The shear deformation of the disk is approximately 4.774605 x 10⁻⁷ meters.
The student is asking about calculating the shear deformation of a disc representing a vertebral disk in the spine. The disc is subjected to a shearing force and has given dimensions and material properties. Using the formula for shear stress and deformation, we calculate shear deformation by dividing the applied force by the product of the shear modulus and the cross-sectional area.
To find the shear deformation (Δx), we use the shear stress formula:
τ = F/A
And the relationship between shear stress (τ), shear strain b, and shear modulus (S):
γ = τ/S
Here, F = 600 N is the force applied, Lo is the original height of the disk (0.007 m), and A is the cross-sectional area calculated from the given diameter (4.00 cm = 0.04 m) of the disk. The area, A, of a circle is πr², where r is the radius.
We first find the area, A, and then substitute τ = F/A into the second formula to find Δx (shear deformation).
Using these equations:
- R = d/2 = 0.04 m / 2 = 0.02 m
- A = π(0.02 m)² = π(0.0004 m²)
- A ≈ 1.2566 x 10⁻³ m²
- τ = F/A = 600 N / 1.2566 x 10⁻³ m²
- τ ≈ 477460.5 N/m²
- Δx = τ/S = 477460.5 N/m² / 1 x 10⁹ N/m²
- Δx ≈ 4.774605 x 10⁻⁷ m
The shear deformation of the disk is approximately 4.774605 x 10⁻⁷ meters.