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Problem : An aircraft hovers over a city awaiting clearnace to land. The aricraft circles with its wings banked at an angle ${\mathrm{tan}}^{-1}\left(0.2\right)$ at a speed of 200 m/s. Find the radius of the loop.
Solution : The aircraft is banked at an angle with horizontal. Since aircraft is executing uniform circular motion, a net force on the aircraft should act normal to its body. The component of this normal force in the radial direction meets the requirement of centripetal force, whereas vertical component balances the weight of aircraft. Thus, this situation is analogous to the banking of road.
$$\begin{array}{l}\Rightarrow \mathrm{tan}\theta =\frac{{v}^{2}}{rg}\\ \Rightarrow r=\frac{{v}^{2}}{g\mathrm{tan}\theta}\end{array}$$
$$\begin{array}{l}\Rightarrow r=\frac{{200}^{2}}{10x0.2}=20000\phantom{\rule{2pt}{0ex}}m\end{array}$$
The moot question is whether banking of road achieves the objectives of banking? Can we negotiate the curve with higher speed than when the road is not banked? In fact, the expression of speed as derived in earlier section gives the angle of banking for a particular speed. It is the speed for which the component of normal towards the center of circle matches the requirement of centripetal force.
If speed is less than that specified by the expression, then vehicle will skid "down" (slip or slide) across the incline as there is net force along the incline of the bank. This reduces the radius of curvature i.e. "r" is reduced - such that the relation of banking is held true :
$$\begin{array}{l}\Rightarrow v=\surd \left(rg\mathrm{tan}\theta \right)\end{array}$$
In reality, however, the interacting surfaces are not smooth. We can see that if friction, acting "up" across the bank, is sufficient to hold the vehicle from sliding down, then vehicle will move along the circular path without skidding "down".
What would happen if the vehicle exceeds the specified speed for a given angle of banking? Clearly, the requirement of centripetal force exceeds the component of normal force in the radial direction. As such the vehicle will have tendency to skid "up" across the bank.
Again friction prevents skidding "up" of the vehicle across the bank. This time, however, the friction acts downward across the bank as shown in the figure.
In the nutshell, we see that banking helps to prevent skidding "up" across the bank due to the requirement of centripetal force. The banking enables component of normal force in the horizontal direction to provide for the requirement of centripetal force up to a certain limiting (maximum) speed. Simultaneously, the banking induces a tendency for the vehicle to skid "down" across the bank.
On the other hand, friction prevents skidding "down" as well as skidding "up" across the bank. This is possible as friction changes direction opposite to the tendency of skidding either "up" or "down" across the bank. The state of friction is summarized here :
1: $v=0;\phantom{\rule{1em}{0ex}}{f}_{S}=mg\mathrm{sin}\theta ,\phantom{\rule{1em}{0ex}}\text{acting up across the bank}$
2: $v=\sqrt{rg\mathrm{tan}\theta};\phantom{\rule{1em}{0ex}}{f}_{S}=0$
3: $v<\sqrt{rg\mathrm{tan}\theta};\phantom{\rule{1em}{0ex}}{f}_{S}>\mathrm{0,}\phantom{\rule{1em}{0ex}}\text{acting up across the bank}$
4: $v>\sqrt{rg\mathrm{tan}\theta};\phantom{\rule{1em}{0ex}}{f}_{S}>\mathrm{0,}\phantom{\rule{1em}{0ex}}\text{acting down across the bank}$
Friction, therefore, changes its direction depending upon whether the vehicle has tendency to skid "down" or "up" across the bank. Starting from zero speed, we can characterize friction in following segments (i) friction is equal to the component of weight along the bank," $\sqrt{mg\mathrm{sin}\theta}$ ", when vehicle is stationary (ii) friction decreases as the speed increases (iii) friction becomes zero as speed equals " $\sqrt{rg\mathrm{tan}\theta}$ " (iv) friction changes direction as speed becomes greater than " $\sqrt{rg\mathrm{tan}\theta}$ " (v) friction increases till the friction is equal to limiting friction as speed further increases and (vi) friction becomes equal to kinetic friction when skidding takes place.
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