To determine the shear modulus of the block of gelatin, we will need to calculate three quantities: shear stress, shear strain, and shear modulus. Let's break down the problem step by step.
### Step 1: Given Information
- Dimensions of the gelatin block: \(60 \text{ mm} \times 60 \text{ mm} \times 20 \text{ mm}\)
- Force applied: \(0.245 \text{ N}\)
- Displacement caused by the force: \(5 \text{ mm}\)
### Step 2: Convert Units
First, it’s often useful to convert the given measurements into meters (SI units) for consistency, as force is given in Newtons (N) and we typically want to work in meters (m).
- Displacement: \(5 \text{ mm} = 0.005 \text{ m}\)
- Height (thickness) of the gelatin block: \(20 \text{ mm} = 0.02 \text{ m}\)
- Area of the surface to which the force is applied: \(60 \text{ mm} \times 60 \text{ mm} = 3600 \text{ mm}^2 = 0.0036 \text{ m}^2\)
### Step 3: Calculate Shear Stress
Shear stress (\(\tau\)) is defined as the force (\(F\)) applied tangentially to an area (\(A\)):
[tex]\[
\tau = \frac{F}{A}
\][/tex]
Plug in the given values:
[tex]\[
\tau = \frac{0.245 \text{ N}}{0.0036 \text{ m}^2} = 68.0556 \text{ N/m}^2
\][/tex]
### Step 4: Calculate Shear Strain
Shear strain (\(\gamma\)) is the displacement (\(x\)) divided by the height (or thickness) (\(h\)) of the material:
[tex]\[
\gamma = \frac{x}{h}
\][/tex]
Plug in the given values:
[tex]\[
\gamma = \frac{0.005 \text{ m}}{0.02 \text{ m}} = 0.25
\][/tex]
### Step 5: Calculate Shear Modulus
Shear modulus (\(G\)) is the ratio of shear stress (\(\tau\)) to shear strain (\(\gamma\)):
[tex]\[
G = \frac{\tau}{\gamma}
\][/tex]
Plug in the calculated shear stress and shear strain:
[tex]\[
G = \frac{68.0556 \text{ N/m}^2}{0.25} = 272.2222 \text{ N/m}^2
\][/tex]
### Conclusion
The calculated shear modulus \(G\) is approximately \(272.22 \text{ N/m}^2\). Therefore, the correct answer is:
b) [tex]\(272.2 \text{ N/m}^2\)[/tex]
