Final answer:
The bulk modulus of elasticity for a sphere that contracts by 0.01% when taken 1 km deep in the sea is 9.8 x 10ⁱ² N/m². This is calculated using the pressure exerted by the water column and the given volumetric strain.
Step-by-step explanation:
To calculate the bulk modulus of elasticity for the material of a sphere that contracts by 0.01% when taken to the bottom of the sea 1 km deep, we need to first determine the pressure exerted on the sphere at that depth. Since the density of water is given as 1 g/cm³ (which is 1000 kg/m³ in SI units), and the depth (h) is 1 km (which is 1000 m), we use the formula for pressure due to a fluid column P = ρgh, where ρ is the density, g is the acceleration due to gravity (9.8 m/s²), and h is the height/depth of the fluid column.
This results in a pressure (P) of:
P = 1000 kg/m³ × 9.8 m/s² × 1000 m = 9.8 × 10⁶ N/m²
Bulk modulus (B) is given by the formula B = ∆P/(∆V/V), where ∆P is the change in pressure, ∆V is the change in volume, and V is the original volume. The volumetric strain (∆V/V) for a 0.01% contraction is 0.01/100 = 1 × 10⁻´. Substituting the known values:
B = 9.8 × 10⁶ N/m² / (1 × 10⁻´)
Therefore, the bulk modulus is 9.8 × 10ⁱ² N/m², which is option (a).