Question

A 7180-kg helicopter accelerates upward at$\mathbf{0}\mathbf{.}\mathbf{80}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$While lifting a 1080-kg frame at a construction site, Fig. 4-68. (a) What is the lift force exerted by the air on the helicopter rotors? (b) What is the tension in the cable (ignore its mass) which connects the frame to the helicopter? (c) What force does the cable exert on the helicopter?

Figure 4-68Problem 79

Short answer

(a) The lift force acted by the air on the rotors will be 87556 N.

(b) The tension force in the cable connecting the frame to the helicopter is 11448 N.

(c) The magnitude of the force that the cable exerts on the helicopter in the downward direction is 11448 N.

Step-by-step solution

Step 1. Understanding line of action of friction force

Frictional force acts when there is relative motion between two bodies or the possibility of relative motion arises.

The direction of the frictional force, in this case, is such that it tries to stop that relative motion.

Step 2. Given data and assumptions

The mass of the helicopter,$M=7180\mathrm{kg}$

The mass of the frame,$m=1080\mathrm{kg}$

The acceleration of the system,$a=0.80\mathrm{m}/{\mathrm{s}}^2$

Let the tension in the cable connecting the helicopter to the frame be ${\overrightarrow{F}}_{\mathrm{T}}$, and the lift force be $F_{\mathrm{lift}}$.

Step 3. Calculating the lift force

Applying Newton’s second law on the system of the helicopter and frame in the vertical direction, you get:

$F_{\mathrm{lift}}-\left(Mg+mg\right)=\left(m+M\right)a$

Substituting the values in the above equation, you get:

$\begin{array}{c}{F}_{\mathrm{lift}}=\left(7180+1080\right)\left(9.8+0.80\right)\\ =87556\mathrm{N}\end{array}$

Thus, the magnitude of the lift force is 87556 N.

Step 4. Calculating the tension force in the cable

The tension force in the cable can be found out by analysing the motion of the frame. Given below is the FBD of the frame.

Applying Newton’s second law in the vertical direction, you get:

$\left|{\overrightarrow{F}}_{\mathrm{T}}\right|-mg=ma$

Substituting the values, you get:

$\left|{\overrightarrow{F}}_{\mathrm{T}}\right|-1080\times 9.8=1080\times 0.80$

Solving the above equation, you get:

$\left|{\overrightarrow{F}}_{\mathrm{T}}\right|=11448\mathrm{N}$

Thus, the tension force in the cable will be 11448 N.

Step 5. Calculating the force on the helicopter by the cable

As the cable is massless, the tension force in the cable will be the same throughout its entire length. Thus, the force exerted by the cable on the helicopter will be tension force.

$\left|{\overrightarrow{F}}_{\mathrm{T}}\right|=11448\mathrm{N}$