water in a dam is 15 m deep. a horizontal pipe of 4 cm diameter passes through the wall of the dam below the surface of water. a plug secures the pipe opening. the frictional force between the plug and pipe wall is about?
To calculate the frictional force between the plug and the pipe wall, we need to consider the pressure of the water acting on the plug.
Step 1: Calculate the pressure of water at the depth of the plug.
The pressure of water increases with depth due to the weight of the water column above it. The pressure at any given depth is given by the equation:
Pressure = density of water × acceleration due to gravity × depth
Given:
Depth = 15 m
Density of water = 1000 kg/m³
Acceleration due to gravity = 9.8 m/s²
Pressure = 1000 kg/m³ × 9.8 m/s² × 15 m
Pressure = 147,000 Pascal (Pa)
Step 2: Calculate the area of the pipe.
The area of a pipe is given by the equation:
Area = π × (radius)²
Given:
Diameter = 4 cm
Radius = Diameter / 2 = 2 cm = 0.02 m
Area = π × (0.02 m)²
Area = 0.00126 square meters (m²)
Step 3: Calculate the force acting on the plug.
The force acting on the plug is given by the equation:
Force = Pressure × Area
Force = 147,000 Pa × 0.00126 m²
Force ≈ 185.22 Newtons (N)
So, the frictional force between the plug and the pipe wall is approximately 185.22 N.
To find the frictional force between the plug and pipe wall, we can use the equation for frictional force:
Frictional force = coefficient of friction × normal force
The normal force is the perpendicular force exerted by the plug on the pipe wall. In this case, the normal force is equal to the weight of the water column above the plug.
To calculate the normal force, we need to first find the area of the pipe opening. The diameter of the pipe is given as 4 cm, so the radius (r) of the pipe can be calculated as half of the diameter:
Radius (r) = 4 cm / 2 = 2 cm = 0.02 m
The area of the pipe opening (A) is given by the formula for the area of a circle:
A = πr²
So, A = π × (0.02 m)² = 0.00126 m²
Now, we can calculate the weight of the water column above the plug. The weight of an object is given by its mass multiplied by the acceleration due to gravity (g).
The mass of the water column (m) can be calculated using the density (ρ), which is the mass per unit volume. For water, the density is approximately 1000 kg/m³.
The volume of the water column (V) can be calculated by multiplying the area of the pipe opening (A) by the depth of the water (h).
V = A × h
V = 0.00126 m² × 15 m = 0.0189 m³
The mass of the water (m) can be calculated as:
m = ρ × V
m = 1000 kg/m³ × 0.0189 m³ = 18.9 kg
Finally, the weight of the water column (W) can be calculated as:
W = m × g
W = 18.9 kg × 9.8 m/s² = 185.22 N
This is the normal force acting on the plug. To find the frictional force, we need to know the coefficient of friction between the plug and the pipe wall, which is not provided in the question. Without this information, we cannot determine the exact value of the frictional force.