Airplane flying in a horizontal circle

[dorkymichelle]

Homework Statement

An airplane is flying in a horizontal circle at a speed of 410 km/h (Fig. 6-41). If its wings are tilted at angle a = 42° to the horizontal, what is the radius of the circle in which the plane is flying? Assume that the required force is provided entirely by an “aerodynamic lift” that is perpendicular to the wing surface.

Homework Equations

A_c = v² / r
F=ma

The Attempt at a Solution

I did this prob with a friend today and then again on my own still didn't get it right... Please help, I have a test tomorrow lol
I am going to use L as the force that's providing the lift.
and a as the angle.
verticle component of L has to equal to mg or else the plane will fall down.
horizontal component of L is making the plane go in a circle
so L cosa - mg = 0
L sin a = m*v² /r
Lcos a = mg
L = mg/cos a
plugging back into L sin a = m*v² /r
i got (mg/cosa)*sina = m*v²/r
the m's cancel
so tan a*g = v²/r
r = v²/(tan a *g)
plugging in numbers
i got r = 410²/tan42*9.8
= 168100/8.8239 = 19050.5 round to 3 significant figures, 19000
please look over and correct mistakes

 

[jhae2.718]

Check your units. Your final expression currently is: $$r = \frac{(410\,\rm km/hr)^2}{\tan(42^{\circ}) \cdot 9.8 \, \rm m/s^2}$$. This gives your answer in $$\frac{\rm km^2 \cdot s^2}{\rm m \cdot hr^2}$$

 

[dorkymichelle]

ah! units are going to be the death of me! thankyou!

 

[jhae2.718]

No prob. It doesn't hurt to put units in your calculations. In physics, quantities without units are meaningless. (Unless they're dimensionless!)

 

[rwisz]

or explain where im going wrong

I would like to point out that this solution assumes that there are no external forces acting on the plane, such as wind or air resistance. In reality, these external forces would affect the flight path and require additional calculations to accurately determine the radius of the circle. Additionally, the solution assumes that the lift force is solely perpendicular to the wing surface, which may not always be the case. It is important to consider all factors and assumptions when solving a physics problem.