When a load of 0.89 106 N is placed on a battleship, the ship sinks only 1.9 cm in the water. Estimate the cross-sectional area of the ship at water level.
To estimate the cross-sectional area of the ship at water level, we will need to use the formula for pressure, which is:
Pressure = Force / Area
Given information:
Load = 0.89 × 10^6 N (Force)
Sinkage = 1.9 cm or 0.019 m
We know that the pressure on the ship is equal to the pressure exerted by the water to support the load of the ship. We can assume that the weight of the ship is evenly distributed over the cross-sectional area.
Now, to calculate the cross-sectional area, we need to rearrange the formula:
Area = Force / Pressure
First, we need to calculate the pressure exerted on the ship:
Pressure = Load / Area
Rearranging the equation:
Area = Load / Pressure
To solve for the pressure, we can use the concept of fluid pressure. The pressure exerted by a fluid is related to its density and the depth under which it is submerged. In this case, the depth is the sinkage of the ship in the water.
The formula for pressure exerted by a fluid is:
Pressure = Density × g × h
Where:
Density = Density of water (typically 1000 kg/m^3)
g = Acceleration due to gravity (approximately 9.8 m/s^2)
h = Depth of sinkage (0.019 m)
Let's calculate the pressure:
Pressure = Density × g × h
Given that:
Density = 1000 kg/m^3
g = 9.8 m/s^2
h = 0.019 m
Calculating the pressure:
Pressure = 1000 kg/m^3 × 9.8 m/s^2 × 0.019 m
Now that we have the pressure, we can calculate the cross-sectional area of the ship:
Area = Load / Pressure
Given that:
Load = 0.89 × 10^6 N
Pressure = Calculated value
Calculating the area:
Area = (0.89 × 10^6 N) / Calculated Pressure
By substituting the values, we can find the estimated cross-sectional area.
rho g *Area * change in draft = weight added
1000 kg/m^3 * 9.8 * A *.019 = .89*10^6