dr. a louro - physics 211 notes (1) - kinematics

8/14/2019 Dr. A Louro - Physics 211 Notes (1) - Kinematics

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Physics 211 Lecture Notes

Part 1: Kinematics

A. A. Louro

Fall 2001


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2


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Contents

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Some functions of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Kinematics in 1 dimension 5

2.1 Motion in a straight line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Displacement along x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Average velocity along x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.4 Instantaneous velocity along x . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.5 Uniform motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.6 Acceleration along x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Uniformly accelerated motion (UAM) . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Calculus concepts 11

3.1 The instantaneous velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Calculating the instantaneous velocity an example . . . . . . . . . . . . . . . . . . 12

3.3 Generalizing the procedure to any time t . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Other functions x(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.5 The concept of derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.6 The time derivatives of some special functions . . . . . . . . . . . . . . . . . . . . . . 14

3.6.1 Rule 1: The derivative of a sum . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.6.2 Rule 2: The derivative of a function multiplied by a constant . . . . . . . . . 14

3.7 The connection with kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3


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4 CONTENTS

4 Oscillators 17

4.1 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 A prototype simple harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 The velocity and acceleration of a simple harmonic oscillator . . . . . . . . . . . . . 19

4.3.1 Time derivatives of trigonometric functions . . . . . . . . . . . . . . . . . . . 194.3.2 The velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.3 The acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Free Fall 23

5.1 Free fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Motion in 2 dimensions - Projectile motion . . . . . . . . . . . . . . . . . . . . . . . 24

6 Vectors 256.1 Vector algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.1.1 Representing a 2D vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Operations with vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.2.1 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2.2 Multiplication by a scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2.3 Time derivative of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.3 Using vector notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29


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Chapter 1

Introduction

1


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2 CHAPTER 1. INTRODUCTION

1.1 Overview

Physics lays claim to a vast territory. Some of the provinces of physics are

the motion of celestial bodies, from interstellar dust grains to clusters of galaxies;

machines;

all things electrical, and all things magnetic;

light;

heat;

the vanishingly small - atoms, nuclei, elementary particles;

the immensely huge, up to the universe itself;

Underlying all these phenomena, there are some simple, general behaviours:

Things interact with each other, affecting each others state. For example, the Earthand the Moon are interacting, forcing each other to revolve about a common centre; ora balloon that has been vigorously rubbed may be attracted to the nearest wall. Thesevery different types of interaction obey the same basic rules.

Mechanics deals with how interactions affect the way things move. In the first section of thiscourse, we look at kinematics, a mathematical description of how things move.

Next, we ask why things move the way they do, which comes under the heading of dynamics.There are several ways of answering this question. One approach is due to Isaac Newton, so itscalled Newtonian mechanics. It is based on the idea that interacting objects apply forces toeach other.

An alternative approach makes use of less intuitive notions, like energy or momentum, andwell reserve this to the end.

1.2 Preliminaries

From the HDOP1:

Classical mechanics The motion of objects that are not too small atoms are barely OK, if youcan disregard their internal structure , or too energetic, that is, moving much more slowlythan light. Historically, the classical period in physics research ranges from the early 1600sto the early 1900s.

1Hitchhikers Dictionary of Physics; non-existent, yet useful!


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1.3. SOME FUNCTIONS OF INTEREST 3

Particle Particle means more that very small, point-like thing. Any object that is moving

rigidly, without spinning, can be considered as a particle. This is because all points in theobject move in the same way, so one point can represent the entire object.

In this course, we will deal almost exclusively with particles. At the end, we willdiscuss briefly rotational equilibrium of large objects.

Observer The motion of an object can only be described relative to an observer. Relative toitself2, of course, the observer is stationary.

Space In classical mechanics, an invisible framework where material objects are located. To keeptrack of a moving object, we can imagine a grid extending throughout space, attached to anobserver. This is called the observers reference frame.

In classical mechanics, the distance between two points in space is independent ofthe observer.

Time What a clock measures.

In classical mechanics, the time interval between two events is measured the sameby all observers.

SI units An ingenious system of units, which we shall build up gradually, called Systeme Interna-tionale. It became official in 1960, and is preferred over all other systems of units in science.For the moment, we are only concerned with distances, measured in meters (m), and times,

measured in seconds (s).

Function A relationship between two quantities, for example air temperature and time. All thefunctions we will consider here involve quantities that may be measured with real numbers.If T stands for temperature and t stands for time, the function T(t) is a listing of the valuesof T at different times t. This is best visualized with a graph. We develop this mathematicalconcept more below.

