Question 3 of 13intermediate🔧 ApplyNumerical4 marks

The edge of an aluminium cube is 10 cm long. One face of the cube is firmly fixed to a vertical wall. A mass of 100 kg is then attached to the opposite face of the cube. The shear modulus of aluminium is 25 GPa. What is the vertical deflection of this face?

Correct Answer

The vertical deflection of the face is approximately 4×104 \times 10⁻⁷ m (or 0.0004 mm).

Exercise: EXERCISES | Q: 8.6
For More Understanding

Explanation

This problem involves shear deformation. When a mass is attached to the free face of the cube fixed to a wall, the weight of the mass acts as a shearing force parallel to the face. The shear modulus relates shear stress to shear strain. From Example 8.4 in the context, we see the formula: shearing strain = (Δx/L) = Stress/G, which gives Δx = (Stress × L)/G. The stress is calculated as force per unit area, where force equals mg.

Solution Steps

  1. Step 1: Calculate the shearing force. F = mg = 100 kg × 9.8 m/s² = 980 N

  2. Step 2: Calculate the area of the face. A = (10 cm)² = (0.1 m)² = 0.01 m²

  3. Step 3: Calculate shear stress. Stress = F/A = 980 N / 0.01 m² = 9.8×109.8 \times 10⁴ N m⁻²

  4. Step 4: Apply the shear modulus formula. Shearing strain = (Δx/L) = Stress/G

  5. Step 5: Calculate vertical deflection. Δx = (Stress × L)/G = (9.8×109.8 \times 10⁴ × 0.1)/(25×1025 \times 10⁹) = 9.8×109.8 \times 10³/(25×1025 \times 10⁹) = 3.92×103.92 \times 10⁻⁷ m ≈\approx 4×104 \times 10⁻⁷ m

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