Radial Densities on Circular Wires

Why Radial Density Simplifies a Circular Wire

A wire can have a density that varies from point to point. If the wire is circular and its density depends only on distance from the circle’s center, however, there is an important simplification: every point on the wire is the same distance from that center. The density is therefore constant along the wire, even if the density function changes with distance elsewhere.

This lets us calculate the mass of a circular arc using its length and the density at just one radius. The same idea also gives the average density directly. The key is to measure distance from the circle’s center, which need not be the origin.

Radial Density on a Circle

Let a circular wire have center $(a,b)$ and radius $R>0$. A point $(x,y)$ has distance

r=\sqrt{(x-a)^2+(y-b)^2}

from the center. A center-relative radial density is one that depends on $(x,y)$ only through this distance. Write it as $\rho(x,y)=\phi(r)$, where $\phi$ is the radial density profile. Here $\rho$ is mass per unit length, and we assume $\phi$ is defined at the radius being considered.

Definition: A center-relative radial density has the form $\rho(x,y)=\phi\!\left(\sqrt{(x-a)^2+(y-b)^2}\right)$, where $(a,b)$ is the circle’s center. On the circular wire of radius $R$, its density is the constant $\phi(R)$.

The last statement follows because points on the circle satisfy $(x-a)^2+(y-b)^2=R^2$. Their distance from the center is always $R$. It does not follow that the density is constant throughout the surrounding plane: points at other distances can have different densities.

Mass and Average Density

For a curve $C$ carrying a continuous density $\rho$, its mass is the scalar line integral $M=\int_C \rho\,ds$, where $ds$ is an element of arc length. The average density along a curve of positive length $L$ is the mass divided by that length, $\overline{\rho}=M/L$. On a circular arc, both quantities simplify because the radial density is constant.

For an arc of radius $R$ whose angular span is $\Delta\theta>0$, traversed once, its length is $L=R\Delta\theta$. The span is measured in radians. A full circle has $\Delta\theta=2\pi$. For an arc described by an increasing angle $t$ from $\alpha$ to $\beta$, $\Delta\theta=\beta-\alpha$.

Theorem: If a circular arc of radius $R>0$ carries a continuous center-relative radial density $\rho(x,y)=\phi\!\left(\sqrt{(x-a)^2+(y-b)^2}\right)$, and the arc is traversed once with angular span $\Delta\theta>0$, then $$ M=\phi(R)R\Delta\theta, \qquad L=R\Delta\theta, \qquad \overline{\rho}=\phi(R). $$

For a full circle, these formulas give $M=2\pi R\phi(R)$ and $\overline{\rho}=\phi(R)$. The average is defined only when the curve has positive length. If density is measured in kilograms per metre and length in metres, mass is in kilograms.

Why the Formula Works

Parameterize the circle by $\mathbf{r}(t)=(a+R\cos t,\ b+R\sin t)$, for $\alpha\leq t\leq\beta$, with $\beta>\alpha$. The distance from its center is $R$ at every parameter value:

\sqrt{(a+R\cos t-a)^2+(b+R\sin t-b)^2} = \sqrt{R^2\cos^2t+R^2\sin^2t} = R.

The velocity is $\mathbf{r}'(t)=(-R\sin t,\ R\cos t)$, so its speed is

\|\mathbf{r}'(t)\| = \sqrt{R^2\sin^2t+R^2\cos^2t} = R.

For a parametrized curve, $ds=\|\mathbf{r}'(t)\|\,dt$. Substituting the constant distance and speed into the mass integral gives

M = \int_\alpha^\beta \rho(\mathbf{r}(t))\|\mathbf{r}'(t)\|\,dt = \int_\alpha^\beta \phi(R)R\,dt = \phi(R)R(\beta-\alpha).

Since the arc length is $L=R(\beta-\alpha)$, this proves $M=\phi(R)L$. Dividing by $L>0$ proves $\overline{\rho}=\phi(R)$. The simplification comes from two constants on the arc: the density and the speed of this circular parametrization.

Worked Examples

Worked Example: Mass of a Full Circle

A full circular wire is centered at $(1,-2)$, has radius $2$ metres, and has density $\rho(x,y)=3+\sqrt{(x-1)^2+(y+2)^2}$ kilograms per metre. Find its mass and average density.

The radial profile is $\phi(r)=3+r$. On the wire, $r=R=2$, so the density is $3+2=5$ kilograms per metre. The full circumference is $L=2\pi R=2\pi(2)=4\pi$ metres. Therefore,

M=(5\text{ kg/m})(4\pi\text{ m})=20\pi\text{ kg}.

