Chapte 5 Foce and Motion In Chaptes 2 and 4 we have studied kinematics, i.e., we descibed the motion of objects using paametes such as the position vecto, velocity, and acceleation without any insights as to what caused the motion. This is the task of Chaptes 5 and 6, in which the pat of mechanics known as dynamics will be developed. In this chapte we will intoduce Newton s thee laws of motion which ae at the heat of classical mechanics. We must note that Newton s laws descibe physical phenomena of a vast ange. Fo example, Newton s laws explain the motion of stas and planets. We must also note that Newton s laws fail in the following two cicumstances: 1. When the speed of objects appoaches (1% o moe) the speed of light in vacuum (c = 8 10 8 m/s). In this case we must use Einstein s special theoy of elativity (1905). 2. When the objects unde study become vey small (e.g., electons, atoms, etc.). In this case we must use quantum mechanics (1926). (5-1)
Newton s Fist Law Scientists befoe Newton thought that a foce (the wod influence was used) was equied in ode to keep an object moving at constant velocity. An object was thought to be in its natual state when it was at est. This mistake was made befoe fiction was ecognized to be a foce. Fo example, if we slide an object on a floo with an initial speed v 0 vey soon the object will come to est. If on the othe hand we slide the same object on a vey slippey suface such as ice, the object will tavel a much lage distance befoe it stops. Newton checked his ideas on the motion of the moon and the planets. In space thee is no fiction, theefoe he was able to detemine the coect fom of what is since known as Newton s fist law : If no foce acts on a body, the body s velocity cannot change; that is, the body cannot acceleate. Note: If seveal foces act on a body (say FA, FB, and FC) the net foce Fnet is defined as F = F + F + F, i.e., F is the vecto sum of F, F, and F. net A B C net A B C (5-2)
Foce: The concept of foce was tentatively defined as a push o pull exeted on an object. We can define a foce exeted on an object quantitatively by measuing the acceleation it causes using the following pocedue. We place an object of mass m = 1 kg on a fictionless suface and measue the acceleation a that esults fom the application of a foce F. The foce is adjusted so that a = 1 m/s 2. We then say that F = 1 newton (symbol: N). Note: If seveal foces act on a body (say FA, FB, and FC) the net foce F is defined as Fn et = FA + FB + FC, i.e., Fn et is the vecto sum of F, F, and F. A B C net (5-3)
m 0 m X F a 0 F a X Mass: Mass is an intinsic chaacteistic of a body that automatically comes with the existence of the body. But what is it exactly? It tuns out that the mass of a body is the chaacteistic that elates a foce F applied on the body and the esulting acceleation a. Conside that we have a body of mass m 0 = 1 kg on which we apply a foce F = 1 N. Accoding to the definition of the newton, F causes an acceleation a 0 = 1 m/s 2. We now apply F on a second body of unknown mass m X, which esults in an acceleation a X. The atio of the acceleations is invesely popotional to the atio of the masses: mx a a = m = m m a a 0 0 0 X 0 X X Thus by measuing a X we ae able to detemine the mass m X of any object. (5-4)
Newton s Second Law The esults of the discussions on the elations between the net foce F net applied on an object of mass m and the esulting acceleation a can be summaized in the following statement known as Newton s second law. m F net a The net foce on a body is equal to the poduct of the body s mass and its acceleation. In equation fom Newton s second law can be witten as: F net = ma The above equation is a compact way of summaizing thee sepaate equations, one fo each coodinate axis: F net,x = max Fnet, y = may net,z z F = ma (5-5)
In this section we descibe some chaacteistics of foces we will commonly encounte in mechanics poblems. F g y The Gavitational Foce: It is the foce that the Eath exets on any object (in the pictue a cantaloupe). It is diected towad the cente of the Eath. Its magnitude is given by Newton s second law. F = ma = mgĵ F = mg g g g W mg y Weight: The weight of a body is defined as the magnitude of the foce equied to pevent the body fom falling feely. F = ma = W mg = W = mg net, y y 0 Note: The weight of an object is not its mass. If the object is moved to a location whee the acceleation of gavity is diffeent (e.g., the moon, whee g m = 1.7 m/s 2 ), the mass does not change but the weight does. (5-6)
Contact Foces: As the name implies, these foces act between two objects that ae in contact. The contact foces have two components: one that is acting along the nomal to the contact suface (nomal foce) and a second component that is acting paallel to the contact suface (fictional foce). Nomal Foce: When a body pesses against a suface, the suface defoms and pushes on the body with a nomal foce pependicula to the contact suface. An example is shown in the pictue to the left. A block of mass m ests on a table. F = ma = F mg = F = mg net, y y N 0 N Note: In this case F N = mg. This is not always the case. Fiction: If we slide o attempt to slide an object ove a suface, the motion is esisted by a bonding between the object and the suface. This foce is known as fiction. Moe on fiction in Chapte 6. (5-7)
Tension: This is the foce exeted by a ope o a cable attached to an object Tension has the following chaacteistics: 1. It is always diected along the ope. 2. It is always pulling the object. 3. It has the same value along the ope (fo example, between points A and B). The following assumptions ae made: a. The ope has negligible mass compaed to the mass of the object it pulls. b. The ope does not stetch. If a pulley is used as in fig.(b) and fig.(c), we assume that the pulley is massless and fictionless. A B (5-8)
Newton s Thid Law: When two bodies inteact by exeting foces on each othe, the foces ae equal in magnitude and opposite in diection. Fo example, conside a book leaning against a bookcase. We label FBC, the foce exeted on the book by the case. Using the same convention we label FCB, the foce exeted on the case by the book. Newton's thid law can be witten as FBC = FCB. The book togethe with the bookcase ae known as a " thid-law foce pai. " A second example is shown in the pictue to the left. The thid-law pai consists of the Eath and a cantaloupe. Using the same convention as above we can expess Newton's thid law as F = F. CE EC (5-9)
The Eath otates about its axis once evey 24 hous and thus it is acceleating with espect to an inetial efeence fame. Thus we ae making an appoximation when we conside the Eath to be an inetial efeence fame. This appoximation is excellent fo most small-scale phenomena. Nevetheless, fo lage-scale phenomena such as global wind systems, this is not the case and coections to Newton s laws must be used. (5-10) Inetial Refeence Fames: We define a efeence fame as inetial if Newton s thee laws of motion hold. In contast, efeence fames in which Newton s law ae not obeyed ae labeled noninetial. Newton believed that at least one such inetial efeence fame R exists. Any othe inetial fame R' that moves with constant velocity with espect to R is also an inetial efeence fame. In contast, a efeence fame R" that acceleates with espect to R is a noninetial efeence fame.
Applying Newton s Laws / Fee-Body Diagams Pat of the pocedue of solving a mechanics poblem using Newton s laws is dawing a fee-body diagam. This means that among the many pats of a given poblem we choose one that we call the system. Then we choose axes and ente all the foces that ae acting on the system and omit those acting on objects that wee not included in the system. An example is given in the figue below. This is a poblem that involves two blocks labeled A and B on which an extenal foce F is exeted. We have the following "system" choices: a. System = block A + block B. The only hoizontal foce is Fapp. b. System = block A. Thee ae now two hoizontal foces: Fapp and FAB. c. Sy stem = block B. The only hoizontal foce is F. app BA (5-11)
Recipe fo the Application of Newton s Laws of Motion 1. Choose the system to be studied. 2. Make a simple sketch of the system. 3. Choose a convenient coodinate system. 4. Identify all the foces that act on the system. Label them on the diagam. 5. Apply Newton s laws of motion to the system. (5-12)