To determine the angle of the water surface relative to the horizontal when a rectangular container is placed on an inclined plane, we need to analyze the forces acting on the water inside the container. The key here is to understand how the incline affects the water surface due to gravity and the friction between the container and the plane.
Understanding the Forces at Play
When the container is placed on an incline, the water inside will adjust its surface to remain level with the gravitational pull acting on it. The angle of the water surface will be influenced by the incline of the plane and the forces acting on the container.
Step-by-Step Analysis
- Container Dimensions and Volume: The base dimensions of the container are 0.4 m by 0.2 m, and the height is 0.4 m. The container is filled with water to a depth of 0.2 m. The volume of water can be calculated as:
Volume = Base Area × Height = (0.4 m × 0.2 m) × 0.2 m = 0.016 m³
- Mass of Water: The density of water is approximately 1000 kg/m³. Therefore, the mass of the water is:
Mass = Density × Volume = 1000 kg/m³ × 0.016 m³ = 16 kg
- Total Mass of the System: The total mass of the container when filled with water is:
Total Mass = Mass of Empty Container + Mass of Water = 10 kg + 16 kg = 26 kg
Calculating Forces on the Inclined Plane
When the container is on the inclined plane, the gravitational force acting on the system can be broken down into two components: one parallel to the incline and one perpendicular to it.
- Gravitational Force: The total weight (W) of the system is:
W = Total Mass × g = 26 kg × 9.81 m/s² = 254.06 N
- Components of Weight: The component of weight acting parallel to the incline (W_parallel) and perpendicular to the incline (W_perpendicular) can be calculated as:
W_parallel = W × sin(30°) = 254.06 N × 0.5 = 127.03 N
W_perpendicular = W × cos(30°) = 254.06 N × (√3/2) ≈ 219.19 N
Frictional Force
The frictional force (F_friction) that opposes the motion can be calculated using the coefficient of friction (μ) and the normal force (N), which is equal to W_perpendicular:
F_friction = μ × N = 0.3 × 219.19 N ≈ 65.76 N
Equilibrium Condition
For the container to remain stationary on the incline, the frictional force must be equal to the parallel component of the weight:
F_friction = W_parallel
Substituting the values we calculated:
65.76 N < 127.03 N
This indicates that the frictional force is not sufficient to prevent the container from sliding down the incline. Therefore, the water surface will tilt to maintain equilibrium.
Determining the Angle of the Water Surface
The angle of the water surface (θ_water) relative to the horizontal can be found using the relationship between the forces acting on the water. The effective angle of the water surface will be equal to the angle of the incline (30°) minus the angle due to the tilt caused by the water's weight. This can be expressed as:
tan(θ_water) = (W_parallel - F_friction) / W_perpendicular
Substituting the values:
tan(θ_water) = (127.03 N - 65.76 N) / 219.19 N
tan(θ_water) = 61.27 N / 219.19 N ≈ 0.279
Now, to find θ_water, we take the arctangent:
θ_water = arctan(0.279) ≈ 15.6°
Final Result
The angle of the water surface relative to the horizontal is approximately 15.6°. This angle indicates how the water adjusts to the incline of the container, demonstrating the principles of fluid dynamics and equilibrium in a gravitational field.
