A rectangular barge 5 m long and 2 m wide floats in fresh water. (a) Find how much deeper it floats when loaded with 500 kg of sand. (b) If the barge can only be pushed 10 cm deeper into the water before water overflows to sink it, how many kilograms of sand can it carry?
To determine how much deeper the rectangular barge floats when loaded with 500 kg of sand, we need to consider the principle of buoyancy.
(a) The buoyant force exerted on the barge is equal to the weight of the water displaced by the submerged portion of the barge. When the barge is not loaded, it displaces a certain volume of water. When loaded with 500 kg of sand, it will displace additional water, causing it to sink deeper. The amount by which it sinks can be determined by comparing the volumes of water displaced in the two situations.
Step 1: Calculate the initial volume of water displaced by the unloaded barge.
The barge has dimensions 5 m (length) and 2 m (width). Since the depth is not given, we can assume it floats totally submerged to simplify the calculation.
Volume = Length × Width × Depth
Given:
Length = 5 m
Width = 2 m
Let's assume the initial depth is h1 (unknown).
Volume1 = 5 m × 2 m × h1 = 10h1
Step 2: Calculate the volume of water displaced by the barge when loaded with sand.
The sand has a weight of 500 kg, which means it displaces the same amount of water in weight. The weight of water displaced will be equal to the weight of the sand. Using the equation W = mg, where W is weight, m is mass, and g is the acceleration due to gravity, we can calculate the mass of water displaced.
Given:
Weight of sand (W) = 500 kg
The weight of water displaced = 500 kg (since weight of sand = weight of water displaced)
Using the formula W = mg, where g = 9.8 m/s², we can find the mass (m) of water displaced.
Weight (water) = mass (water) × g
mass (water) = Weight (water) / g
mass (water) = 500 kg / 9.8 m/s² = 51.02 kg
Now let's calculate the volume of water displaced by the barge when loaded with sand.
Density = mass / volume
Density (water) = 1000 kg/m³ (density of water)
Given:
Density (water) = 1000 kg/m³
Mass (water) = 51.02 kg (calculated above)
Volume2 = Mass (water) / Density (water)
Volume2 = 51.02 kg / 1000 kg/m³ = 0.05102 m³
Step 3: Calculate the depth of the water displaced by the barge when loaded with sand.
In the loaded situation, the barge displaces a volume of water, which is equal to the volume calculated in Step 2.
Volume2 = 5 m × 2 m × h2 = 10h2
Since Volume2 = 0.05102 m³, we can find h2 by rearranging the equation.
10h2 = 0.05102 m³
h2 = 0.05102 m³ / 10 = 0.005102 m
Therefore, the rectangular barge floats around 0.005102 m (or 5.102 cm) deeper when loaded with 500 kg of sand.
(b) To determine how many kilograms of sand the barge can carry before water overflows to sink it, we need to use the given constraint that it can be pushed 10 cm deeper into the water before water overflows.
Let's assume the maximum depth of the barge is h3.
The change in depth = h3 - h1 = 0.10 m (since 10 cm = 0.10 m)
Using the same equation as above, we can calculate the volume of water displaced in the overloaded situation.
Volume3 = 5 m × 2 m × h3 = 10h3
Since the difference in depth is 0.10 m, we have:
10h3 - 10h1 = 0.10 m³
10(h3 - h1) = 0.10 m³
Substituting the values of h1 and solving the equation will give us the maximum depth (h3) the barge can have before sinking.
Once we have h3, we can calculate the weight of water displaced (W) using the formula W = mg, with g = 9.8 m/s², and determine the maximum weight of sand the barge can carry by equating it to the weight of water displaced.
Please note that since the depth of the barge is not given, determining the maximum weight of sand it can carry using the given constraint might require additional information about the dimensions or the maximum depth of the barge.