Determine the work expended when driving a nail to depth

Determine the work expended when driving a nail to depth $d$ into a homogeneous material, given that the resistance offered by the material at depth $x$ is proportional to $x$.

Step 1: Resistance model

The problem states that resistance grows in proportion to the depth already driven (a common approximation, since deeper penetration puts more of the nail's lateral surface in frictional contact with the material):

\[F(x)=kx,\qquad 0\le x\le d,\]

where $k>0$ is a constant depending on the material and the nail.

Step 2: Work as the integral of force over distance

\[W=\int_0^dF(x)\,dx=\int_0^dkx\,dx=k\left[\frac{x^2}{2}\right]_0^d=\frac{kd^2}{2}.\]

\[\boxed{W=\frac{kd^2}{2}}\]

Step 3: Consequence of the quadratic dependence

Since $W\propto d^2$, driving the nail twice as deep requires four times the work — consistent with the everyday observation that a nail becomes progressively harder to drive further in.

Step 4: More general resistance laws

If instead the resistance combines a constant tip resistance $F_{\text{tip}}$ with the linear friction term, $F(x)=F_{\text{tip}}+kx$, the same integration gives $W=F_{\text{tip}}d+\dfrac{kd^2}{2}$; and for a general power-law resistance $F(x)=ax^n$ (covering, e.g., $n=0$ for a constant-resistance hard material or $n>1$ for a hardening one), $W=\dfrac{ad^{n+1}}{n+1}$. In every case, the constant(s) in $F(x)$ must be determined experimentally for the specific nail and material.