A 150 N picture hangs from a nail in the wall. The maximum shear stress the nail can sustain is 2.00 × 10⁶ N/m². The smallest diameter the nail can be is A) 9.77 mm. B) 8.22 mm. C) 7.97 mm. D) 7.52 mm. E) 6.03 mm.

Question

Quizplus · Accepted Answer

The shear stress ($\tau$) is given by the formula $\tau = \frac{F}{A}$, where $F$ is the force applied, and $A$ is the cross-sectional area of the nail. The force applied by the picture is 150 N, and the maximum shear stress the nail can sustain is $2.00 \times 10^6$ N/m$^2$. The cross-sectional area of a circle (which is the shape of the nail's cross-section) is given by $A = \pi r^2$, where $r$ is the radius of the circle. To find the smallest diameter, we first find the smallest radius that can sustain the force without exceeding the maximum shear stress, and then double that radius to get the diameter.Given:- $F = 150$ N- $\tau_{max} = 2.00 \times 10^6$ N/m$^2$We rearrange the shear stress formula to solve for the area $A$:$$A = \frac{F}{\tau_{max}} = \frac{150}{2.00 \times 10^6} = 7.5 \times 10^{-5}$$ m$^2$.Then, we solve for the radius $r$ using the area formula for a circle:$$A = \pi r^2 \Rightarrow r^2 = \frac{A}{\pi} = \frac{7.5 \times 10^{-5}}{\pi}$$.$$r = \sqrt{\frac{7.5 \times 10^{-5}}{\pi}} \approx 4.887 \times 10^{-3}$$ m.The diameter $d = 2r \approx 9.77 \times 10^{-3}$ m or 9.77 mm.

A 150 N Picture Hangs from a Nail in the Wall