Interactive Lecture Demonstration 8 — Toy Airplane

In this demonstration, a toy airplane hanging from a string will move in a horizontal circle at a constant speed. By measuring the period and radius of the airplane's motion, we will be able to calculate the angle that the string should make with the vertical, and compare our calculated angle with the angle we measure directly.

1. Draw your diagram below. In order to show the angle that the string makes with the vertical direction, you will need to draw a SIDE VIEW of the situation, and you will need to make your diagram a snapshot of the airplane as it moves either directly into or directly out of the paper. The airplane should therefore be at either the left or right side of your diagram. Let's choose to draw the plane at the right side of the diagram. Label on your diagram the angle of the string with the vertical, the center of the circle that the airplane makes, the radius of the circle, the direction of the airplane's acceleration, and the direction of the airplane's motion.

Next to your diagram, draw the force diagram for the airplane when it is at the location you drew in the diagram.

2. We hope you realized that the direction of the acceleration of the airplane is towards the center of the circle, and that for this situation the center of the circle is directly to the left of the airplane's location and directly underneath the point where the string is attached to the ceiling. The acceleration is thus in a horizontal direction. You must therefore choose one of your positive directions to be in that same horizontal direction. Your other positive direction must be vertical, either up or down. Label your positive directions next to your force diagram. Then draw a force triangle (vector diagram) and break into components any force that does not point along one of your positive directions. Label on your force triangle the angle that is the same as the angle that the string makes with the vertical direction.

3. Now follow the usual procedure to write the net force equations for the airplane. Remember to use the correct formula for the centripetal acceleration of the airplane. What will you put for the vertical acceleration ?

By this point, this problem should look very familiar. The force diagram and net force equations are identical to those for the BB's in the horizontal accelerometer that we already discussed. This problem is also identical to Example 7.7 in the textbook. All this means you should be finding this problem to be very easy.

4. We want to determine the angle of the string. Just as we did with the horizontal accelerometer problem, we can divide our two net force equations to eliminate the unknown variables T and mg. Do that now to find an equation for the angle in terms of the speed of the airplane (v), radius of the circle (R) and Earth's gravitational field strength (g).

5. No we want to predict the angle, so we will need to measure v and R. The speed can be determined by measuring the radius of the circle and the time it takes the airplane to complete one revolution (i.e. that time is called the period P of the motion). Record below your measurements of R and P. Then use your measurements to calculate the speed and the centripetal acceleration of the airplane. Finally, use your formula from #4 to calculate the angle the string makes with the vertical.    

6. Now we will measure the actual angle the string makes with the vertical, using the horizontal accelerometer as a protractor. Record your measurement below and compare it with the value you calculated.