Dr. A Louro - Physics 211 Notes (2) - Dynamics

8/14/2019 Dr. A Louro - Physics 211 Notes (2) - Dynamics

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Physics 211 Lecture Notes

Part 2: Dynamics

A. A. Louro

Fall 2001


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Contents

1 Forces 1

1.1 An introduction to dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Newtonian mechanics and the concept of force . . . . . . . . . . . . . . . . . . . . . 21.3 Newtons three laws of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Forces in nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.2 Contact forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Applying Newtons laws: Free-body diagrams . . . . . . . . . . . . . . . . . . . . . . 61.6 Apparent weightlessness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Elastic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.8 Systems of interacting bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Circular motion 11

2.1 Kinematics of circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.1 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Non-uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Forces in circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Gravitation 15

3.1 Newtons Law of Universal Gravitation . . . . . . . . . . . . . . . . . . . . . . . . 163.2 g again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Orbital motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Keplers Third Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

A Mass and gravitational charge 19A.1 What do we mean by mass? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3


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4 CONTENTS


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Chapter 1

Forces

1


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2 CHAPTER 1. FORCES

1.1 An introduction to dynamics

As we have mentioned before, kinematics is the description of how things move. Given certaininformation, we can describe the motion of an object by giving expressions for its position andvelocity as functions of time. That way, we can predict, for example, where an ob ject is going tobe at a given time, as well as how fast it will be moving and in which direction.

To be able to do this, we need to know certain things:

The initial conditions, that is the position and velocity of the object at some particularinstant, r0 and v0; and

The acceleration of the object.

It is a basic physical principle that the velocity of an object is only altered if it is interacting withsomething else. So, the next step is to determine what the acceleration of an object is, given itsinteractions with other ob jects.

1.2 Newtonian mechanics and the concept of force

Mechanics was put on a solid logical foundation by Isaac Newton (1642-1727). Newtons pro-gram was to find a small set of very basic principles on which the science of mechanics could beconstructed, rather like Euclids Elements of geometry built upon a small set of axioms, orself-evident truths.

In Newtonian mechanics, objects interact by applying forces to each other. Our intuitive notion

of what a force is, a push or a pull, is sufficient to understand this, provided we allow action ata distance. The Moon and the Earth interact by pulling on each other with gravitational forces.Of course, the Moon and the Earth are not in direct contact with each other, but they interactnevertheless.

One important consideration about forces that is not so intuitive perhaps is that force is a vectorquantity. The magnitude of a force measures how strong it is, and it also has direction: when youthrow a ball, you can throw it hard or gently, and you always throw it in a certain direction. Oragain, the gravitational pull by the Earth on all nearby objects is directed towards the centre ofthe Earth.

1.3 Newtons three laws of motion

The basic principles that Newton found could serve to explain all kinds of motion were three. Theycan be illustrated by considering a very simple interaction, of which a simulation may be foundhere. Two gliders move on a level airtrack with no friction. These precautions are taken in order toeliminate all interactions except the one that takes place when the gliders collide. By investigating
http://www.phas.ucalgary.ca/physlets/newtonslaws.htmhttp://aleph0.clarku.edu/~djoyce/java/elements/elements.htmlhttp://www.newton.org.uk/


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1.3. NEWTONS THREE LAWS OF MOTION 3

their accelerations, that is the change in their velocities over time, during the collision and at other

times when they are apart, you should find results comsistent with Newtons three laws of motion:

N1: Law of inertia If an object is not interacting with anything, then it moves with constant ve-locity. For example, before and after the collision, each glider has constant velocity, includingthe one that is stationary, since its velocity is zero!

N2: While two objects interact, each one has an acceleration proportional to the force, and inverselyproportional to the mass of the body, a measure of how much matter it contains:

a =F

m

The SI unit of mass is the kilogram (kg). Notice that although we usually think of thekilogram as a measure of weight, gravity is not involved in its definition. Mass has also beendescribed as a measure of an objects inertia, that is, its resistance to change in its stateof motion. Newtons 2nd law shows this: Apply the same force F to two bodies, one with asmall mass and the other with a large mass. The one with the large mass will have a smalleracceleration, that is, a smaller change in tis velocity. Well consider some examples of thissituation shortly, after discussing Newtons 3rd. law.

