Between each pair of vertebrae in the spinal column is a cylindrical disc of cartilage. Typically, this disc has a radius of about 2.6 x 10-2 m and a thickness of about 1.4 x 10-3 m. The shear modulus of cartilage is 1.2 x 107 N/m2. Suppose a shearing force of magnitude 18 N is applied parallel to the top surface of the disc while the bottom surface remains fixed in place. How far does the top surface move relative to the bottom surface?
To find how far the top surface moves relative to the bottom surface, we can use the formula for shear strain.
Shear strain (γ) is defined as the change in shape of an object under shear stress. It is given by the formula:
γ = )
where F is the applied shear force, A is the area over which the force is applied, and G is the shear modulus.
In this case, the applied shear force (F) is 18 N. The area of the disc (A) can be calculated using its radius (r) and thickness (t) as follows:
A = π × r^2
The shear modulus (G) is given as 1.2 x 10^7 N/m^2.
Now we can calculate the area of the disc:
A = π × (2.6 x 10^-2 m)^2
A = π × (6.76 x 10^-4 m^2)
Plugging in the values, we get:
A ≈ 2.123 x 10^-3 m^2
Now we can calculate the shear strain using the formula:
γ = )
γ = 18 N / (2.123 x 10^-3 m^2 × 1.2 x 10^7 N/m^2)
Simplifying the equation, we get:
γ ≈ 2.373 x 10^-7 m
Therefore, the top surface moves relative to the bottom surface by approximately 2.373 x 10^-7 meters.
To find how far the top surface moves relative to the bottom surface, we can use the formula for shear strain:
Shear strain = Shear stress / Shear modulus
The shear stress is the force applied per unit area, so we need to find the area of the top surface of the disc.
Area = π * radius^2
Let's calculate the area first:
radius = 2.6 x 10^-2 m
Area = π * (2.6 x 10^-2 m)^2
Now we can calculate the shear strain:
Shear stress = 18 N
Shear modulus = 1.2 x 10^7 N/m^2
Shear strain = 18 N / (1.2 x 10^7 N/m^2)
Finally, we can calculate the displacement of the top surface:
Displacement = Shear strain * thickness
thickness = 1.4 x 10^-3 m
Displacement = (18 N / (1.2 x 10^7 N/m^2)) * (1.4 x 10^-3 m)