During a rescue operation, a 5600-kg helicopter hovers above a fixed point. The helicopter blades send air downward with a speed of 69 m/s. What mass of air must pass through the blades every second to produce enough thrust for the helicopter to hover?
Strictly speaking, you also need to know the vertical velocity component of the air entering the blades from above. It is not zero.
They probably want you to use
Weight = V * dm/dt,
and solve for dm/dt, the air mass flow through the blades, but this is a flawed problem.
I do not endorse their answer
To find the mass of air that must pass through the helicopter blades every second, we can use the concept of thrust.
Thrust is the force that propels an object forward, and it is produced by the reaction of expelling mass in the opposite direction. According to Newton's third law of motion, for every action, there is an equal and opposite reaction.
The thrust force generated by the helicopter blades is equal to the rate of change of momentum of the air mass passing through them. The momentum of an object is given by its mass multiplied by its velocity.
In this case, the momentum change will be equal to the mass of air passing through the blades every second multiplied by the change in velocity of the air.
Given:
Mass of the helicopter, m = 5600 kg
Velocity of the air, v = 69 m/s
To calculate the mass of air passing through the blades every second, we need to determine the thrust force required to keep the helicopter hovering.
Since the helicopter is at rest vertically, the thrust force must be equal to the weight of the helicopter to balance it.
Weight, W = m*g
where g is the acceleration due to gravity (approximately 9.8 m/s^2)
To calculate the thrust force, we can use the formula:
Thrust = mass of air passing through the blades per second * change in velocity of the air
Thrust = m_air * ∆v
Since the helicopter is hovering, its weight is balanced by the upward thrust generated by the blades. Therefore, the thrust force is equal to the weight of the helicopter:
Thrust = Weight = m*g
Now, we can equate it with the mass of air passing through the blades per second multiplied by the change in velocity:
m*g = m_air * ∆v
Re-arranging the equation to solve for the mass of air passing through the blades per second:
m_air = (m*g) / ∆v
Substituting the given values:
m_air = (5600 kg * 9.8 m/s^2) / (69 m/s)
Calculating the mass of air passing through the helicopter's blades per second:
m_air = 784 kg/s
Therefore, to keep the helicopter hovering, approximately 784 kg of air must pass through the blades every second.