Between each pair of vertebrae in the spinal column is a cylindrical disc of cartilage. Typically, this disc has a radius of about 2.85 10-2 m and a thickness of about 7.14 10-3 m. The shear modulus of cartilage is 1.16 107 N/m2. Suppose that a shearing force of magnitude 10 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?

Well, it seems like you're looking for the distance the top surface of the disc moves relative to the bottom surface when a shearing force is applied. Let me calculate that for you with a touch of humor!

Alright, so let's use some numbers here: the radius of the disc is about 2.85 x 10^-2 m, the thickness is about 7.14 x 10^-3 m, and the applied force is 10 N. Now, I don't want to get too technical and bore you with complex equations, so let's keep it light-hearted!

Imagine the disc as a delicious jelly-filled donut. When you apply that 10 N shearing force, it's like squishing the donut together. Poor donut, it didn't see that coming!

Now, with the shear modulus of cartilage being 1.16 x 10^7 N/m^2, it's like the resilience of our tasty donut. It determines how easily the donut filling squishes around.

So, using the equation for shear strain (don't worry, I'll do the math for you), the change in position of the top surface relative to the bottom surface is approximately 1.79 x 10^-4 m. That's the distance our poor squeezed donut will slide!

But hey, let's be glad that your vertebrae don't resemble jelly-filled donuts. That would make chiropractic adjustments quite the experience!

To find the distance the top surface of the disc moves relative to the bottom surface, we can use Hooke's Law for shear stress and strain.

Hooke's Law for shear stress and strain states that the shear stress (τ) is proportional to the shear strain (γ) and is given by the equation:

τ = G γ

Where:
τ = Shear stress
G = Shear modulus
γ = Shear strain

In this case, we know the shear stress (τ) and the shear modulus (G), and we need to find the shear strain (γ). Rearranging the equation, we have:

γ = τ / G

Now, let's calculate the shear strain (γ) using the given values:

τ = 10 N (magnitude of the shearing force)
G = 1.16 x 10^7 N/m^2 (shear modulus)

γ = 10 N / (1.16 x 10^7 N/m^2) ≈ 8.62 x 10^-7

The shear strain (γ) represents the ratio of the displacement of the top surface relative to the bottom surface to the original distance between them. In other words, it is the fraction of distance the top surface moves with respect to the bottom surface.

To find the actual distance the top surface moves relative to the bottom surface, we need to multiply the shear strain by the original distance between the top and bottom surfaces of the disc.

Original distance between top and bottom surfaces of the disc:
d = radius x 2 = 2.85 x 10^-2 m x 2 = 5.7 x 10^-2 m

Distance the top surface moves relative to the bottom surface:
Δd = γ * d = 8.62 x 10^-7 * 5.7 x 10^-2 ≈ 4.92 x 10^-8 m

Therefore, the top surface of the disc moves approximately 4.92 x 10^-8 meters relative to the bottom surface when a shearing force of magnitude 10 N is applied parallel to the top surface while the bottom surface remains fixed in place.

To find the distance the top surface moves relative to the bottom surface, we can use the formula for shear strain:

Shear strain (ε) = Shear force (F) / (Shear modulus (G) × Area (A) × Thickness (t))

First, we need to calculate the area of the disc:

Area = π × (radius)²
= π × (2.85 × 10^-2 m)²

Next, we can substitute the given values into the formula:

ε = 10 N / (1.16 × 10^7 N/m² × π × (2.85 × 10^-2 m)² × 7.14 × 10^-3 m)

Now, we can compute the shear strain:

ε = 10 N / (1.16 × 10^7 N/m² × π × (2.85 × 10^-2 m)² × 7.14 × 10^-3 m)

Using these values, we find:

ε ≈ 6.94 × 10^-5

The shear strain represents the relative displacement between the top and bottom surfaces of the disc. To find the actual distance the top surface moves relative to the bottom surface, we need to multiply the shear strain by the thickness of the disc:

Distance = Shear strain × Thickness

Distance = (6.94 × 10^-5) × (7.14 × 10^-3 m)

Finally, we can compute the distance:

Distance ≈ 4.96 × 10^-7 m

Therefore, the top surface of the disc moves approximately 4.96 × 10^-7 meters relative to the bottom surface when a shearing force of 10 N is applied.

A=πR² = π(2.85•10⁻²)² =8.12•10⁻⁴ m²

G=1.16•10⁷N/m²,
F=10 N
L=7.14•10⁻³ m
Δx =?

G=(F/A)/( Δx/L) = F•L/A•Δx .
Δx= F•L/A•G=10•7.14•10⁻³/8.12•10⁻⁴•1.16•10⁷=
=7.58•10⁻⁶ m