Find the mass of a square plate if at each point
Find the mass of a square plate if at each point the surface density is proportional to the sum of its distances from the diagonals.
Explanation of Method
Placing the square $[-a,a]^2$ with its diagonals along $y=\pm x$, the two distances combine via the clean identity $|x+y|+|x-y|=2\max(|x|,|y|)$, turning the density into $\mu=k\sqrt2\max(|x|,|y|)$. The 8-fold symmetry of both the square and this integrand then reduces the mass integral to one elementary octant.
- Set up the distances to the two diagonals and combine them.
- Reduce via symmetry to one octant, where $\max(|x|,|y|)=x$.
- Evaluate the resulting integral.
Step 1 · Density function
With the square $[-a,a]^2$ and diagonals $x-y=0,\,x+y=0$: $d_1=\frac{|x-y|}{\sqrt2},\,d_2=\frac{|x+y|}{\sqrt2}$, so $\mu=k(d_1+d_2)=\frac k{\sqrt2}(|x-y|+|x+y|)$. Using $|x+y|+|x-y|=2\max(|x|,|y|)$ (verified by direct case-check):
$$ \mu(x,y) = k\sqrt2\max(|x|,|y|). $$
Step 2 · Reduce by symmetry
By the 8-fold symmetry of the square and of $\max(|x|,|y|)$, integrate over the octant $0\le y\le x\le a$ (where $\max(|x|,|y|)=x$) and multiply by $8$:
$$ m = 8k\sqrt2\int_0^a\int_0^xx\,dy\,dx. $$
Step 3 · Evaluate
$$ \int_0^a\int_0^xx\,dy\,dx = \int_0^ax^2\,dx = \frac{a^3}3, \qquad m = 8k\sqrt2\cdot\frac{a^3}3 = \frac{8\sqrt2\,ka^3}3. $$
Additional Notes
- This derivation matches the source exactly. In terms of the full side length $\ell=2a$: $m=\dfrac{\sqrt2\,k\ell^3}3$.
- Verified numerically (e.g. $a=1.6,\,k=1$): direct double integration gives $m\approx15.4470$, matching $\dfrac{8\sqrt2\,ka^3}3$ exactly.
Result
The mass of the square plate (half-side $a$) is: