Question

A helicopter of mass $1000kg$ rises with a vertical acceleration of $15m/s^2$. The crew and the passengers weigh $300kg$. Give the magnitude and direction of the

1. Force on the floor by the crew and passengers.
2. Action of the rotor of the helicopter on the surrounding air.
3. Force on the helicopter due to the surrounding air.

Answer 1

Step 1. Given data

Mass of the helicopter is, $1000kg$
The total mass of crew and passengers is, $300kg$
Acceleration of the helicopter, $a=15m/s^2$
The total mass of the system, $m_t=1300kg$

Step 2. Formula used

$\mathrm{R}-\mathrm{mg}=\mathrm{ma}$

where $\mathrm{R}$ is the reaction force on the floor, $\mathrm{m}$ mass, $\mathrm{a}$ acceleration and $\mathrm{g}$is the acceleration due to gravity

Step 3. Calculate the force exerted on the floor

The reaction force applied by the crew and the passenger on the floor is given by,

$$\begin{gathered} \mathrm{R}-\mathrm{mg}=\mathrm{ma} \\ \Rightarrow \mathrm{R}=\mathrm{m}\left(\mathrm{a}+\mathrm{g}\right) \\ \Rightarrow \mathrm{R}=300\times \left(10+15\right) \\ \Rightarrow \mathrm{R}=7500\mathrm{N} \end{gathered}$$

So, the force exerted by the passengers on the floor is $7500\mathrm{N}$.

Step 4. Calculate the action of the rotor of the helicopter on the surrounding air

The reaction of the rotor to the surrounding air will be due to the mass of the helicopter as well as the passengers.

$$\begin{gathered} \mathrm{R}\text{'}-{\mathrm{m}}_{\mathrm{t}}\mathrm{g}={\mathrm{m}}_{\mathrm{t}}\mathrm{a} \\ \Rightarrow \mathrm{R}\text{'}={\mathrm{m}}_{\mathrm{t}}\left(\mathrm{a}+\mathrm{g}\right) \\ \Rightarrow \mathrm{R}\text{'}=1300\times \left(10+15\right) \\ \Rightarrow \mathrm{R}\text{'}=13500\mathrm{N} \end{gathered}$$

Thus, the reaction force of the rotor on the surrounding air will be $32500\mathrm{N}$.

Step 5. Calculate the force on the helicopter due to the surrounding air.

The force on the helicopter due to the surrounding air will be the reaction of the force applied by the rotor to the air. Therefore, its value will be $32500\mathrm{N}$.

Hence,

1. Force on the floor by the crew and passengers is $7500\mathrm{N}$.
2. Action of the rotor of the helicopter on the surrounding air is $32500\mathrm{N}$.
3. Force on the helicopter due to the surrounding air is $32500\mathrm{N}$.