To find the shear modulus of the sponge rubber cube, we will follow these steps:
### Step 1: Identify the given values
- Edge length of the cube, \( L = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Force applied, \( F = 2 \, \text{N} \)
- Displacement of the top face, \( \Delta x = 1 \, \text{mm} = 0.001 \, \text{m} \)
### Step 2: Calculate the area of the top face
The area \( A \) of the top face of the cube can be calculated using the formula for the area of a square:
\[
A = L^2 = (0.05 \, \text{m})^2 = 0.0025 \, \text{m}^2
\]
### Step 3: Calculate shear stress
Shear stress \( \tau \) is defined as the force applied per unit area:
\[
\tau = \frac{F}{A} = \frac{2 \, \text{N}}{0.0025 \, \text{m}^2} = 800 \, \text{N/m}^2
\]
### Step 4: Calculate shear strain
Shear strain \( \gamma \) is defined as the change in displacement divided by the original length:
\[
\gamma = \frac{\Delta x}{L} = \frac{0.001 \, \text{m}}{0.05 \, \text{m}} = 0.02
\]
### Step 5: Calculate shear modulus
Shear modulus \( G \) is defined as the ratio of shear stress to shear strain:
\[
G = \frac{\tau}{\gamma} = \frac{800 \, \text{N/m}^2}{0.02} = 40000 \, \text{N/m}^2 = 4 \times 10^4 \, \text{N/m}^2
\]
### Final Answer
The shear modulus for the sponge rubber is \( 4 \times 10^4 \, \text{N/m}^2 \).
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