A remote control model airplane connected to a string is flown in a circle with a radius of 0.348 m about a fixed axis with an angular velocity of +1.47 rad/s. The plane’s throttle is increased, so that it acquires an angular acceleration of 2.18 rad/s2. After 0.490 s have elapsed since the plane sped up, what is the total acceleration ( ar) (in m/s2) of the airplane?
To find the total acceleration of the airplane, we need to consider two components of acceleration: centripetal acceleration and tangential acceleration.
1. Centripetal acceleration (ac):
The centripetal acceleration is the acceleration that keeps an object moving in a circular path. It is given by the formula ac = (v^2) / r, where v is the linear velocity and r is the radius of the circular path.
To calculate the linear velocity (v), we can use the formula v = r * ω, where ω is the angular velocity. Given ω = 1.47 rad/s and r = 0.348 m, we can calculate v as:
v = 0.348 m * 1.47 rad/s = 0.51156 m/s
Now, we can find the centripetal acceleration (ac):
ac = (0.51156 m/s)^2 / 0.348 m = 0.750297 m/s^2
2. Tangential acceleration (at):
The tangential acceleration is the acceleration due to the change in angular velocity. It is given by the formula at = r * α, where α is the angular acceleration.
Given α = 2.18 rad/s^2 and r = 0.348 m, we can find the tangential acceleration (at):
at = 0.348 m * 2.18 rad/s^2 = 0.75864 m/s^2
3. Total acceleration (ar):
The total acceleration is the vector sum of the centripetal and tangential acceleration. Since the two accelerations are perpendicular to each other, we can find the total acceleration (ar) using the Pythagorean theorem:
ar = √(ac^2 + at^2)
ar = √(0.750297 m/s^2)^2 + (0.75864 m/s^2)^2
ar = √0.562985719 m^2/s^4 + 0.575945049 m^2/s^4
ar = √1.138930768 m^2/s^4
ar = 1.067 m/s^2
Therefore, the total acceleration (ar) of the airplane after 0.490 s is approximately 1.067 m/s^2.