A team of 6 scouts plans to cross a lake on a raft they designed. The scouts have wooden beams with an average relative density of 0.80. The beams measure 30cmx30cmx3m. The average scouts' mass is about 65kg and for their safety, they want the top of the raft to be at least 3cm above the surface of the water. How many beams do they need to complete their projects successfully?
To determine how many beams the scouts need for their raft, we need to calculate the total mass of the scouts and compare it to the buoyancy force provided by the wooden beams.
1. Calculate the volume of one wooden beam:
Since the dimensions of the beam are 30cm x 30cm x 3m (converted to cm), the volume can be calculated as:
Volume = 30cm x 30cm x 300cm = 270,000 cm^3
2. Calculate the mass of one wooden beam:
The relative density of the wooden beams is given as 0.80, which means that they are 80% as dense as water.
The density of water is approximately 1 g/cm^3. Since the beams are 80% as dense as water, their density is 0.80 g/cm^3.
Mass = Density x Volume = 0.80 g/cm^3 x 270,000 cm^3 = 216,000 g
3. Convert the mass of one beam from grams to kilograms:
1 kg = 1000 g
Mass = 216,000 g / 1000 = 216 kg
4. Calculate the total mass of the scouts:
Since there are 6 scouts and the average mass is given as 65 kg, the total mass of the scouts can be calculated as:
Total Mass = 6 scouts x 65 kg = 390 kg
5. Calculate the buoyancy force provided by one wooden beam:
The buoyancy force is equal to the weight of the water displaced by the raft. It can be calculated using Archimedes' principle:
Buoyancy Force = Density of Water x Volume of Displaced Water x Gravitational Acceleration
Since the density of water is 1000 kg/m^3 and the volume of one beam is 0.3m x 0.3m x 3m = 0.27 m^3 (converted from cm^3),
Buoyancy Force = 1000 kg/m^3 x 0.27 m^3 x 9.8 m/s^2 = 264.6 N
6. Determine the number of beams required:
To ensure the raft is safely above the water, the buoyancy force must be greater than or equal to the total mass of the scouts.
Number of Beams = Total Mass of Scouts / Buoyancy Force per Beam
Number of Beams = 390 kg / 264.6 N = 1.473 beams
Since we cannot have a fraction of a beam, the scouts would need to use at least 2 beams to ensure their raft remains safely afloat.