Why Does a Toy Airplane on a String Tilt When It Flies in Circles?

[kirafreedom]

Homework Statement

A particular type of toy airplane available in stores have little motors attached to them. Consider hanging one on a string from the ceiling and starting its motor. As it moves around in a circle with constant speed the stringwill make an angle with the vertical as shown in the diagram.
a) Assuming the plane has a mass m, find T, the tension in the string, in terms of m, θ, and g, by balancing the vertical forces on the plane.
b) Now use the equation for circular motion to solve for T in terms of m, θ, and v, (the speed of the plane), and l, (the length of the string).
c) By setting these two expressions for T equal to each other, find v in terms of θ, g, and l.
d) If you were to try this, you would find v difficult to measure directly. Write the period of rotation T, (not to beconfused with the tension), in terms of θ.

Homework Equations

T*cos(θ)=mg
r=l*sin(θ)
T*sin(θ)=m(v^2/r)

The Attempt at a Solution

a) T*cos(θ)=mg
T=mg/cos(θ)

b) T*sin(θ)=m(v^2/r)
T=[m(v^2/r)]/sin(θ)
r= [T*sin(θ)v^2]/m

r=l*sin(θ)
[T*sin(pheta)v^2]=l*sin(θ)
T=[l*sin(θ)m]/v^2

c) [l*sin(θ)m]/v^2=[m(v^2/r)]/sin(θ)
v=squart[(l*(sinθ)^2)/r]

d)
I am not sure

 

[tia89]

Ok for a)
BUT: pay attention to the math in b)... I didn't understand what you did but you did something wrong. Hint: it is simpler than that... you got the expression for $T$ right, then just substitute the expression for $r$ in terms of $l$ and $\theta$
c) Need correct b) first
d) Use the relation between period and velocity (keep in mind for you $v$ is the speed of the plane, therefore the tangential velocity) http://en.wikipedia.org/wiki/Circular_motion#Uniform_circular_motion

 

[haruspex]

b) T*sin(θ)=m(v^2/r)
T=[m(v^2/r)]/sin(θ)
r= [T*sin(θ)v^2]/m

Try that last step again.