Newtons 2nd. law may be read in a couple of different ways. First, it me seen as a prescrip-tion for calculating the acceleration of an object once we know what forces it is subject to.Kinematics then takes over and predicts the objects motion over time given its acceleration.Second, Newtons 2nd. law provides with a way of measuring forces: If we observe a tennisball with a given mass, and find that it has a certain acceleration, the product of the twoquantities is a measure of the force. It follows that the SI unit of force is kg m / s2. Thiscombination of units is called, not surprisingly, the Newton, abbreviated N.

N3: Law of action and reaction The glider simulation shows that in the course of a collision,

m1a1 = m2a2

By Newtons 2nd. law, the left-hand side is the force applied on glider 1 by glider 2, whilethe right-hand side is the force applied on glider by glider 1. Newtons 3rd. law postulatesthat each force is equal in magnitude to the other one, and opposite in direction. In Newtons

own words, To every action, there is an equal and opposite reaction.

As a consequence of Newtons 3rd. law, if two objects with dissimilar masses interact witheach other, the one with the larger mass will have the smaller magnitude acceleration. Thisis easily seen with the glider simulation; try it!
http://physics.nist.gov/cuu/Units/kilogram.html


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4 CHAPTER 1. FORCES

1.4 Forces in nature

There are surprisingly few different possible types of interaction between physical objects. Withinthe realm of classical physics, we shall consider gravitational, electrical and magnetic forces.Of these, we shall discuss gravity initially. We will also refer to contact forces later, which arereally electrical interactions between nearby objects.

1.4.1 Gravity

It was Isaac Newton (again!) who proposed that all bodies attract each other with a force that isproportional to their masses and diminishes in strength with the distance separating them. Specif-ically, any two masses m1 and m2 separated by a distance r attract each other with a gravitationalforce of magnitude

|Fg

| |Fg| m1m2r2

In particular, if one of the masses is the Earth itself lets call it Me and the other is anobject of mass m very near the Earths surface, so that its distance to the centre of the Earth isthe radius of the Earth, Re, then

|Fg| MemR2e

so that in fact the only variable that |F g| depends on is the mass of the object m. We call thegravitational force on a 1-kg mass the gravitational field at the Earths surface, g. Then theforce on any mass m is

Fg = mg

Interestingly, |g| = 9.8 N/kg. According to Newtons 2nd. law, for an object in free fall near thesurface of the Earth, only subject to Earths gravitational pull,

Fg = ma mg = ma

so the numerical value of what we have previously called the acceleration due to gravity is equal tothe value of the gravitational field. This is consequence of a very important assumption, though:the mass that appears in Newtons 2nd. law is the same as the mass that the gravitationalforce acts upon. For a further discussion of this, see Appendix A.

1.4.2 Contact forces

When two solid objects come into contact at a common surface, new forces come into play, whichare actually very short-range electrical interactions between the atoms on the surface of the twobodies.


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1.4. FORCES IN NATURE 5

The normal force

If every object on the Earths surface is being pulled towards the centre of the Earth, why doesnteverything continue burrowing down into the Earth?! Clearly, the ground exerts another force thateffectively prevents objects from penetrating into the Earth.

This force is perpendicular to the contact surface. Hence the rather unfortunate label normalforce.

The word normal comes from the Latin norma, meaning a carpenters square,

an L-shaped metal piece used as a standard right angle in woodworking. In everyday

speech, we have come to associate the word with standard; but in mathematics,

normal means perpendicular.

Friction forces

The ground can also exert forces on objects along the surface. Although the exact nature of theseforces is not yet well understood, a simple model exists that dates back to the late 1700s whichgives a reasonably accurate description of how friction forces work.

According to this model, there are two different regimes: One thing to consider is the frictionforce that prevents us (up to a point) from moving an object that is stationary, relative to thecontact surface.

Try the following simple experiment: Push sideways on a heavy object, like a couch,

very lightly at first, then gradually increasing your force. You will observe that at first,

nothing happens; the couch stays put. This means that the ground must be applying

to the couch a force equal in magnitude to your push, but in the opposite direction, so

that the net force on the couch is zero, and it doesnt accelerate. Eventually, however,

as you gradually increase your force, the couch suddenly begins to slide, and as you

continue to push, you feel a considerable resistance.

During the first stage, when the couch remained unmoved, static friction was at work. Experimentshows that there is an upper limit to the static friction force, which depends on two things:

the nature of the materials that are in contact: You wont have to push as hard to get thecouch moving if its resting on a smooth, polished surface;

the force with which the couch presses against the ground, or equivalently, the force by theground on the couch. This isnt necessarily equal to the weight: If the couch is on a sloping

surface, the upper limit to the static friction force only depends on the normal force by theground.

We can summarize these two dependencies as follows:

| Fs| s| N| (1.1)


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6 CHAPTER 1. FORCES

where Fs is the static friction force and s is the static friction coefficient, a dimensionless

number generally between 0 and 1. A small s means the couch is easy to shift, a large s meansyou better get help!Once the couch begins to slide, the forward motion of the couch is resisted by a kinetic friction

force, which is approximately constant, and equal in magnitude to

|Fk| = k| N| (1.2)where k is the kinetic friction coefficient, which also depends on the nature of the two surfacesin contact, and is usually slightly smaller than s for the same materials.

1.5 Applying Newtons laws: Free-body diagrams

Armed with Newtons laws, we can predict how an object is going to move, by simply observinghow it interacts with its environment. From this, we can deterine what the total force on the bodyis, and from Newtons 2nd. law determine its acceleration; the equations of kinematics allow us tocalculate its trajectory.

In this section, well focus on the first part of this process, determining the total force on anobject. A very useful visual aid is the free-body diagram, where we isolate the object wereinterested in, and draw all the forces acting on it. Also, to make the work of using Newtons 2nd.law easier, we will label all the forces and make a list of the labels we have chosen, so there is noconfusion.

As an example, consider a skier travelling downhill, shown in Figure 1.1. We can represent theskier with a point: Assuming she is not rotating, the motion of the skier is well represented by a

single point. Figure 1.1 shows the three forces that are acting on the skier, and a listing of thelabels. Notice that in our description of each force, we state explicitly what is applying it. Theonly exception to this rule is the gravitational force by the Earth on the skier, which well refer asthe skiers weight.

1.6 Apparent weightlessness

We have already called weight the gravitational force on a mass. So its puzzling to see a bodyin free fall, floating in its environment as if gravity were suspended. Astronauts, for example, haveto deal with this apparent weightlessness, which can have quite serious physiological effects. Andyet, gravity has not been turned off!

The key is that since all objects in free fall have the same acceleration, two objects fallingtogether, like an astronaut and the spacecraft, have no acceleration relative to each other. Therefore,relative to the spacecraft, the astronaut seems to be floating.

Consider a person inside a free-falling elevator. If this subject is standing on a scale, the scalereading would be zero. In this sense, the person is apparently weightless.


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1.7. ELASTIC FORCES 7

F_g

F_f

N

F_f

N

F_g

: Kinetic friction force by the ground

: Normal force by the ground

: The skiers weightx

y

Figure 1.1: Free-body diagram of a skier

Remember that when you stand on the ground, the upward normal force by the ground balancesyour weight. If youre actually standing on a balance, you are applying a downward force on itequal in magnitude to its force on you (Newtons 3rd. law). This is what the scale reads. Onlyif you are standing on level ground, is this reading equal to your weight. If you are standing on ascale inside an elevator with vertical acceleration ax (lets take +x downwards), Newtons 2nd lawgives

mg

|N

|= max (1.3)

(draw a free-body diagram and confirm this). The scale reading is equal to | N|, which is

| N| = m(g ax) (1.4)

So for example, if you and the elevator are in free fall, ax = g and therefore | N| = 0. On the otherhand, if the elevator was accelerating upwards, ax would be negative, and m(g ax) > mg: Youwould feel heavier.

