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 momentum of an object can be calculated by multiplying its mass by its velocity. Since the mass of water per second that strikes the blade is given as 33.6 kg/s, we can calculate the momentum before and after the interaction:

Momentum before = mass of water per second * velocity of incident water stream
= 33.6 kg/s * 17.4 m/s

Momentum after = mass of water per second * velocity of exiting water stream
= 33.6 kg/s * (-15.0 m/s)

According to the conservation of momentum, the momentum before the interaction must be equal to the momentum after the interaction:

Momentum before = Momentum after

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

From this equation, we can solve for the mass of water per second:

33.6 kg/s * 17.4 m/s = 33.6 kg/s * (-15.0 m/s)
584.64 kg m/s = -504 kg m/s

Next, we can find the magnitude of the average force exerted on the water by using the formula:

Force = (Change in momentum) / (Time)

Since the mass of water per second is constant, the force can be calculated as:

Force = [(mass of water per second) * (change in velocity)] / (Time)

The change in velocity is the difference between the velocity of the incident water stream and the velocity of the exiting water stream:

Change in velocity = |velocity of incident water stream| - |velocity of exiting water stream|
= 17.4 m/s - (-15.0 m/s)
= 32.4 m/s

Finally, we can substitute the values into the formula to calculate the magnitude of the average force:

Force = [(33.6 kg/s) * (32.4 m/s)] / (Time)

Please provide the value for the time of interaction so that we can calculate 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 turbine blade, we can use the principle of conservation of momentum.

The momentum of an object is the product of its mass and velocity. In this case, the momentum of the incoming water is given by the product of its mass per second (33.6 kg/s) and its velocity (+17.4 m/s):

Momentum of incoming water = mass of water per second * velocity of incoming water
= 33.6 kg/s * 17.4 m/s
= 583.44 kg * m/s

Similarly, the momentum of the outgoing water is given by the product of its mass per second (33.6 kg/s) and its velocity (–15.0 m/s):

Momentum of outgoing water = mass of water per second * velocity of outgoing water
= 33.6 kg/s * (-15.0 m/s)
= -504 kg * m/s

According to the principle of conservation of momentum, the sum of the momenta before and after the interaction must be equal:

Momentum before interaction = Momentum after interaction

Therefore:

Momentum of incoming water = Momentum of outgoing water

583.44 kg * m/s = -504 kg * m/s

Since momentum is a vector quantity, the negative sign indicates the opposite direction of the velocity. To find the magnitude of the average force exerted on the water by the blade, we can take the absolute value of the difference in momentum and divide it by the time interval over which the change happens.

Average force = (change in momentum) / (time interval)

In this case, the change in momentum is the difference between the momentum of the incoming water and the momentum of the outgoing water:

Change in momentum = |583.44 kg * m/s - (-504 kg * m/s)|
= |583.44 kg * m/s + 504 kg * m/s|
= |1077.44 kg * m/s|

The time interval is not given in the problem, so we cannot calculate the actual force without that information. However, by using the given formulas and values, we have determined the magnitude of the average force exerted on the water by the blade to be 1077.44 kg * m/s.