The helicopter in the drawing is moving horizontally to the right at a constant velocity. The weight of the helicopter is W=41600 N. The lift force L generated by the rotating blade makes an angle of 21.0° with respect to the vertical. (a) What is the magnitude of the lift force? (b) Determine the magnitude of the air resistance R that opposes the motion.
find
Lx = Lsin21
Ly = Lcos21
y direction,
Lcos21 - 55600 = 0
L =44559.63N
x direction
Lsin21 - Fair = 0
where F air= 15968.74N
To solve this problem, we will use trigonometry to find the magnitudes of the lift force and air resistance.
Given:
Weight of the helicopter, W = 41600 N
Angle of the lift force with respect to the vertical, θ = 21.0°
(a) Magnitude of the lift force:
The weight acts vertically downward, so the vertical component of the lift force will balance the weight.
The vertical component of the lift force, L_vertical = W
The magnitude of the lift force, L = L_vertical / sin(θ)
L = W / sin(θ)
L = 41600 N / sin(21.0°)
Using a calculator, we can find:
L ≈ 122483 N
Therefore, the magnitude of the lift force is approximately 122483 N.
(b) Magnitude of the air resistance:
Since the helicopter is moving horizontally to the right at a constant velocity, the air resistance will balance the horizontal component of the lift force.
The horizontal component of the lift force, L_horizontal = L * cos(θ)
L_horizontal = 122483 N * cos(21.0°)
Using a calculator, we can find:
L_horizontal ≈ 114619 N
Therefore, the magnitude of the air resistance is approximately 114619 N.
To answer these questions, we need to break down the forces acting on the helicopter.
(a) To find the magnitude of the lift force, we can use trigonometry. Since the lift force makes an angle of 21.0° with respect to the vertical, we can decompose it into two components: one along the vertical direction and one along the horizontal direction.
The vertical component of the lift force can be calculated using the equation:
L_vertical = L * sin(angle),
where L is the magnitude of the lift force and angle is the angle the lift force makes with respect to the vertical.
Therefore, L_vertical = L * sin(21.0°).
(b) To determine the magnitude of the air resistance, we need to consider that it opposes the motion of the helicopter. In this case, since the helicopter is moving horizontally to the right, the air resistance will act in the opposite horizontal direction.
The air resistance can be calculated using the equation:
R = W - L_horizontal,
where W is the weight of the helicopter, and L_horizontal is the horizontal component of the lift force.
Since the helicopter is moving at a constant velocity, the net force acting on it is zero. Therefore, the magnitude of the air resistance is equal to the magnitude of the horizontal component of the lift force.
To find the horizontal component of the lift force, we can use trigonometry again:
L_horizontal = L * cos(angle),
where L is the magnitude of the lift force and angle is the angle the lift force makes with respect to the vertical.
Therefore, L_horizontal = L * cos(21.0°).
Now we can substitute the values into the equations to find the answers.
(a) The magnitude of the lift force, L, is given as 41600 N. Therefore, we can find the vertical component as:
L_vertical = 41600 N * sin(21.0°).
(b) To find the magnitude of the air resistance, R, we need to calculate the horizontal component of the lift force, L_horizontal, and subtract it from the weight of the helicopter, W:
R = 41600 N - L_horizontal.
By plugging in the values obtained in step (a) and using the equations given above, we can find the magnitude of the air resistance, R, that opposes the motion of the helicopter.