(a) To calculate the flow rate in liters per second, we need to convert the given measurements to the appropriate units and use the formula:
Flow rate = Cross-sectional area × Velocity
First, we need to convert the internal diameter of the hose from centimeters to meters:
Internal diameter = 1.60 cm = 0.016 m
Next, we can calculate the cross-sectional area of the hose using the formula:
Cross-sectional area = π × (radius)^2
Radius = Internal diameter / 2 = 0.016 m / 2 = 0.008 m
Now we can calculate the flow rate:
Flow rate = π × (0.008 m)^2 × 1.50 m/s
= 0.000302 m^3/s
To convert the flow rate to liters per second, we multiply by 1000 (since 1 liter = 0.001 cubic meters):
Flow rate in liters per second = 0.000302 m^3/s × 1000
= 0.302 L/s
Therefore, the flow rate in liters per second is 0.302 L/s.
(b) To find the nozzle's inside diameter, we can use the equation of continuity, which states that the product of the cross-sectional area and velocity of a fluid remains constant along a streamline:
A1 × v1 = A2 × v2
Where A1 and v1 are the cross-sectional area and velocity at the hose, and A2 and v2 are the cross-sectional area and velocity at the nozzle.
Given:
Velocity at the hose (v1) = 1.50 m/s
Velocity at the nozzle (v2) = 16.0 m/s
We already calculated the cross-sectional area at the hose (A1) in part (a) as:
A1 = π × (0.008 m)^2
Now we can rearrange the equation and solve for the cross-sectional area at the nozzle (A2):
A2 = (A1 × v1) / v2
= (π × (0.008 m)^2 × 1.50 m/s) / 16.0 m/s
Calculating this expression will give us the cross-sectional area at the nozzle. To find the diameter, we can use the formula:
Diameter = 2 × √(A2 / π)
Substituting the calculated value of A2, we can find the diameter of the nozzle.