Force and Motion: An Illustrated Guide to Newton's Laws

8 Focus on a-Perp

CHAPTER 8 Focus on a-Perp As the concept map in Chapter 7 (Figure 108) illustrates, there are two basic ways to view the acceleration vector. . As we discussed in Chapter 6, we can view the acceleration vector as an indication of whether an object is speeding up, slowing down, or turning. . Equally well, as we discussed in Chapter 7, we can look at the acceleration vector as an indication of how fast the velocity vector itself is changing with time. At the intersection of these two basic perspectives lies a marvelous formula for a⊥, the turning part of the acceleration. This is a formula you will use all the time in your study of physics. The formula first appeared in 1658, in a book called Horologium Oscillatorium, written by Christian Huygens, a Dutch physicist. Huygens considered a very specific kind of motion: namely, an object moving in a circular path at a constant speed. This pattern of motion is called “uniform circular motion.” In uniform circular motion, the speed is constant, so a is zero. The only acceleration present is the turning part, a⊥. Huygens derived the following formula for a⊥: a⊥ = v2 r . (83) 127 128 PART I DESCRIBING MOTION Here, v is the speed of the object as it moves around the circle, and r is the radius of the circular path. Notice that a⊥ has the correct units for acceleration: (m/s)2 m = m2 s2 . m = m s2 . Where Huygens’s Formula Comes From The basic insight behind Huygens’s Formula is that—all other things being equal—if you move in a very tight circle (a circle with a small value of r), your velocity vector will be changing at a rapid rate. The reason for this is that when you move in a small circle, your velocity vector rotates through large angles in short times. Compare the small circular track in Figure 111 with the larger one. Figure 111 shows snapshots of two cars moving at equal speeds but driving on circular tracks of very different size. The snapshots are shown at equally spaced moments of time t1, t2, and t3. The velocity vectors are shown at these moments. For the car on the small track, the velocity vector changes a lot in the given time interval, meaning that the rate of change of velocity (acceleration) is large. For the car on the large track, the velocity vector changes much less in the given time interval, meaning that the rate of change of velocity (acceleration) is small. Therefore, the acceleration will be greater when the radius is smaller. This is why there is an r in the denominator of Huygens’s Formula. Of course, the radius of the circle isn’t everything. Your speed matters, too, because the faster you move around the circle, the faster your velocity vector will be rotating through those angles. This is why there is a v2 in the numerator of the formula: A large value of v2 means that you are traveling the circle quickly, so your velocity vector is changing fast; hence we should have a large acceleration . If you want to know more of the details leading to Huygens’s Formula, the Appendix gives a mathematical derivation. Beyond Uniform Circular Motion Huygens derived Equation 83 for the particular case of motion in a circular path at a constant speed. But the wonderful thing about Huygens’s Formula is that it works even when the speed is Chapter 8 Focus on a-Perp 129 v1 v2 v3 v1 v2 v3 v3 very different from v1 large acceleration v3 more similar to v1 small acceleration v1 v2 v3 t1 t2 t3 v1 v2 v3 t1 t2 t3 Small track Large track FIGURE 111. Snapshots of two cars moving at equal speeds on circular tracks of different size. not constant and even when the trajectory is not circular! (I don’t know if Huygens himself knew this or not.) You can use a⊥ = v2 r to find the perpendicular component of the acceleration vector pretty much no matter how an object is moving. You just have to know how to interpret the “r” in the formula when the path is not a circle. In general, the r in Huygens’s Formula is the radius of curvature of the object’s trajectory. See Figure 112. When you’re moving in a curved path, the radius of...