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Chapter 4: Problem 54

What force do the blades of a \(4300-\mathrm{kg}\) helicopter exert on the air when the helicopter is (a) hovering at constant altitude; (b) dropping at \(21 \mathrm{m} / \mathrm{s}\) with speed decreasing at \(3.2 \mathrm{m} / \mathrm{s}^{2} ;(\mathrm{c})\) rising at \(17 \mathrm{m} / \mathrm{s}\) with speed increasing at \(3.2 \mathrm{m} / \mathrm{s}^{2} ;\) (d) rising at a steady \(15 \mathrm{m} / \mathrm{s} ;\) (e) rising at \(15 \mathrm{m} / \mathrm{s}\) with speed decreasing at \(3.2 \mathrm{m} / \mathrm{s}^{2} ?\)

Short Answer

Expert verified
The force the helicopter blades exert on the air is (a) \(4.2 \times 10^4 N\) for hovering, (b) \(2.972 \times 10^4 N\) when dropping with speed decreasing, (c) \(5.462 \times 10^4 N\) when rising with speed increasing, (d) \(4.2 \times 10^4 N\) when rising steadily, and (e) \(2.972 \times 10^4 N\) when rising with speed decreasing.

Step by step solution

01

Identify the given values

Firstly, you have to identify all the given values in the problem. For example, the mass of the helicopter \(m = 4300 kg\), the first acceleration \(a1 = -3.2 m/s^2\), the second acceleration \(a2 = 3.2 m/s^2\) and the gravitational acceleration \(g = 9.81 m/s^2\).
02

Apply Newton's Second Law for hovering helicopter

The force exerted by the blades of the hovering helicopter is equal to the weight of the helicopter. Use the equation \(F = m \cdot g\) where \(g\) is the acceleration due to gravity. This will be beneficial to find the force exerted on the helicopter in the first scenario.
03

Calculate force for dropping helicopter

In the second scenario, we need to calculate the force when the helicopter is dropping but speed is decreasing. This means, helicopter is being accelerated upwards. Hence total acceleration of the helicopter will be an addition of gravitational acceleration and the upwards acceleration. So overall force will be \(F = m \cdot (g - a1)\).
04

Calculate force for rising helicopter with increasing speed

In the third scenario, the helicopter is rising and its speed is increasing. The force exerted can be calculated by using \(F = m \cdot (g + a2)\).
05

Calculate force for rising helicopter at a steady speed

As the helicopter rises steadily in the fourth scenario, its force equals its weight because the speed is constant meaning that there is no acceleration. So, the force is \(F = m \cdot g\).
06

Calculate force for rising helicopter with decreasing speed

In the last scenario, helicopters is rising but the speed is decreasing. This equals to accelerating downwards. The force exerted by the helicopter when rising can be calculated by \(F = m \cdot (g - a1)\).

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Most popular questions from this chapter

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