Answer to Question #141545 in Mechanics | Relativity for Fletcher Madden

Question

A special prototype model airplane of mass 400g has a control wire 8cm long attached to its body. The other end of the control line is attached to a fixed point. When the airplane flies with its wings horizontal in a horizontal circle, making one revolution every 4seconds, the control wire is elevated to 30 degrees above the horizontal. Draw a diagram showing the forces exerted on the plane and determine (i) the tension in the wire (ii) the lift on the plane ( Assume acceleration of free fall, g= 10m/s and 2=10)

Answer 1

Explanations & Calculations

• The length of the control wire should be 8m not 8cm
• $\small \omega=\small0.25rps =\small2\pi\times 0.25 rads^{-1}$

1. Horizontal component of the tension provides the centripetal force needed to move in the circle while vertical component balances the weight of the aircraft to keep it horizontal

Therefore, applying Newton's second law towards the center of the circular path,

$\qquad\qquad \begin{aligned} \small T\cos30 &= \small mr\omega^2=m(L\cos 30)\omega^2 \\ \small T &= \small mL\omega^2 \\ &=\small0.4kg\times 8m\times (2\pi\times 0.25)^2\\ &= \small \bold{7.896\,N} \end{aligned}$

• T can be calculated considering the vertical equilibrium also but as the g is approximated to be 10ms^-2 it gives T's value as 8N. If g is taken as 9.8ms^-2 T gets the normal value as calculated above.

2, The lift on the plane = $\small T\sin 30=7.896\times 0.5= 3.948\,N$