The average density is $M/L=(20\pi\text{ kg})/(4\pi\text{ m})=5$ kilograms per metre. As a check, substituting the radius into the density gives $3+2=5$ kilograms per metre, exactly the computed average. The mass also equals density times circumference.

Worked Example: Mass and Average Density on an Arc

A circular arc has center $(-1,3)$, radius $2$ metres, and angular parameter from $t=\pi/6$ to $t=5\pi/6$. Its density is $\rho(x,y)=1+(x+1)^2+(y-3)^2$ kilograms per metre. Find the arc’s mass and average density.

On this circle, $(x+1)^2+(y-3)^2=R^2=4$, so the density is $1+4=5$ kilograms per metre everywhere on the arc. Its angular span is

\Delta\theta=\frac{5\pi}{6}-\frac{\pi}{6}=\frac{4\pi}{6}=\frac{2\pi}{3}.

The arc length is $L=R\Delta\theta=2(2\pi/3)=4\pi/3$ metres. Hence its mass is

M=(5\text{ kg/m})\left(\frac{4\pi}{3}\text{ m}\right) =\frac{20\pi}{3}\text{ kg}.

The average density is $(20\pi/3\text{ kg})/(4\pi/3\text{ m})=5$ kilograms per metre. Directly in the parameter integral, the integrand is $5\cdot2=10$, so $M=\int_{\pi/6}^{5\pi/6}10\,dt=10(2\pi/3)=20\pi/3$ kilograms, confirming the result.

Worked Example: A Radial Profile with a Fraction

A quarter-circle wire is centered at $(2,-1)$, has radius $3$ metres, and is described by an angular interval of length $\pi/2$. Its density is $\rho(x,y)=12/(1+\sqrt{(x-2)^2+(y+1)^2})$ kilograms per metre. Find its mass and average density.

The radial profile is $\phi(r)=12/(1+r)$. At the wire’s radius, $\phi(3)=12/(1+3)=3$ kilograms per metre. The arc length is $L=3(\pi/2)=3\pi/2$ metres, so

M=(3\text{ kg/m})\left(\frac{3\pi}{2}\text{ m}\right) =\frac{9\pi}{2}\text{ kg}.

Dividing mass by length gives $(9\pi/2\text{ kg})/(3\pi/2\text{ m})=3$ kilograms per metre. This agrees with the density at radius $3$, and substitution in the integral gives $\int \phi(3)\,ds=3(3\pi/2)=9\pi/2$ kilograms.

A Common Mistake: Measuring from the Wrong Center

The radial distance must be measured from the center used in the density formula. It is not automatically the distance from the origin. For example, consider a circle centered at $(2,-1)$ with radius $3$ and density $\rho(x,y)=2+(x-2)^2+(y+1)^2$ kilograms per metre. The correct distance is $r=\sqrt{(x-2)^2+(y+1)^2}$, so on the wire $r=3$ and the density is $2+3^2=11$ kilograms per metre. Its full-circle mass is $11(2\pi\cdot3)=66\pi$ kilograms.

A wrong calculation might use the density at the center, $\rho(2,-1)=2$, throughout the wire, and claim $M=2(6\pi)=12\pi$ kilograms. That value is wrong: the center is not on the wire, and the wire’s points have distance $3$, not $0$, from the center. Substituting $r=3$ into the density and multiplying by the circumference corrects the calculation. A related error is to use $\sqrt{x^2+y^2}$ for distance when the stated density uses distance from $(2,-1)$.

When This Shortcut Applies

The shortcut depends on the specific relationship between the wire and its density: the curve must be a circle centered at the same point used to define the radial distance. Then every point on the wire has the same radius, and the density restriction is constant. If the density also depends on position in some other way, it may vary along the circle, so it cannot generally be replaced by a single value. Likewise, a circular curve paired with distance measured from a different point does not automatically have constant density.

For a qualifying arc or full circle, the calculation has a reliable sequence: identify the center and radius; evaluate the radial profile at that radius; find the arc length from the radius and angular span; multiply density by length for mass; and divide mass by length for average density. This turns the line integral into a short calculation while preserving the reason the simplification is valid.

Check Your Understanding

Use the fixed-radius property to answer each question. Assume the stated density is continuous on the wire.

A circle has radius $4$ metres and radial density profile $\phi(r)=2r+1$ kilograms per metre. What are the density and average density on the full circle?
An arc has radius $3$ metres and angular span $\pi/3$. If its density is $7$ kilograms per metre, what is its length and mass?
Why is it important to identify the center in a formula for radial density?
A full circle has radius $R$ and profile $\phi(r)=r^2$. Write its mass and average density in terms of $R$.
What condition makes the average-density formula $M/L$ well-defined for an arc?