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 69.0 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 apply the principle of conservation of linear momentum.
The principle of conservation of linear momentum states that the total momentum before an event is equal to the total momentum after the event, assuming no external forces are acting on the system.
In this case, the water stream acts as a system, and we can consider the turbine blade as the event.
Let's denote the mass of the water stream as m (m = 69.0 kg/s), the initial velocity as v_i (+18.0 m/s), and the final velocity as v_f (-18.0 m/s).
The initial momentum of the system is given by:
p_i = m * v_i
The final momentum of the system is given by:
p_f = m * v_f
According to the principle of conservation of linear momentum, p_i = p_f.
Therefore we have:
m * v_i = m * v_f
Since we want to find the magnitude of the average force exerted on the water by the blade, we need to find the change in momentum of the water stream.
The change in momentum is given by:
Δp = m * (v_f - v_i)
Substituting the given values:
Δp = 69.0 kg/s * (-18.0 m/s - 18.0 m/s)
Simplifying:
Δp = 69.0 kg/s * (-36.0 m/s)
Calculating:
Δp = -2484.0 kg * m/s
The negative sign indicates that the change in momentum is in the opposite direction.
Now, the average force exerted on the water by the blade is equal to the change in momentum divided by the time it takes for the change to occur.
Since the time is not given in the problem, we cannot directly calculate the average force. However, we have all the components required to calculate it:
Average force (F_avg) = Δp / Δt
To find Δt, we need to know the distance over which the change in velocity occurs or the time taken to accelerate/decelerate from one velocity to another. If this information is not provided in the problem, we cannot find the exact average force.
So, without the value of Δt or the distance, we cannot calculate the magnitude of the average force exerted on the water by the blade.