Calculating the Mass of Air Needed for a Helicopter to Hover

How can we determine the mass of air required for a helicopter to hover?

During a rescue operation, a 5300 kg helicopter hovers above a fixed point. The helicopter blades send air downward with a speed of 62.0 m/s. What mass of air must pass through the blades every second to produce enough thrust for the helicopter to hover?

Answer:

837.74 kilograms of air must pass through the blades every second to produce enough thrust for the helicopter to hover.

To determine the mass of air that must pass through the blades every second to produce enough thrust for the helicopter to hover, we can use the principle of conservation of momentum.

The thrust force exerted by the helicopter's blades is equal to the rate of change of momentum of the air. Mathematically, we can express this as:

Thrust = Rate of change of momentum

The momentum of an object is defined as the product of its mass and velocity. In this case, the mass of air passing through the blades every second will have a velocity equal to the downward speed of the air expelled by the blades.

Let's denote the mass of air passing through the blades every second as "m" (in kilograms) and the downward speed of the air as "v" (in meters per second).

The momentum change of the air per second is given by:

Rate of change of momentum = mv

Since the helicopter is hovering, the thrust force must be equal to the weight of the helicopter. The weight of an object is given by the formula:

Weight = mass x gravity

In this case, the weight of the helicopter is equal to its mass (5300 kg) multiplied by the acceleration due to gravity (9.8 m/s²).

Now, equating the thrust force to the weight of the helicopter:

m v = 5300 kg x 9.8 m/s²

Solving for "m":

m = (5300 kg x 9.8 m/s²) / v

Substituting the given value for the downward speed of the air (v = 62.0 m/s) into the equation:

m = (5300 kg x 9.8 m/s²) / 62.0 m/s

Therefore, approximately 837.74 kilograms of air must pass through the blades every second to produce enough thrust for the helicopter to hover.