One cm of a 10-cm-long rod is made of metal, and the rest is wood. The metal has a density of 5000 kg/m3 and the wood has a density of 500 kg/m3. When the rod is set into pure water, the metal part points downward. How much of the rod is underwater?
I got it
the answer is 9.5 cm underwater
To find out how much of the rod is underwater, we need to compare the density of each material to the density of the water.
Given:
Density of metal = 5000 kg/m^3
Density of wood = 500 kg/m^3
Density of water = 1000 kg/m^3 (assumed)
Now, let's analyze the situation:
1. Calculate the volume of the metal part:
Since 1 cm of the rod is made of metal and the entire rod is 10 cm long, the volume of the metal can be calculated as:
Volume of metal = (1 cm) * (10 cm) * (1 cm) = 10 cm^3
2. Calculate the mass of the metal part:
Using the formula:
Mass = Density * Volume
Mass of metal = Density of metal * Volume of metal
Mass of metal = 5000 kg/m^3 * 10 cm^3 * (1 m/100 cm)^3
Mass of metal = 0.005 kg
3. Calculate the volume of the wood part:
Since the rest of the rod is wood, we can calculate the volume of the wood part as:
Volume of wood = Total volume of rod - Volume of metal
Volume of wood = (10 cm) * (10 cm) * (1 cm) - 10 cm^3 = 990 cm^3
4. Calculate the mass of the wood part:
Using the same formula as before:
Mass of wood = Density of wood * Volume of wood
Mass of wood = 500 kg/m^3 * 990 cm^3 * (1 m/100 cm)^3
Mass of wood = 0.495 kg
5. Calculate the buoyancy force:
The buoyancy force is the weight of the water displaced by the rod. It can be calculated as:
Buoyancy force = Density of water * Volume of rod
Buoyancy force = 1000 kg/m^3 * 10 cm * 10 cm * 10 cm * (1 m/100 cm)^3
Buoyancy force = 1 kg
6. Calculate the total weight of the rod:
The weight of the rod is the sum of the weight of the metal and wood parts:
Total weight of rod = Weight of metal + Weight of wood
Total weight of rod = Mass of metal * Gravity + Mass of wood * Gravity
Total weight of rod = (0.005 kg + 0.495 kg) * 9.8 m/s^2
Total weight of rod = 4.9 N
7. Determine the submerged length:
The rod will be fully submerged if the buoyancy force is equal to the weight of the rod. Since the buoyancy force is 1 kg and the weight of the rod is 4.9 N, it means that only a fraction of the rod is submerged.
To find the submerged length of the rod, we can use the principle of ratios:
Submerged length of rod / Total length of rod = Buoyancy force / Total weight of rod
Substituting the values:
Submerged length of rod / 10 cm = 1 kg / 4.9 N
Simplifying:
Submerged length of rod / 10 = 0.204 cm
Therefore, approximately 0.204 cm of the rod is submerged in water.