The shear modulus (also known as the modulus of rigidity) relates the shear stress to the shear strain in a material. It is given by:
G = τ/γ
where G is the shear modulus, τ is the shear stress, and γ is the shear strain.
In this problem, we are given the shear deformation, which is a measure of the shear strain. We can use this information to find the shear stress by rearranging the equation for the shear modulus:
τ = Gγ
We are also given the height of the cylinder (which represents the disk), which we can use to calculate the cross-sectional area A of the disk:
A = πr^2
where r is the radius of the disk. Since we are not given the radius, we cannot calculate the stress directly. However, we can use the fact that the volume of the disk is constant to relate the change in height to the change in radius:
h1 = h2
πr1^2 = πr2^2
r2 = sqrt(r1^2 - Δr^2)
where h1 is the original height of the cylinder, h2 is the height after the deformation, r1 is the original radius of the cylinder, r2 is the radius after the deformation, and Δr is the change in radius.
We are given that the shear deformation is 3.34 micrometers (or 3.34 × 10^-6 meters), so:
γ = Δr / h1 = Δr / (3 cm) = 3.34 × 10^-6 / 0.03 = 1.113 × 10^-7
Substituting this value and the given shear modulus into the equation for shear stress, we get:
τ = (1 × 10^9 N/m^2) × (1.113 × 10^-7) = 0.111 N/m^2
Note that this answer is independent of the dimensions of the cylinder, since we used the fact that the volume is constant to relate the change in height to the change in radius.