A hollow copper tube (diameter = 4.01 cm) is sealed at one end and loaded with lead shot to give a total mass of 0.218 kg. When the tube is floated in pure water, what is the depth, z, of its bottom end?
What's the answer?
To find the depth, z, of the bottom end of the hollow copper tube when it is floated in pure water, we can use Archimedes' principle. Archimedes' principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the displaced fluid.
Let's break down the steps to calculate the depth:
1. Determine the volume of the hollow copper tube:
- The tube is in the shape of a cylinder, so we can use the formula for the volume of a cylinder: V = πr²h, where r is the radius and h is the height.
- The diameter of the tube is given as 4.01 cm. Therefore, the radius, r, is half of that: r = 4.01 cm / 2 = 2.005 cm = 0.02005 m.
- We need to find the height, h, which is the depth, z, plus the length of the sealed end of the tube. Since the tube is sealed at one end, h will be equal to z plus the length of the sealed end.
- Let's denote the length of the sealed end as L.
2. Calculate the mass of water displaced by the tube:
- The volume of water displaced will be equal to the volume of the tube (V).
- The density of water (ρ_water) is approximately 1000 kg/m³.
- So, the mass of water displaced (m_water) can be obtained using the formula: m_water = ρ_water * V.
3. Calculate the buoyant force on the tube:
- The buoyant force (F_buoyant) is equal to the weight of the displaced water.
- F_buoyant can be calculated using the formula: F_buoyant = m_water * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
4. Calculate the weight of the lead shot:
- The total mass of the tube and lead shot combined is 0.218 kg.
- Let's denote the mass of the lead shot as m_shot.
- The weight of the lead shot (W_shot) can be calculated using the formula: W_shot = m_shot * g.
5. Determine the equilibrium condition:
- At equilibrium, the weight of the tube and lead shot combined (W_total) should be equal to the buoyant force (F_buoyant).
- So, W_total = W_shot = F_buoyant.
6. Calculate the depth, z:
- Rearrange the equation from step 5 to solve for z: L + z = V / (πr²) = W_total / (ρ_water * g).
- Substitute the values into the equation to find the depth, z.
By following these steps, you should be able to calculate the depth, z, of the bottom end of the hollow copper tube when it is floated in pure water.