During a rescue operation, a 5350 kg helicopter hovers above a fixed point. The helicopter blades send air downward with a speed of 59.0 m/s.
What mass of air must pass through the blades every second to produce enough thrust for the helicopter to hover?
impulse*time= mass*velocity
weight= mass/time*velocity
mass/time=weight/velocity= mg/59
Well, to get to the bottom of this, let's take things step by step. First, we need to determine the velocity of the air being pushed by the blades. Given that it's 59.0 m/s, I hope those blades have a good sense of direction, or else we could have some serious wind-related mishaps!
Now, thrust is essentially the force exerted by the moving air, which is equal and opposite to the force exerted by the helicopter to stay aloft. To calculate the mass of air passing through the blades every second, we need to divide the force exerted by the velocity of the air.
The force, in this case, would be the weight of the helicopter, which we can calculate by multiplying its mass (5350 kg) by the acceleration due to gravity (9.8 m/s^2). Let's call this force F.
So, now we have F divided by the velocity of the air (59.0 m/s), which gives us the mass of air passing through the blades every second. Voilà!
To find the mass of air passing through the blades every second, we can use the principle of conservation of momentum.
The thrust force generated by the helicopter is equal to the rate of change of momentum of the air.
Thrust Force = Rate of Change of Momentum
The momentum of the air is given by the mass of the air (m_air) multiplied by its velocity (v_air).
Momentum of Air = m_air * v_air
Since the helicopter is hovering, the change in momentum of the air each second is equal to zero, as there is no acceleration in the vertical direction.
Thrust Force = Rate of Change of Momentum = 0
Therefore, the thrust force generated by the helicopter is zero.
To hover, the helicopter must generate enough upward thrust force to balance its weight.
Thrust Force = Weight of the Helicopter
Weight of the Helicopter = mass of the helicopter (m_helicopter) * acceleration due to gravity (g)
Let's assume the acceleration due to gravity, g, is approximately equal to 9.8 m/s^2.
Thrust Force = m_helicopter * g
Now, we can equate the thrust force to the weight of the helicopter and solve for the mass of the air passing through the blades every second.
m_air * v_air = m_helicopter * g
Rearranging the equation, we get:
m_air = (m_helicopter * g) / v_air
We're given:
- Mass of the helicopter (m_helicopter) = 5350 kg
- Velocity of the air (v_air) = 59.0 m/s
- Acceleration due to gravity (g) ≈ 9.8 m/s^2
Substituting the given values into the equation:
m_air = (5350 kg * 9.8 m/s^2) / 59.0 m/s
Calculating the value using the equation:
m_air ≈ 885.593 kg
Therefore, approximately 885.593 kg of air must pass through the helicopter blades every second to produce enough thrust for the helicopter to hover.
To find the mass of air that must pass through the helicopter blades every second to produce enough thrust for the helicopter to hover, we can use the concept of conservation of momentum.
The thrust force produced by the helicopter blades is equal to the rate of change of momentum of the air mass being moved downward. Mathematically, it can be expressed as:
Thrust = Rate of change of momentum
The momentum of an object is given by the product of its mass and velocity:
Momentum = mass * velocity
For the air being moved downward by the helicopter blades, the velocity is given as 59.0 m/s.
To calculate the mass of the air, we rearrange the equation:
Mass = Thrust / Velocity
The thrust force required for the helicopter to hover is equal to the weight of the helicopter, which can be calculated as:
Weight = mass * gravity
The mass of the helicopter is given as 5350 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
Therefore, the thrust force is:
Thrust = Weight = mass * gravity
Now, substituting the given values:
Thrust = 5350 kg * 9.8 m/s^2
Next, we can substitute the value of thrust into the mass equation:
Mass = Thrust / Velocity
Mass = (5350 kg * 9.8 m/s^2) / 59.0 m/s
Simplifying the expression:
Mass = 87700 kg⋅m/s / 59.0 m/s
Finally, calculating the value:
Mass ≈ 1486 kg
So, the mass of air that must pass through the helicopter blades every second to produce enough thrust for the helicopter to hover is approximately 1486 kg.