This leads to a surprising interpretation of what the normal force is: In magnitude,

its the apparent weight of a body.

1.7 Elastic forces

If you stretch a spring, it pulls on your hands; if you compress it, it pushes out. The spring exertsa force that depends on its length, or more precisely, on its extension, the difference between itspresent length and its natural length.


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8 CHAPTER 1. FORCES

In fact, Robert Hooke found experimentally that the force applied by the spring is proportional

to its extension. (Click here for a nice graphical illustration of this). If we callFel the force by thespring, and x its extension, then along the x direction,

Fx = kx (1.5)

where k is called the elastic constant of the spring, and the sign indicates that the force isdirected opposite to the extension.

1.8 Systems of interacting bodies

So far, we have looked at how single bodies are affected by forces. But a single body may in

fact be composed of several units linked together, which interact among themselves as well as withtheir environment. Some examples of systems of bodies are

the atoms in a large, macroscopic object.

Exercise 1.8.1 Estimate how many atoms fit on the head of a pin. Data: The volume

occupied by an atom of iron is about 1029 m3, and the head of a pin has a volume of about1 mm3.

a train

the solar system

a galaxy.

Consider for example the cart and horse shown in Figure 1.2. Only the horizontal forces are shown.Fhc is the force applied by the horse to the cart; the cart pulls on the horse with Fch. The horse ispropelled forward by the external force F (originating outside the system, applied by the ground).N2 applied to the horse in the horizontal direction (x) gives

|F| |Fch| = mhax (1.6)

and applied to the cart it gives

|Fhc

|= mcax (1.7)

Notice that since the horse and the cart are llinked together, they both have the same accelerationalong x, ax.

N3 gives an additional equation:

|Fch| = |Fhc| (1.8)
http://www.montypython.net/scripts/galaxy.php3http://www.phas.ucalgary.ca/phys211/fall/hooke.htmhttp://www.bbc.co.uk/education/archive/local_heroes97/bioghooke.shtml


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1.8. SYSTEMS OF INTERACTING BODIES 9

F_hc

F_ch

F

Figure 1.2: Cart and horse.

Adding the two equations from N2 gives

| F| = (mc + mh)ax (1.9)

which is the same thing as N2 applied to the horse-and-cart system as a whole. As for the internalforces Fch and Fhc, they can be obtained from equation 1.7 and N3:

|Fch| = |Fhc| = mcmc + mh

ax (1.10)


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Chapter 2

Circular motion

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12 CHAPTER 2. CIRCULAR MOTION

2.1 Kinematics of circular motion

2.1.1 Uniform circular motion

Click on this link to see an animation of an object in a circular orbit, moving with constant speed.In a coordinate system with the origin at the centre of the circle, its position at any instant of timeis (see Figure 2.1:

x = r cos() (2.1)

y = r sin() (2.2)

and since varies with time as the object goes round the circle, as

x

y

x

y

r

Figure 2.1: Coordinates of an object in circular motion.

=2

Tt = t (2.3)

so

x = r cos(t) (2.4)

y = r sin(t) (2.5)
http://www.phas.ucalgary.ca/physlets/circmotion.htm


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2.1. KINEMATICS OF CIRCULAR MOTION 13

Notice that each coordinate behaves like a 1-dimensional simple harmonic oscillator! (Click here

to see this in action).To get the components of the velocity and the acceleration of this object, we have to take timederivatives. Earlier, we saw that

d

dt(cos(t)) = sin(t) (2.6)

d

dt(sin(t)) = cos(t) (2.7)

so

vx =dx

dt= r sin(t) (2.8)

vy =

dy

dt = r cos(t) (2.9)

and

ax =dvx

dt= 2r cos(t) (2.10)

ay =dvx

dt= 2r sin(t) (2.11)

Comparing this with equations (2.4) and (2.5) above, we see that

ax = 2x (2.12)ay = 2y (2.13)

or even more succinctly in vector form,

a = 2r (2.14)where r is the position vector of the object.

