A 4600 kg helicopter accelerates upward at 2.0 m/s2. The force of gravity acting upon the helicopter is 46,000 N. Determine the lift force exerted on the propellers by the air. Make a free-body diagram. Write a net force equation for the axis along which forces are not balanced.
The Correct Answer and Explanation is:
Correct Answer:
Given:
- Mass of helicopter, m = 4600 kg
- Acceleration upward, a = 2.0 m/s²
- Gravitational force (weight), W = 46,000 N
- Lift force exerted by air = F_lift
- Gravitational acceleration, g ≈ 10 m/s² (Note: confirmed from W = mg → 4600 kg × 10 m/s² = 46,000 N)
Free-Body Diagram Description:
- Upward Force: Lift force (F_lift)
- Downward Force: Gravitational force (W = 46,000 N)
scssCopyEdit ↑ F_lift
│
│
[HELICOPTER]
│
↓ 46,000 N (Weight)
Net Force Equation (Vertical Axis):
The net force is given by Newton’s Second Law:
F_net = m × a
The net force is the difference between the upward lift force and the downward gravitational force:
F_lift – W = m × a
Rearranged for F_lift:
F_lift = W + m × a
Substitute known values:
F_lift = 46,000 N + (4600 kg × 2.0 m/s²)
F_lift = 46,000 N + 9,200 N
F_lift = 55,200 N
Final Answer:
The lift force exerted on the propellers by the air is 55,200 N.
Explanation:
In this problem, a helicopter with a mass of 4600 kg is accelerating upward at 2.0 m/s². To find the lift force generated by the air acting on the helicopter’s propellers, we apply Newton’s Second Law, which states that the net force on an object equals its mass multiplied by its acceleration.
First, calculate the gravitational force acting downward. The problem already provides this value as 46,000 N, which aligns with the calculation using the approximation for gravitational acceleration, where:
W = m × g = 4600 kg × 10 m/s² = 46,000 N
Since the helicopter accelerates upward, the lift force must be greater than the gravitational force to overcome the weight and provide additional upward acceleration. The net force is the difference between the upward lift force and the downward gravitational force:
F_net = F_lift – W
By rearranging the formula and substituting known values, the lift force becomes:
F_lift = W + m × a
Plug in the numbers:
F_lift = 46,000 N + (4600 kg × 2.0 m/s²) = 46,000 N + 9,200 N = 55,200 N
Thus, to achieve an upward acceleration of 2.0 m/s², the air must exert a lift force of 55,200 N on the helicopter’s propellers. This ensures the helicopter not only balances its own weight but also accelerates upward as required.
