To solve this problem, we need to analyze the forces acting on the uniform rod AB and the lead weight at point B. The rod is floating in water, with half of its length submerged, and we need to find the tension in the cord supporting the rod and the total volume of the rod.
Understanding the Setup
We have a uniform rod AB that is 4 meters long and weighs 12 kg. It is supported at one end (point A) and has a 6 kg lead weight attached at the other end (point B). The rod is floating, meaning the forces acting on it are balanced. The buoyant force acting on the submerged part of the rod must equal the total weight of the rod and the lead weight.
Calculating the Total Weight
The total weight of the system can be calculated as follows:
- Weight of the rod = 12 kg
- Weight of the lead = 6 kg
- Total weight = 12 kg + 6 kg = 18 kg
Determining the Buoyant Force
Since the rod is floating with half of its length submerged, we can find the buoyant force. The buoyant force is equal to the weight of the water displaced by the submerged part of the rod. The volume of the submerged part of the rod can be calculated as:
- Length submerged = 4 m / 2 = 2 m
- Volume of the rod = length × cross-sectional area
Let’s denote the cross-sectional area of the rod as A. Therefore, the volume of the submerged part is:
Volume submerged = 2 m × A
The buoyant force (B) can be expressed as:
B = Volume submerged × Density of water × g
Assuming the density of water is approximately 1000 kg/m³ and g (acceleration due to gravity) is about 9.81 m/s², we have:
B = (2 m × A) × 1000 kg/m³ × 9.81 m/s²
Setting Up the Equation
For the rod to float, the buoyant force must equal the total weight of the system:
B = Total weight
Substituting the values we have:
(2 m × A) × 1000 kg/m³ × 9.81 m/s² = 18 kg × 9.81 m/s²
We can cancel out 9.81 m/s² from both sides:
2 m × A × 1000 kg/m³ = 18 kg
Now, solving for A:
A = 18 kg / (2 m × 1000 kg/m³) = 0.009 m²
Finding the Total Volume of the Rod
The total volume of the rod can be calculated as:
Total Volume = Length × Cross-sectional Area = 4 m × A
Substituting the value of A:
Total Volume = 4 m × 0.009 m² = 0.036 m³
Calculating the Tension in the Cord
The tension in the cord (T) can be determined by considering the forces acting on the rod. The upward buoyant force and the tension in the cord must balance the weight of the rod and the lead weight:
T + B = Total weight
We already know B and the total weight:
T + (2 m × A × 1000 kg/m³ × 9.81 m/s²) = 18 kg × 9.81 m/s²
Substituting the value of B:
T + 18 kg = 18 kg
Thus, T = 0 kg. This means that the buoyant force alone is sufficient to support the weight of the rod and the lead weight, and there is no additional tension in the cord.
Summary of Results
In summary, we found that:
- The total volume of the rod is 0.036 m³.
- The tension in the cord is 0 N, indicating that the buoyant force is sufficient to support the entire weight of the rod and the lead weight.
