A model airplane is flying in a horizontal circle with a constant speed. The initial radius of the circle is R. The boy holding the cord to which the airplane is attached, then decides to increase the length of the cord so that the radius of the circle increases to 2R. The speed of the airplane does not change. How does the final centripetal acceleration of the airplane when the radius is 2R compare to the initial centripetal acceleration of the airplane when the radius is R?
The final centripetal acceleration is one-fourth the initial centripetal acceleration.
The final centripetal acceleration is two times the initial centripetal acceleration.
The final centripetal acceleration is four times the initial centripetal acceleration.
The final and the initial centripetal accelerations have the same value.
The final centripetal acceleration is one-half the initial centripetal acceleration.
To determine how the final centripetal acceleration of the airplane when the radius is 2R compares to the initial centripetal acceleration of the airplane when the radius is R, we need to understand the relationship between centripetal acceleration, speed, and radius.
Centripetal acceleration can be calculated using the formula:
a = v^2 / r
Where:
a is the centripetal acceleration
v is the speed of the object
r is the radius of the circle
Given that the speed of the airplane remains constant, we can see that the centripetal acceleration is inversely proportional to the radius of the circle. As the radius increases, the centripetal acceleration decreases, and vice versa.
In this case, when the initial radius is R and the final radius is 2R, we can see that the final radius is twice the initial radius. Therefore, the final centripetal acceleration should be half the initial centripetal acceleration.
Therefore, the correct answer is:
The final centripetal acceleration is one-half the initial centripetal acceleration.