As your plane speeds down the runway during takeoff, you decide to determine the acceleration of the plane. you take your yo-yo and note that when you suspend it, the string makes an angle of 22 degrees with the vertical. What is the acceleration of the plane?
g*tan(22°)
That is assuming one has a way to find the true vertical in an accelerating plane and which is possibly taking off. Most high-rises are not allowed near an airport.
To determine the acceleration of the plane, we first need to understand the forces acting on the yo-yo when it is suspended. The angle between the string and the vertical direction can provide us with this information.
When the yo-yo is suspended, two forces act on it: the tension in the string (T) and the force of gravity (mg), where m represents the mass of the yo-yo and g represents the acceleration due to gravity (approximately 9.8 m/s²).
The vertical component of the tension force (T_vertical) balances the force of gravity, while the horizontal component of the tension force (T_horizontal) provides the centripetal force necessary to keep the yo-yo in circular motion.
Since the yo-yo is not accelerating vertically (it remains at rest), the vertical component of the tension force is equal in magnitude but opposite in direction to the force of gravity:
T_vertical = mg
Using trigonometry, we can express T_vertical and T in terms of the angle 22 degrees:
T_vertical = T * cos(22°)
Furthermore, since the yo-yo is in circular motion, the centripetal force T_horizontal can be expressed as:
T_horizontal = m * a,
where a is the acceleration of the yo-yo in the horizontal direction.
The centripetal force required to sustain circular motion can also be expressed as:
T_horizontal = m * v² / r,
where v is the linear velocity of the yo-yo, and r represents the radius of the circular path.
Since the yo-yo is stationary (suspended), its linear velocity (v) is zero, which means the centripetal force is also zero:
T_horizontal = 0.
Now, combining the equations for T_vertical and T_horizontal, we have:
T * cos(22°) = 0.
Simplifying the equation, we find:
T = 0.
Thus, we can conclude that when the yo-yo is suspended and the string makes an angle of 22 degrees with the vertical, the tension in the string is zero.
This implies that there is no upward force acting on the plane. The only vertical force acting on the plane during takeoff is the force of gravity. Therefore, the acceleration of the plane can be assumed to be 9.8 m/s², which is the acceleration due to gravity.