1.3 Some functions of interest

Although there are infinitely many possible mathematical functions, we can do with a small set ofessential types of function for our purposes in this course. Again, using temperature as a functionof time as an example, the functions of interest are:

2The observer need not be human, or even alive. The observer is really just particle that acts as a reference pointfor measuring position and velocity.


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4 CHAPTER 1. INTRODUCTION

1. A constant:

T(t) = C (1.3.1)A graph of this function is a flat straight line. In this case, the temperature simply does notvary with time.

2. : A linear function:T(t) = A + Bt (1.3.2)

A graph of this function is also a straight line, but with a slope determined by the constantB.

3. A quadratic function:T(t) = A + Bt + Ct2 (1.3.3)

A graph of this function is a curve known as a parabola.

4. A sine or cosine function:

T(t) = sin or T(t) = cos (1.3.4)

It is well known that these functions are periodic: A graph of a sine or cosine functionlooks like a wave, repeating as the angle passes through a full circle. We will encounter suchfunctions when we study the motion of an oscillator, like a weight suspended from a spring,as a function of time. What does time have to do with an angle?

Consider an analog clock. The pointer rotates over time, forming an angle that varies withtime relative respect to some reference direction. If the pointer rotates at a steady rate, the

angle is a linear function of time (see above): = constant t (1.3.5)

So for example, if the ambient temperature T is a sine function of time (which is not a badapproximation to the variation of temperature throughout the day), it could be written asfollows:

T(t) = Tav + A sin(constant t) (1.3.6)

Here, Tav is the average temperature. Some more jargon: The constant A is called theamplitude of the oscillation. Since the sine function oscillates between +1 and 1, thetemperature T oscillates between the two extreme values of Tav + A and Tav A.

Exercise 1.3.1 Looking up the local weather report for September 11th. 2001, I find thatthe low for the day was 5.5 o C, and the high was 9.3 o C. Assuming that the temperature

as a function of time throughout the day was a simple sine function, what was the average

temperature Tav, and what was the amplitude of the temperature oscillation A?


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Chapter 2

Kinematics in 1 dimension

5


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6 CHAPTER 2. KINEMATICS IN 1 DIMENSION

2.1 Motion in a straight line

We start our study of kinematics with 1-dimensional motion, that is, motion along a straight line.Theis has not only the advantage of simplicity, but is also very useful for studying more generalmotion in 2 or 3 dimensions: We will see later that in this case the motion can be separated into1-D motions along independent directions.

We begin with some definitions.

2.1.1 Position

The position of an object moving only on a straight line can be measured with an infinitely longruler, or coordinate axis. The zero of the ruler coincides with the observer. The position of theobject is represented by the distance to the zero point, or origin, with a + or - sign to indicate onwhich side of the origin the object lies. The position on a straight line is usually labelled x. Forexample, if the x axis is vertical, with the origin at your eye level, and positive means upwards,your feet might be at x = 1.60 m.

2.1.2 Displacement along x

Delta notation A compact way of denoting a change in a quantity Q. It is written Q, and isread as The change in Q.

If the object is moving, its position will change over an interval of time t from an initial valuex0 at time t0 to a final value x at time t = t0 + t. The objects displacement along x isx = x x0.

Since position is measured in meters, so is displacement which is a difference betweentwo positions.

Notice that x can be positive or negative. What does the sign of x tell you?

2.1.3 Average velocity along x

The objects average velocity along x is

vav,x = xt

(2.1.1)

Because the displacement is a signed quantity, so is the average velocity. What doesthe sign of the average velocity tell you?


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2.1. MOTION IN A STRAIGHT LINE 7

t

x

x

t

Figure 2.1: Average velocity along x over an interval t.

t

x

t

Figure 2.2: Instantaneous velocity along x at time t.


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2.1.4 Instantaneous velocity along x

The average velocity along x has a simple graphic interpretation. On a graph of position versustime, join the initial point (x0, t0) and the end point (x0 + x, t0 + t) with a straight line. (SeeFigure 2.1). The slope of this straight line is given by x/t, which is just the average velocity ofthe object along x, which is denoted simply by vx.

But how do we measure the instantaneous velocity at a particular time t? Graphically, theprocedure is a natural extension of the way we find an average velocity: Draw a straight linetangent to the curve representing x(t) at the time of interest t. The slope of this straight line isthe objects instantaneous velocity along x at time t. (See Figure 2.2).

Of course, the instantaneous velocity along x can change over time. If you draw several tangentlines to the curve of Figure 2.2 at different times, you will notice that as time progresses the objectmoves initially in the +x direction, slows down and comes to a full stop, and starts moving in the

x direction with increasing speed.