Notice that the acceleration vector points in the opposite direction to the position

vector, namely towards the centre of the circle. The magnitude of the acceleration is

|a| = 2r (2.15)We can also write the acceleration in terms of the speed by noting that

= 2T r = 2r

T= |v| (2.16)

so that

|a| = |v|2

r(2.17)
http://-/?-http://www.phas.ucalgary.ca/physlets/circmotion.htmhttp://www.phas.ucalgary.ca/physlets/circmotion.htmhttp://-/?-


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14 CHAPTER 2. CIRCULAR MOTION

2.1.2 Non-uniform circular motion

An object could move in a circle, but not with constant speed. It could be speeding up, or slowingdown. In that case, in addition to the acceleration towards the centre of the circle that we justsaw, or centripetal acceleration, it would have a component of the acceleration in the directionin which its moving, tangent to the circle. The first component is the radial acceleration, and thesecond is the tangential acceleration. See Figure 2.2.

a_t

a_r

a

Figure 2.2: Radial and tangential components of the acceleration.

2.2 Forces in circular motion

2.2.1 Uniform circular motion

According to Newtons 2nd. law, if an object has an acceleration in a certain direction, there is aforce in that direction. So if the acceleration of an object in uniform circular motion is towards thecentre, it must be produced by a force pointing towards the centre of the circle.

Examples abound. For instance, consider a stone of mass m being whirled around on the end of

a string of length l at speed |v|. The force on the stone is provided by the string, and its magnitudeis| F| = m |v|

2

l(2.18)


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Chapter 3

Gravitation

15


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3.4. KEPLERS THIRD LAW 17

equation (3.1), and according to Newtons 2nd. law,

GMSm

r2= m

v2

r(3.3)

where v is the speed of the planet as it orbits the Sun. If we assume this speed is constant, itfollows that

v =2r

T(3.4)

so

GMSm

r2= m

42r2

T2

r(3.5)

which can be reduced to

T =

2

GMSr3/2

(3.6)

This is Keplers 3rd. law (K3). In fact, Newton derived his law of universal gravitation by reasoningin a similar way from Keplers 3rd. law, in the more general case of an elliptical orbit. (See thesimulation at http://www.phas.ucalgary.ca/physlets/kepler3.htm).

Although we have derived K3 for the special case of planets orbiting the Sun, it can be gener-alized to any satellites with a common parent body, e.g. satellites of the Earth.

Notice also that if T and r can be measured independently, the mass of the parent body canbe determined from the constant in K3. It is interesting to note that historically, G was measuredby Henry Cavendish about a century after Newton presented his law, precisely for the purpose ofweighing the Earth, as it was the last vital piece of information missing from equation (3.6).
http://www.phas.ucalgary.ca/physlets/kepler3.htmhttp://www.phas.ucalgary.ca/physlets/kepler3.htm


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18 CHAPTER 3. GRAVITATION


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Appendix A

Mass and gravitational charge

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20 APPENDIX A. MASS AND GRAVITATIONAL CHARGE

A.1 What do we mean by mass?

We have rather carelessly used the word mass in two different contexts, which are really inde-pendent of each other:

1. We first used the word mass in the context of Newtons 2nd. law, which says that whentwo bodies interact, while they exert equal magnitude forces on each other, the accelerationof each one is inversely proportional to the amount of matter it contains, which we called itsmass;

2. On the other hand, Newton tells us that the gravitational force between two bodies is pro-portional to their masses. This need not be so; the gravitational force could be proportionalto some other property. Indeed, the electrical force between two objects is proportional totheir electric charge, which has nothing to do with their mass.

Strictly speaking, we should say that the gravitational force is proportional to the interactingobjects gravitational charge; and it is a remarkable fact that the gravitational charge is thesame thing as the mass in Newtons 2nd. law.

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Dr. A Louro - Physics 211 Notes (2) - Dynamics