# Chapter 4. Dynamics: Newton s Laws of Motion

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1 Chapter 4 Dynamics: Newton s Laws of Motion

2 The Concepts of Force and Mass A force is a push or a pull. Contact forces arise from physical contact. Action-at-a-distance forces do not require contact and include gravity and electrical forces.

3 The Concepts of Force and Mass Force is a vector quantity (e.g. like velocity or acceleration). Arrows are used to represent forces. The length of the arrow is proportional to the magnitude of the force. 15 N 5 N

4 The Concepts of Force and Mass Mass is a measure of the amount of stuff contained in an object. It is some intrinsic property of an object. mass weight --> discuss later

5 Isaac Newton (ca. 1687) came up with three laws of motion which form the basis of (classical) mechanics. They describe the effects of forces on objects with mass. Let s discuss each of the three laws separately:

6 Newton s First Law of Motion Newton s First Law An object continues in a state of rest or in a state of motion at a constant speed along a straight line, unless compelled to change that state by a net force. The net force is the vector sum of all of the forces acting on an object.

7 Newton s First Law of Motion The net force on an object is the vector sum of all forces acting on that object. The SI unit of force is the Newton (N). Individual Forces Net Force 4 N 10 N 6 N = 10 N - 4 N

8 Newton s First Law of Motion Individual Forces Net Force 5 N 3 N 37 o 4 N

9 Newton s First Law of Motion Inertia is the natural tendency of an object to remain at rest or in motion at a constant speed along a straight line. The mass of an object is a quantitative measure of inertia. SI Unit of Mass: kilogram (kg)

10 Newton s First Law of Motion Inertial reference frame a frame of reference in which Newton s first law of motion is valid, i.e. a reference frame that is not accelerating Examples of inertial reference frames: Ø Inside a spacecraft moving at constant velocity in space Ø Inside an aircraft moving at constant velocity in the air Ø At the surface of the Earth (approximately) Examples of non-inertial reference frames: Ø Inside an accelerating spacecraft or airplane Ø On a revolving merry-go-round Ø At the surface of the Earth

11 Newton s Second Law of Motion Mathematically, the net force is written as F = F + F + F where the Greek letter sigma denotes the vector sum.

12 Newton s Second Law of Motion Newton s Second Law When a net external force acts on an object of mass m, the acceleration that results is directly proportional to the net force and has a magnitude that is inversely proportional to the mass. The direction of the acceleration is the same as the direction of the net force.! a =! F m F = a m Valid in an Inertial reference frame

13 Newton s Second Law of Motion F = ma kg m s 2 Units of force g cm s 2 slug ft s 2 = newton N ( ) SI units = dyne dyn ( ) cgs units = pound lb ( ) BE units

14 Newton s Second Law of Motion A free-body-diagram is a diagram that represents the object and the forces that act on it.

15 Newton s Second Law of Motion The net force in this case is: 275 N N 560 N = +110 N and is directed along the + x axis of the coordinate system.

16 Newton s Second Law of Motion If the mass of the car is 1850 kg then, by Newton s second law, the acceleration is a = F m = +110 N 1850 kg = m s2

17 The Vector Nature of Newton s Second Law The direction of force and acceleration vectors can be taken into account by using x and y components. F = a m is equivalent to F = y ma y F = ma x x

18 The Vector Nature of Newton s Second Law Example. A man is stranded on a raft (mass of man and raft = 1300 kg). By paddling, he causes an average force P of 17 N to be applied to the raft in a direction due east (the +x direction). The wind also exerts a force A on the raft. This force has a magnitude of 15 N and points 67 o north of east. Ignoring any resistance from the water, find the x and y components of the raft s acceleration.

19 The Vector Nature of Newton s Second Law

20 The Vector Nature of Newton s Second Law The net force on the raft can be calculated in the following way: Force X Y P +17 N 0 N A F +(15 N) cos67 = 6 N +(15 N) sin67 = 14 N +23 N +14 N

21 The Vector Nature of Newton s Second Law a x = F x m = +23 N 1300 kg = m s2 a y = F y m = +14 N 1300 kg = m s2 a = m/s 2 θ = 31 o

22 The Vector Nature of Newton s Second Law Assume that initially the velocity of the raft is 0.15 m/s in a direction due east (the + x direction). Assuming that the forces are maintained for 65 s, find the x and y components of the raft s displacement during this time interval. x v 0x v x a x t y v 0y v y a y t? 0.15 m/s m/s 2 65 s? 0 m/s m/s 2 65 s x = v 0x t + 1/2 a x t 2 = (0.15)(65) + 1/2 (0.018)(65) 2 = 48 m y = v 0y t + 1/2 a y t 2 = (0)(65) + 1/2 (0.011)(65) 2 = 23 m r = 53 m θ = 26 o

23 Newton s Third Law of Motion Newton s Third Law of Motion Whenever one body exerts a force on a second body, the second body exerts an oppositely directed force of equal magnitude on the first body.

24 Newton s Third Law of Motion An astronaut pushes on the spacecraft with some force P. Suppose that the magnitude of the force is 36 N. If the mass of the spacecraft is 11,000 kg and the mass of the astronaut is 92 kg, what are the accelerations?

25 Newton s Third Law of Motion On the spacecraft On the astronaut F = P. F = P. a s = P m s = + 36 N 11,000 kg = m 2 s a A = P m A = 36 N 92 kg = 0.39 m 2 s

26 Another Newton s 2 nd Law example Example. The figure below shows two forces with magnitudes F 1 = N and F 2 = N acting on an object, the plus signs indicating that the forces act along the +x axis. A third force F 3 also acts on the object but is not shown in the figure. The object is moving with a constant velocity of +750 m/s along the x axis. Find the magnitude and direction of F 3. F 3? v = +750 m/s F 1 F 2 +x

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I do not know what I may appear to the world/ but to myself I seem to have been only like a boy playing on the sea shore, and diverting myself in now and then finding a smoother pebble or a prettier shell