In other words, in general the instantaneous velocity along x is also a function of time,vx(t).

2.1.5 Uniform motion

Uniform motion is motion with constant velocity. In this case, the average velocity along x is thesame regardless of which time interval we choose to average over. Say that the moving object is atx0 at time t0 and at x at a later time t. The average velocity along x coincides with the constantinstantaneous velocity along x, vx, so we can write

vx =x

t=

x x0t t0

(2.1.2)

This allows us to write x as a function of t:

x = x0 + vx(t t0) (2.1.3)

This is a linear function of time.

We can now give a complete description of the motion of an object, at least in the simple caseof uniform motion. The objects position and velocity along x may be obtained at any time t fromthe following equations:

vx = constant (2.1.4)

x = x0 + vx(t t0) (2.1.5)

if we are given the value of the constant vx and the initial condition x0 at time t0.


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2.1. MOTION IN A STRAIGHT LINE 9

t

x

t

v

vx

Figure 2.3: Average acceleration along x.

2.1.6 Acceleration along x

The term acceleration has a broader meaning in physics than in everyday language. Usuallyacceleration means an increase in speed. In physics we use the same word acceleration tomean any change in velocity.

For an object that is moving in a straight line, this implies that slowing down whiletravelling in a certain direction is also an acceleration. And changing direction is alsoan acceleration.

Acceleration is analogous to velocity, in that acceleration is the rate of change of velocity with time,just as velocity is the rate of change of position with time. So, similar to the definitions of averageand instantaneous velocity we have

Average acceleration along x If an objects velocity along x changes by vx over an intervalof time t, its average acceleration along x is

aav,x =vxt

(2.1.6)

Graphically, it is found just like the average velocity. See Figure 2.3; it looks identical toFigure 2.1, but notice this is a graph of velocity as a function of time !

Instantaneous acceleration along x is also found as the slope of a straight line tangent to thecurve representing vx(t). (See Figure 2.4).

From these definitions, we see that the SI unit of acceleration is (m/s)/s. This can also beexpressed as (m/s2), although some physics instructors prefer the more intuitive (m/s)/s.


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t

x

t

v

Figure 2.4: Instantaneous acceleration along x.

2.2 Uniformly accelerated motion (UAM)

2.2.1 Velocity

If ax is constant over time, then the velocity along x is a linear function of time, since

ax =vxt

=vx v0x

t t0(2.2.1)

from which

vx = v0x + ax(t t0) (2.2.2)

2.2.2 Position

The position as a function of time in UAM is similar to the case of uniform motion (see equation(2.1.3), with the addition of a new term because of the acceleration. We present the expression forx(t), and postpone the discussion of the acceleration term until after we have discussed derivatives:

x(t) = x0 + v0xt +1

2axt

2 (2.2.3)


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Chapter 3

Calculus concepts

11


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12 CHAPTER 3. CALCULUS CONCEPTS

3.1 The instantaneous velocity

We have seen that the procedure to find the instantaneous velocity of an object thats moving alongthe x axis at any desired time t is this:

Draw a graph of x(t);

At the point on the curve corresponding to the instant t, draw a straight line tangent to thecurve;

Then the slope of the line gives the instantaneous velocity there.

We can always draw a graph and estimate the velocity at any instant through careful measurement.But at best we get an estimate, and its a tedious procedure to do many times. Instead, well tryto reproduce symbolically this procedure, and arrive at a method for calculating (exactly!) theinstantaneous rate of change of any variable that changes with time.

We begin with a concrete example.

3.2 Calculating the instantaneous velocity an example

Consider the following situation: An object moves along x in a way that its position as a functionof time is

x(t) = t2

if t is measured in seconds, and x in meters. We want to know its instantaneous velocity along xat t = 2.0 s.

We might begin by estimating the instantaneous velocity there as the average velocity betweensay t = 2.0 s and t = 3.0 s. This would just be an approximation, but hopefully a reasonably goodone. Here is the calculation:

vx x(3.0 s) x(2.0 s)

3.0 s 2.0 s=

9.0 m 4.0 m

1.0 s= 5.0 m/s

Of course, if we take a smaller interval of time, we would expect the approximation to improve.Lets repeat the calculation for the interval between t = 2.0 s and t = 2.1 s:

vx

x(2.1 s) x(2.0 s)

2.1 s 2.0 s =

4.41 m 4.0 m

1.0 s = 4.1 m/s

In fact, lets pick successively smaller time intervals starting at t = 2.0 s and see if the approxima-tions approach the same value as they improve. Here is a table summarizing the results, includingthe two approximations we calculated in full:


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3.3. GENERALIZING THE PROCEDURE TO ANY TIMET 13

t (s) x (m) vx xt

1.0 5.0 5.00.1 0.41 4.10.01 0.0401 4.010.001 0.004001 4.001

You get the idea. Clearly, as we make the time interval over which we calculate the average velocityprogressively smaller, the approximation to the instantaneous velocity approaches 4 m/s.

