CLASSICA DYNAMIC OF PARTICLES AND SYSTEMS 2. Newtonian Mechanics-Single Particle

1. CLASSICA DYNAMIC OF PARTICLESAND SYSTEMS2. Newtonian Mechanics-Single Particle Associate Professor: C. H.LIAO

2. Contents • 2.1 Introduction 48 • 2.2 Newton's Laws 49 • 2.3 Frames of Reference 53 • 2.4 The Equation of Motion for a Particle 55 • 2.5 Conservation Theorems 76 • 2.6 Energy 82 • 2.7 Limitations of Newtonian Mechanics 88

3. 2.1 Introduction • The science of mechanics need certain fundamental concepts such as distanceand time. The combination of the concepts of distance and time allows us to define the velocity and acceleration of a particle. • The third fundamental concept, mass, requires some elaboration, which we give when we discuss Newton's laws. • Physical laws must be based on experimental fact. • We cannot expect a priori that the gravitational attraction between two bodies must vary exactly as the inverse square of the distance between them. But experiment indicates that this is so. • The postulate then assumes the status of a physical law. If some experiments disagree with the predictions of the law, the theory must be modified to be consistent with the facts.

4. 2.2 Newton's Laws • We begin by simply stating in conventional form Newton's laws of mechanics : • I. A body remains at rest or in uniform motion unless acted upon by a force. • II. A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force. • III. If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

5. The First Law, for example, is meaninglesswithout the concept of "force," a word Newton used in all three laws. • Infact, standing alone, the First Law conveys a precise meaning only for zero force;that is, a body remaining at rest or in uniform. A body moving in this manner istermed a free body (or free particle).

6. The Second Law provides an explicit statement: Force is related to the timerate of change of momentum.

7. Newton‘s Third Law &the conservation of linear momentum To demonstrate the significance of Newton‘s Third Law, let us paraphrase itin the following way, which incorporates the appropriate definition of mass: • III‘. If two bodies constitute an ideal, isolated system, then the accelerations of these bodiesare always in opposite directions, and the ratio of the magnitudes of the accelerationsis constant. This constant ratio is the inverse ratio of the masses of the bodies. the more general conservation of linear momentum.

8. 2.3 Frames of Reference • A reference frame is called an inertial frame if Newton's laws are indeed valid in that frame. • That is, if a body subject to no external force moves in a straight line with constant velocity (or remains at rest), then the coordinate system establishing this fact is an inertial reference frame. • If Newton's laws are valid in one reference frame, then they are also valid in any reference frame in uniform motion (i.e., not accelerated) with respect to the first system.

9. FIGURE 2-1 We choose to describe the path of a free particle moving along the pathACin a rectangularcoordinate system whose origin moves in a circle.Such a system is not an inertial reference frame.

10. 2.4 The Equation of Motion for a Particle

11. Some useful problem-solving techniques. • 1. Make a sketch of the problem, indicating forces, velocities, and so forth. • 2. Write down the given quantities. • 3. Write down useful equations and what is to be determined. • 4. Strategy and the principles of physics must be used to manipulate the equationsto find the quantity sought. Algebraic manipulations as well as differentiationor integration is usually required. Sometimes numerical calculationsusing a computer are the easiest, if not the only, method of solution. • 5. Finally, put in the actual values for the assumed variable names to determinethe quantity sought.

12. Ex 2.1 If a block slides without friction down a fixed, inclined plane with θ = 30°, whatis the block's acceleration? Sol: The total force Fnet is constant:

14. Ex. 2.2 If the coefficient of static friction between the block and plane in the previous example is μs = 0.4, at what angle θ will the block start sliding if it is initially at rest? Sol:

15. Mter the block in the previous example begins to slide, the coefficient of kinetic (sliding) friction becomes μk = 0.3. Find the acceleration for the angle θ = 30°. Sol:

16. Effects of Retarding Forces where Cw is the dimensionless drag coefficient, ρis the air density, v is the velocity,and A is the cross-sectional area of the object (projectile) measured perpendicularlyto the velocity.

18. Ex. 2.4 As the simplest example of the resisted motion of a particle, find the displacement and velocity of horizontal motion in a medium in which the retarding force is proportional to the velocity.

19. Ex. 2-5 Find the displacement and velocity of aparticle undergoing vertical motion in amedium having a retarding force proportional to the velocity. Sol.:

21. Ex.2-6 Sol.:

22. When y = O if θ= 55 ° we have Vo = 1450 m/s and θ= 55°, the rangebecome Big Bertha's actual range was 120 km. The difference is a result of the real effect of air resistance. For t = T The maximum predicted heightis: The range R is

23. Ex. 2-7 Next, we add the effect of air resistance to the motion of the projectile in the previous example. Calculate the decrease in range under the assumption that the force caused by air resistance is directly proportional to the projectile's velocity. Sol.:

24. Perturbation Method keep only terms in the expansion through k3,

26. p.s. : Read the force every 50 m/ s for Figure 2-3c and every 100 m/ s Sol.:

28. Ex. 2-9 Atwood's machine consists of a smooth pulley with two masses suspended from a light string at each end (Figure 2-11). Find the acceleration of the masses and the tension of the string (a) when the pulley center is at rest and (b) when the pulley is descending in an elevator with constant acceleration α

29. The equations of motion Sol.: (a) when the pulley center is at rest. (b) when the pulley is descending in an elevator with constant acceleration α.

30. Ex. 2-10

31. Sol.: ∵

32. 2.5 Conservation Theorems If the particle is free, that is, if the particle encounters no force I. The total linear momentum p of a particle is conserved when the total force on it is zero. II. The angular momentum of a particle subject to no torque is conserved. III. The total energy E of a particle in a conservative force field is a constant in time. the component of linearmomentum in a direction in which the forcevanishes is constant in time.

34. III. The total energy E of a particle in a conservative force field is a constant in time. The total energy of a particle to be the sum of the kineticand potential energies

35. Ex. 2-11 Sol.: The angular momentum The velocity of the outside edge

36. 2.6 Energy equilibrium point P.S.: At x=x0is an equilibrium point, Equilibrium may be stable, unstable, orneutral.

37. ∵U0at x = 0 can define to be zero

38. Ex. 2-12

39. Sol.:

40. 2.7 Limitations of Newtonian Mechanics • We have implied that these are all measurable quantities and that they can be specified with any desired accuracy, depending only on the degree of sophistication of our measuring instruments. • When we attempt to make precise measurements on microscopic objects,however, we find a fundamental limitation in the accuracy of the results. • This momentum is uncertain by an amount ap. The product • △ x △ p is a measure of the precision with which we can simultaneously determinethe electron‘s position and momentum; △x → 0, △ p → 0 implies a measurementwith all imaginable precision. • for if △x → 0, then wemust have △ p → 0 for Heisenberg's uncertainty principle to be satisfied. • Newtonian mechanics is therefore subject to fundamental limitations whensmall distances or high velocities are encountered.

41. Problem discussion. Thanks for your attention.

42. Problem discussion. Problem: • 2-3, 2-7, 2-12, 2-17, 2-19, 2-23, 2-28, 2-29, 2-36, 2-41, 2-42, 2-49, 2-54

43. P. 2-3

44. P. 2-7

45. P. 2-12

47. P. 2-17

48. P. 2-19

49. P. 2-23