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VECTORS AND CENTRIPETAL ACCELERATION We have just seen that position, displacement, velocity and acceleration each have associated direction, as well as a magnitude. In general, quantities which have both a numerical value and a direction are called vectors. By contrast, quantities which have only numerical value (i.e. no direction) are called scalars. In our previous discussion, distance and speed are scalars. (Other well known scalars are mass, time, and temperature.) For the purpose of visualizing vectors, we often draw them using arrows, with the direction of the arrow (obviously!) showing the direction of the vector, while the length of the arrow conveys its magnitude. (For example, look at the various velocity vectors on p 189 of the textbook). We can also use a positive numerical factor to expand or shrink a particular vector (in this case called V), and a minus sign to reverse its direction, as the following illustrations show. V 2V 2 V 3 V 3 V 2 There is one other important point regarding vectors to be made here. A vector is completely determined by its magnitude (or length) and its direction. In particular, in 2 most applications, the location of a vector is immaterial, i.e. moving a vector to a different location does not change the vector as long as the magnitude and the direction remain unchanged. This point is illustrated in Figures i, ii and iii below. U U U V V i ii V iii Note that V has been positioned in three different locations relative to U (completely separated, “tail-totail”, “tail-to-tip”), but since their length and direction have not been changed, the vector is still V in each case! You won’t have to draw vectors often (if at all) in this course, but it can be useful to visualize vectors occasionally, for the purpose of explanation. Most of the time, you will recognize a vector not by its representation with an arrow, but simply because it is clearly given an associated direction (such as being told that objects have velocities of 5 m/s east or 80 km/h south-west). We will use vector diagrams again in the next discussion, but only to help explain the concept of centripetal acceleration. 3 CENTRIPETAL ACCELERATION Consider a body traveling counterclockwise around a circular path at constant speed, and suppose we look at its position and velocity vectors at two times which are separated by a relatively small time interval t. The initial and final position and velocity vectors (denoted ri, rf, vi and vf) are shown in the following diagram. vi vf rf ri The position vectors ri and rf are certainly different (since their directions aren’t the same) but, since the path is a circle, they have the same length (which we will call simply r). Likewise, the velocity vectors vi and vf point in different directions, but, since the speed is constant, they also have the same length (which we will call simply v). 4 Because vectors can be moved without changing them (as long as the length and direction remain the same, we can produce the following new diagrams: r vi v vf rf ri Note that we have added the vectors representing the displacement (or change in position) r and the change in velocity v. It is easy to see that these two triangles are similar triangles, and hence that their corresponding sides have a constant ratio (remember that the concept of similar triangles was an important historical milestone of science!). As a result, we can write (since the lengths of vi and ri are just v and r): v r v r or, dividing by t, v r v t r t . 5 But, according to the definitions of acceleration and velocity, v r a and v. t t This means that a v v2 , or, multiplying by v, a v r r . Finally, note in the above diagram that the direction of v (and hence the direction of the acceleration a) is always perpendicular to the direction of the displacement r. So, to summarize, when an object travels around a circle of radius r with constant speed v, it experiences an acceleration v2 a r which is directed toward the center of the circle. Because it is “center-pointing”, this special type of acceleration has been given the name centripetal acceleration. Remember, again, that the speed is constant, showing that a body can accelerate without changing speed! This is because the velocity vector is changing direction even though its magnitude stays the same. 6 This means that centripetal acceleration can always be calculated for any object traveling in a circle, if we know: (i) the radius of the circle; and (ii) the speed of the object. For example, a person sprinting around a circle of radius 9 m with speed 6 m/s is accelerating toward the center of the circle with v 2 62 a 4 m/s2. r 9 The notion of centripetal acceleration, and this formula in particular, will be very important later on in our discussion of gravity.