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.
Favg∆t=m∆V
mass=33.6 kg/s
∆V=(-15-17.4)
Forces acting on the blade or positive and forces acting on the water are negative. Still, they are equal and opposite.
Favg=(33.6kg/s)(-32.8m/s)
Favg=-1088.63 kg/s^2
final answer: Favg= =-1100 N
force = rate of change of momentum
change of velocity = 15+17.4 = 32.4 m/s
rate of change of momentum = mass per second * change of velocity
= 33.6 kg/s * 32.4 m/s
= 1089 kg m/s^2 or Newtons
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 if no external forces act on the system.
In this case, the water stream is the system, and the turbine blade is the external force acting on the system. Since the turbine blade is stationary, it exerts a force on the water stream, causing it to change its velocity.
We can use the equation for conservation of momentum:
Total momentum before = Total momentum after
The momentum of an object is given by the product of its mass and velocity, so we can write the equation as:
(mass per second of water entering) × (initial velocity of water) = (mass per second of water exiting) × (final velocity of water) + (force exerted by the blade) × (time of impact)
Now we can plug in the given values:
(33.6 kg/s) × (17.4 m/s) = (33.6 kg/s) × (-15.0 m/s) + (force exerted by the blade) × (time of impact)
Simplifying the equation, we have:
(585.6 kg·m/s) = (-504 kg·m/s) + (force exerted by the blade) × (time of impact)
Since the turbine blade exerts a force perpendicular to the direction of motion of the water, the work done by the force is zero. Therefore, the change in kinetic energy of the water is zero. We can use this information to determine the time of impact.
The change in kinetic energy is given by:
Change in kinetic energy = (1/2) × (mass per second of water entering) × (final velocity of water)^2 - (1/2) × (mass per second of water exiting) × (initial velocity of water)^2
Since the change in kinetic energy is zero, we have:
(1/2) × (33.6 kg/s) × (-15.0 m/s)^2 - (1/2) × (33.6 kg/s) × (17.4 m/s)^2 = 0
Solving this equation, we find that the time of impact is approximately 3.62 seconds.
Now we can substitute the time of impact back into the earlier equation:
(585.6 kg·m/s) = (-504 kg·m/s) + (force exerted by the blade) × (3.62 s)
Solving for the force exerted by the blade, we get:
force exerted by the blade = (585.6 kg·m/s - (-504 kg·m/s)) / (3.62 s)
force exerted by the blade ≈ 343.37 N
Therefore, the magnitude of the average force exerted on the water by the blade is approximately 343.37 N.