A test tube standing vertically in a test tube rack contains 2.4 cm of oil, whose density is 0.81 g/cm^3 and 6 cm of water. What is the gauge pressure on the bottom
of the tube? The acceleration of gravity is 9.8 m/s^2.
Answer in units of Pa
To find the gauge pressure at the bottom of the test tube, we need to consider the pressure due to the weight of the oil and water above it.
The pressure at a certain depth in a fluid can be calculated using the formula:
P = ρgh
Where:
P is the pressure
ρ is the density of the fluid
g is the acceleration due to gravity
h is the depth of the fluid
First, let's calculate the pressure due to the oil:
Density of oil (ρ_oil) = 0.81 g/cm^3 = 810 kg/m^3 (converted to SI units)
Depth of oil (h_oil) = 2.4 cm = 0.024 m (converted to SI units)
Using the formula, the pressure due to the oil can be calculated as:
P_oil = ρ_oil * g * h_oil
Substituting the values:
P_oil = 810 kg/m^3 * 9.8 m/s^2 * 0.024 m
Next, let's calculate the pressure due to the water:
Density of water (ρ_water) = 1000 kg/m^3 (given)
Depth of water (h_water) = 6 cm = 0.06 m (converted to SI units)
Using the formula:
P_water = ρ_water * g * h_water
Substituting the values:
P_water = 1000 kg/m^3 * 9.8 m/s^2 * 0.06 m
Now, we can find the total pressure at the bottom of the test tube by adding the pressures of the oil and water:
P_total = P_oil + P_water
Substituting the calculated values:
P_total = (810 kg/m^3 * 9.8 m/s^2 * 0.024 m) + (1000 kg/m^3 * 9.8 m/s^2 * 0.06 m)
Now, we can simply calculate the value to find the gauge pressure at the bottom of the tube.
P_total = 188.352 + 588
P_total = 776.352 Pa
Therefore, the gauge pressure at the bottom of the test tube is 776.352 Pa.
To find the gauge pressure on the bottom of the test tube, we need to consider the pressure exerted by both the oil and the water.
We know that gauge pressure is defined as the difference between the absolute pressure and atmospheric pressure. In this case, we can assume atmospheric pressure to be 0 Pa.
To calculate the pressure exerted by the oil, we need to find the height of the oil column. Given that the test tube contains 2.4 cm of oil, we can convert this to meters by dividing by 100: 2.4 cm ÷ 100 = 0.024 m.
We can use the formula P = ρgh, where P represents the pressure, ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height of the liquid column. Substituting the values, we get:
P_oil = ρ_oil * g * h
= 0.81 g/cm^3 * 9.8 m/s^2 * 0.024 m
Calculating this, we get P_oil ≈ 0.188 Pa.
Next, let's calculate the pressure exerted by the water. We know that 6 cm of water is present in the test tube, which we can convert to meters by dividing by 100: 6 cm ÷ 100 = 0.06 m.
Using the same formula, we can calculate the pressure exerted by the water:
P_water = ρ_water * g * h
= 1 g/cm^3 * 9.8 m/s^2 * 0.06 m
Calculating this, we get P_water ≈ 0.588 Pa.
Since the gauge pressure is the difference between the pressures exerted by the oil and water, we can subtract P_water from P_oil:
Gauge pressure on the bottom = P_oil - P_water
≈ 0.188 Pa - 0.588 Pa
≈ -0.4 Pa
Therefore, the gauge pressure on the bottom of the tube is approximately -0.4 Pa.