A scuba diver searches for treasure at a depth of 20.0m below the surface of the sea. at what pressure must the scuba device deliver air to the diver? (density of sea water= 1030kg/m^3;atmospheric pressure= 1.013x10^5 pa)
another one
A king's crown is said to be solid gold but may be made of lead and covered with gold. when it is weighed in air, the scale reads 0.475 kg. when it is submerged in water the scale reads 0.437 kg.is it solid gold? if not, what percent is mass is gold? (density of gold= 19.3x10^3 kg/m^3)
1.
p= ρ•g•h + p (atm)=1000•9.8•20 + 101325= =2.97•10^5 Pa.
2.
(a)
m=0.475 kg, m1 =0.437 kg
m1•g = m•g – F(buoyancy) =>
m1•g = m•g – ρ(w)•V•g,
V= (m - m1)/ρ(w) = (0.475-0.437)/1000 =3.8•10^-5 m^3.
m1•g = ρ(x) •V•g – ρ(w)•V•g ,
m•g = ρ(x) •V•g,
ρ(x) = m/V = 0.475/3.8•10^-5 =12.5•10^3 kg/m^3
(b)
19.3•10^3•x +11.4•10^3•(100-x) = 12.5•10^3•100,
19.3•x+11.4•(100-x) = 12.5•100,
7.9•x = 110,
x =13.92% ≈14%.
The crown is roughly 14% gold and 86% lead.
To solve the first problem, we can use Pascal's law which states that pressure in a fluid increases with depth. The pressure due to the weight of the water column above the diver can be calculated using the formula:
Pressure = density x gravitational acceleration x depth
Given:
Density of sea water = 1030 kg/m^3
Depth = 20.0 m
We can substitute these values into the formula to find the pressure:
Pressure = 1030 kg/m^3 x 9.8 m/s^2 x 20.0 m
Calculating this expression, we get:
Pressure = 2,028,800 Pa
Therefore, the scuba device must deliver air to the diver at a pressure of 2,028,800 Pa.
Now let's move on to the second problem. To determine whether the king's crown is made of solid gold or not, we need to use Archimedes' principle, which states that the buoyant force exerted on an object submerged in a fluid is equal to the weight of the displaced fluid.
Given that the crown has a mass of 0.475 kg in air and a mass of 0.437 kg when submerged in water, we can find the buoyant force using the formula:
Buoyant force = weight in air - weight in water
Weight in air = mass x gravitational acceleration
Weight in water = density of water x volume of crown x gravitational acceleration
Since the crown is said to be solid gold, we can assume its density is the same as the density of gold, which is given as 19.3x10^3 kg/m^3.
Calculating the buoyant force:
Weight in air = 0.475 kg x 9.8 m/s^2
Weight in water = (19.3x10^3 kg/m^3) x volume of crown x 9.8 m/s^2
Now, we can set up the equation:
0.475 kg x 9.8 m/s^2 - (19.3x10^3 kg/m^3) x volume of crown x 9.8 m/s^2 = 0.437 kg x 9.8 m/s^2
Simplifying the equation:
4.663 kg - 190.14 kg/m^3 x volume of crown = 4.2786 kg
Solving for the volume of the crown:
190.14 kg/m^3 x volume of crown = 4.663 kg - 4.2786 kg
Volume of crown = (4.663 kg - 4.2786 kg) / 190.14 kg/m^3
Calculating the volume:
Volume of crown = 0.00202053 m^3
Since the crown is said to be solid gold, we can calculate the mass of the gold:
Mass of gold = density of gold x volume of crown
Mass of gold = (19.3x10^3 kg/m^3) x 0.00202053 m^3
Calculating the mass of gold:
Mass of gold = 39.14 kg
To calculate the percentage of gold in the crown, we can use the formula:
Percentage of gold = (Mass of gold / Total mass of the crown) x 100
Percentage of gold = (39.14 kg / 0.475 kg) x 100
Calculating the percentage of gold:
Percentage of gold = 8224%
Therefore, the scale readings suggest that the crown is not made of solid gold, and instead, approximately 8224% of its mass is gold.