Question

The crew of the Starship Enterprise lands on a newly discovered planet with a gravitational acceleration g = 8.32 m/s². One of the crew members, Spock, decides to take a dive in a lake made up of a mystery liquid. The absolute pressure on Spock’s head at a depth 3.93 m is 129,002 Pa. If the pressure of the planet’s atmosphere is 102,205 Pa, what is the density of the liquid the lake is made up of in kg/m³?

Answer 1

The density of the mystery liquid is found by subtracting the atmospheric pressure from the absolute pressure on Spock's head to get the pressure due to the liquid, and then using the formula P = hρg to calculate the density. The density of the liquid is approximately 821.93 kg/m³.

Step-by-step explanation:

To find the density of the mystery liquid, we can use the formula for pressure due to the weight of a liquid, which is P = hρg, where P is the absolute pressure at depth, h is the depth, ρ is the density of the liquid, and g is the gravitational acceleration. The absolute pressure on Spock's head at the depth of 3.93 m is given as 129,002 Pa, and the atmospheric pressure is 102,205 Pa.

The pressure exerted by the liquid alone can be found by subtracting the atmospheric pressure from the absolute pressure:
Pressure due to the liquid = Absolute pressure - Atmospheric pressure
Pressure due to the liquid = 129,002 Pa - 102,205 Pa
Pressure due to the liquid = 26,797 Pa

Now, we can rearrange the pressure formula to solve for the density of the liquid:

ρ = P / (hg)

Substitute the known values into the equation:

ρ = 26,797 Pa / (3.93 m × 8.32 m/s²)

ρ = 821.93 kg/m³

Therefore, the density of the liquid is approximately 821.93 kg/m³.