A 7000 kg helicopter accelerates upward at 3 m/s^2. What lift force is exerted by the air on the helicopter’s propellers?
The Correct Answer and Explanation is:
Correct Answer:
The lift force exerted by the air on the helicopter’s propellers is 77,900 N.
Detailed Explanation:
To calculate the lift force acting on the helicopter, we need to consider both the gravitational force (weight) and the additional force required to accelerate the helicopter upward.
The forces acting on the helicopter are:
- Weight of the helicopter (W)
W=m×gW = m \times gW=m×g
Where:
m=7000 kgm = 7000 \, \text{kg}m=7000kg (mass of the helicopter)
g=9.8 m/s2g = 9.8 \, \text{m/s}^2g=9.8m/s2 (acceleration due to gravity) So:
W=7000×9.8=68,600 NW = 7000 \times 9.8 = 68,600 \, \text{N}W=7000×9.8=68,600N - Net Force Required for Upward Acceleration (F_net)
Fnet=m×aF_{\text{net}} = m \times aFnet=m×a
Where:
a=3 m/s2a = 3 \, \text{m/s}^2a=3m/s2 (upward acceleration) So:
Fnet=7000×3=21,000 NF_{\text{net}} = 7000 \times 3 = 21,000 \, \text{N}Fnet=7000×3=21,000N - Total Lift Force (F_lift)
The lift force must overcome both the weight of the helicopter and provide the additional upward force for acceleration. Using Newton’s Second Law:
Flift−W=FnetF_{\text{lift}} – W = F_{\text{net}}Flift−W=Fnet Rearranging the equation:
Flift=Fnet+WF_{\text{lift}} = F_{\text{net}} + WFlift=Fnet+W Substituting the known values:
Flift=21,000+68,600=89,600 NF_{\text{lift}} = 21,000 + 68,600 = 89,600 \, \text{N}Flift=21,000+68,600=89,600N Correction: There was a slight mistake above, let me correct it properly:
Step-by-step correction:
- Weight:
W=7000×9.8=68,600 NW = 7000 \times 9.8 = 68,600 \, \text{N}W=7000×9.8=68,600N - Net Force required for upward acceleration:
Fnet=7000×3=21,000 NF_{\text{net}} = 7000 \times 3 = 21,000 \, \text{N}Fnet=7000×3=21,000N - Lift force:
Flift=W+Fnet=68,600+21,000=89,600 NF_{\text{lift}} = W + F_{\text{net}} = 68,600 + 21,000 = 89,600 \, \text{N}Flift=W+Fnet=68,600+21,000=89,600N
Final Lift Force:
The correct lift force exerted by the air is 89,600 N.
Conclusion:
The helicopter experiences two forces: its weight pulling it downward and the lift from the air pushing it upward. For it to accelerate upward at 3 m/s23 \, \text{m/s}^23m/s2, the lift force must be greater than the gravitational force. This requires the propellers to exert a lift of 89,600 N, combining the force needed to counteract gravity and the extra force needed for acceleration.
