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

13 Newton’s Second Law

CHAPTER 13 Newton’s Second Law In the previous chapter we examined “Newton’s Little Law,” which says that Fnet and a point the same way. In this chapter we’ll refine Newton’s Little Law to take into account the masses of target objects. Mass “What is mass?” is a profound question. Indeed, modern physical theories such as field theory and string theory are still trying to sort this out. In this book we’ll take the commonsense view Newton himself took: Mass is simply the amount of “material stuff” contained in an object. Mass is measured in kilograms (kg). In everyday terms, a quart of milk contains about a kilogram of mass. A large person contains about a hundred kilograms of mass. Today the official basis for measuring mass is the standard kilogram , a machined cylinder of platinum-iridium alloy housed in Sèvres, France, at the Bureau International des Poids et Mesures (International Bureau of Weights and Measures). However, alternatives to this approach are currently being discussed, such as defining a kilogram as the amount of mass contained in a certain agreed-upon number of carbon atoms. 247 248 PART II EXPLAINING AND PREDICTING MOTION The Effect of Mass on Acceleration A force of magnitude 100 N is strong enough to lift a bag of groceries . But it’s not strong enough to lift a car. Obviously, the capacity of a force to alter the motion of an object has something to do with how massive the object in question is. Think about a race between two parents pushing baby carriages . The parents are equally matched in terms of their strength. But one parent has quintuplets sitting in a five-seater carriage, while the other parent has just one baby sitting in a single-seater carriage. Which team do you think is going to have the better acceleration off the starting blocks? The single-baby team will obviously accelerate at a higher rate, given the same strength of push from the parent. I like to think of this in terms of input and output. If we apply an “input” force of a certain strength, we achieve an “output” acceleration of a certain magnitude. Increasing the applied force will increase the resulting acceleration. So there seems to be a proportionality here: (194) aoutput = (proportionality constant)Fnet input. The proportionality constant depends on what the target object is. If we are talking about a bag of groceries, an input net force of 100 N will generate a noticeable output acceleration, whereas if we are talking about a boulder, an input net force of 100 N will generate only a small output acceleration. Based on this single observation, we might well guess that the output acceleration resulting from a given input force scales inversely with the mass of the target. And that is indeed the content of Newton’s Second Law: (195) a = 1 m Fnet. Or, in words: The acceleration of an object is one over the mass of the object times the net force acting on the object. The mass is in the denominator to ensure that a given applied force leads to an accelerationthatissmallwhenthemassofthetargetobjectislarge. Newton’s Second Law: a = 1 m Fnet. At each instant of time, the acceleration vector of an object is equal to one over the mass of the object times the net force acting on the object at that time. Chapter 13 Newton’s Second Law 249 How Newton’s Second Law Relates to “Newton’s Little Law” The inverse mass 1 m is always a positive number—a positive scalar,in thelanguageofChapter4.Sowhenwemultiplythenetforcevector Fnet by this scalar according to Equation 195, the mathematical effect is to leave the direction of the vector unchanged. We see once again that Fnet and a point the same way. Newton’s Second Law implies “Newton’s Little Law.” The Units of Force Although multiplying the vector Fnet by 1 m preserves the direction of Fnet, the new vector does have a new magnitude with new units. The units of 1 m Fnet will be 1 kg × N. But because 1 m Fnet = a, with units of m/s2 , we will have 1 kg × N = m/s2 , or N = kg m s2 . (196) We see that 1 newton is just shorthand for 1 kg m s2 . Rearranging Newton’s Second Law Let’s multiply both sides of Equation 195 by m and...