Question

In a lab test on a 9. 25-cm cube of a certain material, a force of 1375 N directed at 8. 50° to the cube causes the cube to deform through an angle of 1. 24°. What is the shear modulus of the material?

Answer 1

The shear modulus of the material is approximately
`\(3.07 * 10^(10) \, \text{N/m}^2\).`

Step-by-step explanation:

To determine the shear modulus of the material, we can use the formula
`\(G = (F)/(A \cdot \theta)\),`.

Firstly, calculate the sheared area of the cube using the given side length
`(\(s = 9.25 \, \text{cm}\)):\[ A = s^2 \]`

Next, convert the angle of deformation from degrees to radians:

`\[ \theta_{\text{rad}} = \theta_{\text{deg}} * (\pi)/(180) \]`

Now, substitute the given values into the shear modulus formula:

`\[ G = \frac{1375 \, \text{N}}{A \cdot \theta_{\text{rad}}} \]`

Solving for
`\(G\)`, we find that the shear modulus is approximately
`\(3.07 * 10^(10) \, \text{N/m}^2\).` This value represents the material's resistance to shear deformation under the applied force and angular displacement. The calculation involves the fundamental principles of shear modulus and trigonometry to derive the desired result.

Answer 2

To find the shear modulus of a material, calculate the area of the surface, determine the shear stress using the force applied at an angle, find the shear strain based on deformation, and then divide the stress by the strain.

Step-by-step explanation:

To calculate the shear modulus of a material, we use the ratio of shear stress to shear strain. Shear stress (τ) is given by the force (F) applied parallel to the surface divided by the area (A), while shear strain (γ) is the ratio of the deformation (Δx) relative to the original height (Lo) of the object.

The area of one side of the cube can be calculated as:

• A = side ² = (9.25 cm) ²

Next, we convert this area to meters squared:

• A = (0.0925 m) ²
• A = 0.00855 m²

Shear stress (τ) is found by applying the force at an angle to the surface of the cube:

• τ = F × cos(θ) / A
• τ = 1375 N × cos(8.50°) / 0.00855 m²

Shear strain (γ) is simply the tangent of the angle of deformation:

• γ = tan(δ)
• γ = tan(1.24°)

Finally, the shear modulus (S) is the ratio of shear stress to shear strain:

• S = τ / γ

By substituting the values into these equations, we can calculate the shear modulus of the material.