To determine the lift force exerted by the air on the propellers of a 4500 kg helicopter accelerating upward at [tex]\(2.0 \, \text{m/s}^2\)[/tex], follow these steps:
1. Determine the weight of the helicopter:
- The weight ([tex]\(W\)[/tex]) is the force due to gravity and can be calculated using the formula:
[tex]\[
W = m \cdot g
\][/tex]
where [tex]\(m\)[/tex] is the mass of the helicopter and [tex]\(g\)[/tex] is the acceleration due to gravity.
- Given:
[tex]\[
m = 4500 \, \text{kg}
\][/tex]
[tex]\[
g = 9.8 \, \text{m/s}^2
\][/tex]
- Therefore, the weight is:
[tex]\[
W = 4500 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 44100 \, \text{N}
\][/tex]
2. Determine the net force needed for the upward acceleration:
- The net force ([tex]\(F_{\text{net}}\)[/tex]) required to produce an upward acceleration can be found using Newton's second law:
[tex]\[
F_{\text{net}} = m \cdot a
\][/tex]
where [tex]\(a\)[/tex] is the acceleration of the helicopter.
- Given:
[tex]\[
a = 2.0 \, \text{m/s}^2
\][/tex]
- Therefore, the net force is:
[tex]\[
F_{\text{net}} = 4500 \, \text{kg} \cdot 2.0 \, \text{m/s}^2 = 9000 \, \text{N}
\][/tex]
3. Determine the total lift force:
- The lift force ([tex]\(F_{\text{lift}}\)[/tex]) must overcome both the weight of the helicopter and provide the additional force for the upward acceleration. Hence:
[tex]\[
F_{\text{lift}} = W + F_{\text{net}}
\][/tex]
- We have already calculated:
[tex]\[
W = 44100 \, \text{N}
\][/tex]
[tex]\[
F_{\text{net}} = 9000 \, \text{N}
\][/tex]
- Therefore, the total lift force is:
[tex]\[
F_{\text{lift}} = 44100 \, \text{N} + 9000 \, \text{N} = 53100 \, \text{N}
\][/tex]
Hence, the lift force exerted by the air on the propellers is [tex]\(53,100 \, \text{N}\)[/tex].
