A stream of water strikes a stationary turbine blade horizontally, as the drawing illustrates. The incident water stream has a velocity of + 17.4 m/s, while the exiting water stream has a velocity of – 15.0 m/s. The mass of water per second that strikes the blade is 33.6 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 initial momentum of the water is given by the product of its mass (m1 = 33.6 kg/s) and its velocity (v1 = 17.4 m/s):
p1 = m1 * v1

The final momentum of the water is given by the product of its mass (m2 = 33.6 kg/s) and its velocity (v2 = -15.0 m/s):
p2 = m2 * v2

According to the principle of conservation of momentum, the change in momentum is equal to the average force exerted on the water by the blade multiplied by the change in time. Since the blade exerts a horizontal force, the change in velocity of the water is also horizontal.

The change in momentum can be calculated as:
Δp = p2 - p1

Now, we can calculate the average force using the equation:
F = Δp / Δt

Since the time taken for the water to pass through the blade is not given, we assume it to be 1 second:
Δt = 1 s

Substituting the values, we have:
Δp = (m2 * v2) - (m1 * v1)
F = Δp / Δt

Calculating the values:
Δp = (33.6 kg/s * -15.0 m/s) - (33.6 kg/s * 17.4 m/s)
Δp = -504 kg⋅m/s - 583.2 kg⋅m/s
Δp = -1087.2 kg⋅m/s

F = -1087.2 kg⋅m/s / 1 s
F ≈ -1087.2 N

Since the magnitude of force is always positive, we take the absolute value:
F = 1087.2 N

Therefore, the magnitude of the average force exerted on the water by the blade is approximately 1087.2 N.

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 force is equal to the rate of change of momentum.

First, we need to find the change in momentum of the water stream. The momentum of an object is given by the product of its mass and velocity.

Change in momentum = (mass of water per second that strikes the blade) * (final velocity - initial velocity)

Change in momentum = 33.6 kg/s * (-15.0 m/s - 17.4 m/s)

Next, we need to find the time it takes for the water stream to change its momentum. Since the blade exerts a force horizontally and the water stream is the only external force acting on the water, we can assume that it takes the entire time for the water to change its momentum.

Time taken = (change in momentum) / (force)

Finally, we can find the force exerted on the water by the blade by rearranging the equation:

Force = (change in momentum) / (time taken)

Plugging in the values:

Force = (33.6 kg/s * (-15.0 m/s - 17.4 m/s)) / (time taken)

Note: The above calculation assumes the system is ideal and neglects other factors such as friction or any losses in energy.