Question

Question: A toy airplane hangs from the ceiling at the bottom end of a string. You turn the airplane many times to wind up the string clockwise and release it. The airplane starts to spin counter clockwise, slowly at first and then faster and faster. Take counter clockwise as the positive sense and assume friction is negligible. When the string is entirely unwound, the airplane has its maximum rate of rotation. (i) At this moment, is its angular acceleration (a) positive, (b) negative, or (c) zero? (ii) The airplane continues to spin, winding the string counter clockwise as it slows down. At the moment it momentarily stops, is its angular acceleration (a) positive, (b) negative, or (c) zero?

Short answer

Answer

(i) The angular acceleration of an airplane, when it has maximum rate of rotation, is zero. Hence, option (c) is correct answer.

(ii) The angular acceleration of an airplane, when it winds the string counter clockwise as it slows down is negative. Hence, option (b) is correct answer.

Step-by-step solution

Defining angular acceleration

Angular acceleration is defined as the time rate of change of angular velocity. It is calculated by using the formula given below:

$\alpha =\frac{d\omega }{dt}=\frac{\Delta \omega }{\Delta t}$

Here$\Delta \omega$ is change in angular velocity and is change in time.

Calculating angular acceleration when the airplane has maximum rate of rotation

Given that counter clockwise direction should be taken positive and clockwise direction as negative. At maximum rate of rotation, change in angular velocity is zero because at maximum rate of rotation, initial and final velocities are same for a small interval . Hence, put $\omega =0$, we get

$\alpha =\frac{d\omega }{dt}=0$

Thus, angular acceleration of an airplane is zero when it has maximum rate of rotation. Hence, option(c) is the correct answer.

Calculating angular acceleration when the airplane winds the string counter clockwise as it slows down

Though the angular speed is zero at the given instant, there is angular acceleration since the wound-up string applies a torque on the airplane. It is similar to the ball which is thrown upwards. At the top of its flight, it momentarily comes to rest, but is still accelerating because of the gravitational force acting on it. Hence, option (b) is the correct answer.