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

14 Dynamics

CHAPTER 14 Dynamics The Two Basic Problems of Dynamics Newton’s Second Law permits us to solve two basic kinds of problems . These problems are in a sense the reverse of one another: . ProblemType1. By observing the motion of an object, deduce the nature of the forces at work on it. . Problem Type 2. By knowing something about the forces at work on an object, predict its future motion. In his Principia, Sir Isaac Newton formulated and solved both kinds of problems. For example, Newton used Johannes Kepler’s observation that the planets move in elliptical orbits to conclude thatthegravitationalforceholdingtheplanetsintheirorbitsweakens with distance from the sun, falling off inversely with the square of the distance. By observing the motions of the planets, Newton could infer the nature of the forces at work on them. This was a problem of Type 1. Next, taking it as established that the gravitational forces acting on the planets vary inversely with the square of their distance from the sun, Newton proceeded to predict the future course of their motion. This was a problem of Type 2. All of this, mind you, was in the same book in which Newton proposed the Second Law in the first place! His contemporaries could perhaps be forgiven for needing some time to take all of this in. Oh, by the way, Newton also had to invent calculus in order to solve these problems! 267 268 PART II EXPLAINING AND PREDICTING MOTION Another Look at the Shopping Cart Problem Back in Chapter 13 we looked at the motion of a shopping cart (Worked Example 2). The graphs we drew for that problem— reproduced here in Figures 209 and 210—provide a great illustration of how the two kinds of dynamics problems work. Study these figures before reading further. Finding the Forces from the Motion Let’s try a problem of the first type: finding the forces from the motion. find the rate of change... vx 0 t If you know the trajectory of a target... x 0 t and find it again... then multiply by the mass of the target... and voila! You have a profile of the net force! 0 t Fnet, x ax 0 t FIGURE 209. How the first kind of dynamics problem works. Read the words from the top of the page to the bottom, then from left to right. Chapter 14 Dynamics 269 the velocity, then work backward again.... vx 0 t and voila! You have predicted the future movements of the target! x 0 t to find the acceleration, then work backward to find... then just divide by the mass of the target... If you know the forces that will act on a target over time... 0 t Fnet, x ax 0 t FIGURE 210. How the second kind of dynamics problem works. Read the words from the lower right part of the page to the left, then from bottom to top.» WORKED EXAMPLE 1 A 10 g bullet is fired horizontally into a tree trunk at a speed of 500 m/s. Once the bullet enters the tree trunk, frictional forces act over time to bring the bullet to rest in a distance of 4 cm. Find the magnitude and direction of the frictional force that acts to bring the bullet to rest.» SOLUTION 1 This problem gives us a laundry list of “kinematical” information: data on distances and speeds. We can process this kinematical information to compute the acceleration of the bullet. Then we’ll usetheaccelerationandNewton’sSecondLawtoinferthestrength of the friction force responsible for stopping the bullet. 270 PART II EXPLAINING AND PREDICTING MOTION This is one of those situations in which we are not given any timedataexplicitly.Norareweaskedforanytimedata.Sothemost convenient kinematical formula to use is the one without any time variable in it (see Focused Problem 10 of Chapter 9): (199) v2 f ,x − v2 i,x = 2axx. A formula like Equation 199 always refers to a chosen coordinate system. (Otherwise, what on earth do all of those “x” subscripts mean?) So before proceeding, we’d better draw a cartoon with a coordinate system laid down over it. I’ve done that in Figure 211. (How did I know to draw a the way I did?) x 0 0.04 m a before during after ! FIGURE 211. Solving Worked Example 1. A coordinate system laid onto an image of a bullet shot into a tree shown at three...