A stream of water strikes a stationary turbine blade horizontally, as the drawing illustrates. The incident water stream has a velocity of +18.0 m/s, while the exiting water stream has a velocity of -18.0 m/s. The mass of water per second that strikes the blade is 36.0 kg/s. Find the magnitude of the average force exerted on the water by the blade.
To find the magnitude of the average force exerted on the water by the blade, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, provided no external forces are acting on the system.
In this case, the system consists of the water stream and the turbine blade before and after the interaction. Since there are no external forces acting on the system, the initial momentum of the water stream must be equal to the final momentum of the water stream and the blade.
The momentum, p, is defined as the product of an object's mass, m, and its velocity, v. So, we have:
Initial momentum = Final momentum
(mass of water stream x initial velocity of water stream) = (mass of water stream x final velocity of water stream) + (mass of turbine blade x velocity of turbine blade)
Let's calculate the momentum of the water stream before and after the interaction:
Initial momentum of water stream = (mass of water per second x initial velocity of water stream)
= (36.0 kg/s) x (+18.0 m/s)
= 648 kg·m/s (momentum in the positive direction)
Final momentum of water stream = (mass of water per second x final velocity of water stream)
= (36.0 kg/s) x (-18.0 m/s)
= -648 kg·m/s (momentum in the negative direction)
Let's assume the mass of the turbine blade is M, and its velocity is V. Therefore, the final momentum of the water stream and the blade is (M + 36.0) x V.
Now, applying the principle of conservation of momentum:
648 kg·m/s = -648 kg·m/s + (M + 36.0 kg/s) x V
Simplifying the equation:
2(648 kg·m/s) = (M + 36.0 kg/s) x V
1296 kg·m/s = (M + 36.0 kg/s) x V
Now, we need to calculate the magnitude of the average force exerted on the water by the blade. This can be done by using Newton's second law of motion, which states that force (F) is equal to the rate of change of momentum (Δp) with respect to time (Δt).
Since we know the mass flow rate of the water stream is 36.0 kg/s, the change in momentum Δp = mass flow rate x change in velocity of water stream = (36.0 kg/s) x [(-18.0 m/s) - (+18.0 m/s)] = -36.0 kg·m/s
As the water stream hits the blade, its momentum changes by -36.0 kg·m/s. Assuming this change in momentum happens over a time interval Δt, we can express the force as:
Force (F) = Change in momentum (Δp) / Time interval (Δt)
To calculate the average force, we need to find the change in momentum divided by the time it takes for the change to occur.
However, since the problem does not provide information about the time interval, we cannot calculate the specific value of the magnitude of the average force exerted on the water by the blade without knowing the time interval.
To find the magnitude of the average force exerted on the water by the blade, we can use Newton's second law of motion, which states that the force exerted on an object is equal to its mass times its acceleration.
In this case, the mass of water per second striking the blade is given as 36.0 kg/s. Since the water stream is changing direction, we need to account for the change in velocity. The change in velocity from +18.0 m/s to -18.0 m/s is 36.0 m/s.
Using the equation for force (F = m * a), we can calculate the average force exerted on the water by the blade.
First, we need to find the acceleration. The change in velocity (Δv) divided by the time interval (Δt) will give us the average acceleration (a).
Δv = -18.0 m/s - (+18.0 m/s) = -36.0 m/s
Δt = 1 s
Now we can calculate the average acceleration.
a = Δv / Δt = (-36.0 m/s) / (1 s) = -36.0 m/s²
Next, we can calculate the force.
F = m * a = (36.0 kg/s) * (-36.0 m/s²) = -1296 N
The negative sign indicates that the force acts in the opposite direction to the motion/velocity of the water.
Therefore, the magnitude of the average force exerted on the water by the blade is 1296 N.