A rectangular barge floats in freshwater. When a 400 kg block is loaded on the 5 m long by 2 m wide barge,
the barge sinks a few centimeters deeper. How much deeper does the barge lower?
To find out how much deeper the barge sinks when the block is loaded on it, we need to calculate the change in buoyant force.
The buoyant force on an object submerged in a fluid depends on the volume of the fluid displaced by the object. According to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced.
Here's how you can calculate the change in buoyant force and subsequently the change in depth:
1. Determine the volume of the barge:
Volume = length × width × height
= 5 m × 2 m × initial depth (before loading the block)
2. Calculate the weight of the barge in freshwater:
Weight of Barge = density of water × g (acceleration due to gravity) × Volume of Barge
= 1000 kg/m³ × 9.8 m/s² × Volume of Barge
3. Find the original buoyant force acting on the barge:
Buoyant Force on Barge = Weight of Barge
= Weight of Water Displaced
4. Determine the volume of the block:
Volume of Block = Weight of Block (mass × g) / density of water
= 400 kg × 9.8 m/s² / 1000 kg/m³
5. Calculate the new volume of water displaced with the block on the barge:
New Volume = Volume of Barge + Volume of Block
= 5 m × 2 m × (initial depth + height change)
6. Determine the new weight of the barge with the block:
Weight of Barge with Block = density of water × g (acceleration due to gravity) × New Volume
= 1000 kg/m³ × 9.8 m/s² × New Volume
7. Find the new buoyant force acting on the barge with the block:
New Buoyant Force = Weight of Barge with Block
= Weight of Water Displaced
8. Calculate the change in buoyant force:
Change in Buoyant Force = New Buoyant Force - Buoyant Force on Barge
9. Finally, calculate the change in depth:
Change in Depth = Change in Buoyant Force / (density of water × g)
Plug in the values into each step to calculate the change in depth.