A nail projects $25.0 \mathrm{~mm}$ horizontally outward from a wall. A $400 \mathrm{~N}$ picture is hung from the end of the nail, and the nail deflects downward by $2.00 \times 10^{-5} \mathrm{~m}$. If the shear modulus is $3.40 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}$, then what is the radius of the nail?
A) $2.16 \mathrm{~mm}$
B) $2.65 \mathrm{~mm}$
C) $1.86 \mathrm{~mm}$
D) $2.44 \mathrm{~mm}$
E) $1.60 \mathrm{~mm}$

Question

Quizplus · Accepted Answer

The shear stress ($\tau$) is given by the force applied (F) divided by the area (A), and the shear strain ($\gamma$) is the displacement (x) divided by the original length (L). The shear modulus (G) relates stress and strain by $\tau = G\gamma$. Rearranging to find the area, $A = \frac{F}{\tau} = \frac{FL}{Gx}$. For a circular cross-section, $A = \pi r^2$, so solving for $r$ gives $r = \sqrt{\frac{FL}{\pi Gx}}$. Substituting the given values, $r = \sqrt{\frac{(400)(0.025)}{\pi(3.40 \times 10^{10})(2.00 \times 10^{-5})}}$, results in $r \approx 2.16 \mathrm{~mm}$.

A Nail Projects 25.0 mm25.0 \mathrm{~mm}25.0 mm Horizontally Outward from a Wall 400 N400 \mathrm{~N}400 N

A Nail Projects $25.0 \mathrm{~mm}$ Horizontally Outward from a Wall $400 \mathrm{~N}$