water is pumped steadily out of a flooded basement at a speed of 4.5 m/s through a uniform house of radius 1.0 cm.the hose passes out through a window 3.5 m above the waterline.what is the power of the pump
To find the power of the pump, we need to calculate the amount of work done by the pump per unit time. The power is given by the equation:
Power = Work / Time
First, let's calculate the work done by the pump. Work is defined as the force applied over a distance. In this case, the force is the weight of the water being pumped out of the basement, and the distance is the vertical height the water is lifted.
The weight of the water can be determined using the density of water and the volume of water being pumped per second. The volume of water per second can be calculated by multiplying the cross-sectional area of the hose by the speed at which the water is being pumped.
Given:
- Speed of water being pumped (v) = 4.5 m/s
- Radius of hose (r) = 1.0 cm = 0.01 m
- Height of window above waterline (h) = 3.5 m
- Density of water (ρ) = 1000 kg/m^3
To calculate the cross-sectional area of the hose (A), we can use the formula for the area of a circle:
A = π * r^2
Next, we can calculate the volume of water being pumped per second (V):
V = A * v
Now, we have the mass flow rate (m_dot), which is the volume of water being pumped per second multiplied by the density of water:
m_dot = V * ρ
The weight of the water being pumped per second (W_dot) is given by:
W_dot = m_dot * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Finally, the work done by the pump per second (W_dot) is given by:
Work = W_dot * h
Now that we have the work done per second, we can divide it by time to calculate the power.
Let's plug in the given values and calculate the power of the pump.
- Note: Since we are calculating the power per unit time, the calculated power will be in watts (W).
- Also, please make sure to convert all units to the SI system before calculating.
Now, let's compute the solution.