A nail projects 25.0 mm horizontally outward from a wall. A 400 N picture is hung from the end of the nail, and the nail deflects downward by 2.00 × 10^-5 m. If the shear modulus is 3.40 × 10¹⁰ N/m², then what is the radius of the nail? A) 1.60 mm B) 1.86 mm C) 2.16 mm D) 2.44 mm E) 2.65 mm

Question

Quizplus · Accepted Answer

First, we need to find the shear stress on the nail which can be given by the equation:

Shear stress = (Force * Distance from the wall)/((pi) * (Radius)^3 * (Deflection))

Here, Force = 400 N, Distance = 25.0 mm = 0.025 m, Deflection = 2.00 × 10^-5 m, and Shear modulus = 3.40 × 10^10 N/m^2

Plugging in these values, we get:

Shear stress = (400 N * 0.025 m)/(pi * (Radius)^3 * 3.40 × 10^10 N/m^2 * 2.00 × 10^-5 m) 
             = (2 * 10^5 N/m^2)/(pi * (Radius)^3 * 3.40)

Now, we can use the equation for shear modulus which relates shear modulus (G) to Young's modulus (E) and Poisson's ratio (v):

G = (E/2) * (1 + v)

Here, we know the value of shear modulus (G) and the value of Poisson's ratio for a typical metal (v = 0.33). The value of Young's modulus (E) can be found using the equation:

E = 2G * (1 + v)

Plugging in the values, we get:

E = 2 * 3.40 × 10^10 N/m^2 * (1 + 0.33) 
  = 8.88 × 10^10 N/m^2

Now, we can use the equation for the deformation of an elastic beam to find the radius of the nail:

Deflection = (Force * Length)/(3 * E * (pi) * (Radius)^3)

Here, Length is the length of the nail which we can assume to be 100 times the radius of the nail for simplicity (i.e., Length = 100 * Radius).

Plugging in the values, we get:

2.00 × 10^-5 m = (400 N * 100 * Radius)/(3 * 8.88 × 10^10 N/m^2 * pi * (Radius)^3)

Simplifying and solving for Radius, we get:

Radius = 2.16 mm

Therefore, the answer is C) 2.16 mm.

A Nail Projects 25