Phenomenon: A ball is swinging in a horizontal circle at constant speed supported by a string. Experiment shows that as the ball moves faster, 9 becomes larger. Here's the question we'd like to answer: how does the speed of the hall depend on the angle 8 that the string makes with the vertical Application of Newton's Laws: circular motion direction? 1) Based on the previous activity on circular motion (Activity 13 from DL Ill). what is the direction of acceleration for an object traveling in a circle at constant speed'ill 2) How many forces are acting on the hall? 3) Put a properly scaled force diagram for the ball on the board. Is your diagram consistent with your answer in I)? How do you know? Whole Class Discussion Follow the steps below to develop a mathematical relationship between speed and the angle 6: 4) 5) 6) 7) 3} 9} Use the force diagram and Newton's 2'\"1 Law to develop two equations {one for the vertical direction and one for the horizontal direction). 1 \"tangential The mgoirnde of acceleration for an object moving in a circle is summer\" = r (notice that the velocity in this expression is the tangential speed of the ball). Plug this acceleration into the appropriate relationship from 4). The magnitude of the tension force is hard to measure. so we would like to eliminate it from our equations. Use your two mathematical relationships to eliminate the tension force. leaving you with one equation. The tangential speed 12; is difficult to measure directly. but the angular speed of the bail to is straightforward to measure with a timer. Use the relationship between tit and to (in = no) to eliminate Vi- The radius of the circle 1" is also difcult to measure, but we can easily measure the length of the string. Use geometry to express 1" in terms of string length d and the angle from the vertical 3. After some cancellations and simplications. we should be left with a concise relationship answering the question posed at the beginning of this activity. If you had a ball on a 50 cm string swinging around at one revolution per second, what angle would the string make with the vertical?I Whole Class Discussion Ill} Is there a litnit on the possible values of the angle 9'? What is it? 11} Is there a limit on the possible values of the angular speed to? What is it