To determine the bulk modulus of rubber based on the information provided, we can use the relationship between pressure change, volume change, and bulk modulus. The bulk modulus (K) is defined as the ratio of the change in pressure to the fractional change in volume. Let's break this down step by step.
Understanding the Problem
We know that the rubber ball experiences a volume reduction of 0.1% when submerged in water at a depth of 200 meters. The density of water is given as 1000 kg/m³, and we will use the acceleration due to gravity as 10 m/s².
Step 1: Calculate the Pressure at the Depth
The pressure at a certain depth in a fluid can be calculated using the formula:
Where:
- P₀ = atmospheric pressure (approximately 101,325 Pa at sea level)
- ρ = density of the fluid (1000 kg/m³ for water)
- g = acceleration due to gravity (10 m/s²)
- h = depth (200 m)
Substituting the values:
- P = 101,325 Pa + (1000 kg/m³)(10 m/s²)(200 m)
- P = 101,325 Pa + 2,000,000 Pa
- P = 2,101,325 Pa
Step 2: Calculate the Change in Volume
The volume change of the rubber ball is given as 0.1%. We can express this as a fraction:
Where ΔV is the change in volume and V₀ is the original volume of the ball.
Step 3: Relate Pressure Change to Bulk Modulus
The bulk modulus (K) can be calculated using the formula:
Here, ΔP is the change in pressure, which we can assume is equal to the pressure at the depth since the ball is submerged. Thus, ΔP = 2,101,325 Pa.
Step 4: Calculate the Bulk Modulus
Now we can substitute the values into the bulk modulus formula:
- K = -2,101,325 Pa / (-0.001)
- K = 2,101,325,000 Pa
To express this in more manageable terms, we can convert it to gigapascals (GPa):
Final Result
The bulk modulus of rubber, based on the given conditions, is approximately 2.1 GPa. This value indicates how incompressible the rubber is under the pressure exerted by the water at that depth.
