ above the $x$-axis. Find the centroid. (answer)