3.3 Generalizing the procedure to any time t

OK, so what we did is this. We have an object whose position as a function of time is x(t) = t2 (inSI units). To find its instantaneous velocity at a time t, we calculated the average velocity over a

really tiny time interval, from t to t + , where (the Greek letter epsilon) stands for a very tinyincrement. Technically, it is said to be infinitesimally small, meaning it can be made as smallas we like.

So the velocity is calculated as the tiny well, infinitesimal displacement over the infinitesimaltime interval. In terms of this is

vx(t) x(t + ) x(t)

(t + ) t=

(t + )2 t2

=

t2 + 2t + 2 t2

= 2t +

Now we see that when is made vanishingly small, the instantaneous velocity becomes exactly

vx = 2t

Notice that this is nicely consistent with our first example, where we found that the velocity att = 2.0s was 4.0 m/s.

3.4 Other functions x(t)

It will be left as an exercise to the reader to verify, using the same procedure as above that ifx(t) = t, then vx(t) = 1. And ifx(t) = 1, then vx = 0, which makes perfect sense because if theposition of the object is 1 m without varying, then its not moving, so its instantaneous velocity iszero.

3.5 The concept of derivative

We can summarize what weve done to calculate the instantaneous velocity like this: We found theaverage velocity along x over a small time interval,

vx x

t


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and then we made t arbitrarily small. The result is written

vx = dxdt

(3.5.1)

and pronounced dee-x-dee-t. The d stands for differential, and it means a vanishingly smallchange in a quantity. The function dx/dt is called the time derivative of x(t).

3.6 The time derivatives of some special functions

All this time weve been using x(t) as a model for a function of time whose rate of change we wantedto calculate. But now were ready to let go of that crutch. What we have learned applies to anyfunction of time. Thus, the instantaneous rate of change of the function f(t) = t2 is df/dt = 2t.Lets summarize the time derivatives we know so far in a table:

f(t) dfdt

t2 2tt 11 0

With these, we can build many more different functions, like f(t) = 3 2t + 45t2. To calculate thetime derivative of a function like that (which is not such a futile exercise as it looks), We need toknow a couple of rules about derivation:

3.6.1 Rule 1: The derivative of a sum

The derivative of a sum is the sum of the derivatives. In other words,d

dt[f(t) + g(t)] =

df

dt+

dg

dt

Exercise 3.6.1 What is the time derivative of the function F(t) = t + t2?

3.6.2 Rule 2: The derivative of a function multiplied by a constant

Here, mathematical notation is much better than words. Call the constant const; the rule is

d

dt[const f(t)] = const

df

dt(3.6.1)

OK, now were ready to stretch our muscles. Try these exercises.

Exercise 3.6.2 What is the time derivative of f(t) = 2 + 3t?

Exercise 3.6.3 What is the time derivative of g(t) = 5t 5t2?


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3.7. THE CONNECTION WITH KINEMATICS 15

3.7 The connection with kinematics

We found before that for uniformly accelerated motion, the position as a function of time is

x = x0 + v0xt +1

2axt

2

where the initial position x0, the initial velocity along x v0x and the acceleration ax are all constants.(Weve assumed t0 = 0 for simplicity). So, the velocity as a function of time should be given bythe time derivative of x(t). Verify and I mean that that the time derivative of x(t) is

v0x + axt

which is indeed the expression of the instantaneous velocity in UAM. But we can go further. The

acceleration is after all the instantaneous rate of change of the velocity with time, so we shouldrecover the acceleration if we take the time derivative of the velocity. Again, verify that the timederivative of v0x + axt is the acceleration ax.


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Chapter 4

Oscillators

17


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18 CHAPTER 4. OSCILLATORS

4.1 Oscillators

Many physical systems may be described by a quantity that varies periodically with time, from thependulum, where the variable is the angle between the string and the vertical direction, to chemicaloscillators where the variable is the concentration of a chemical substance. If the variation followsthe simple form of a sine or a cosine function of time, the system is called a simple harmonicoscillator. In this course, we shall concentrate on simple harmonic oscillators, partly because oftheir simplicity, and also because of their wide applicability.

4.2 A prototype simple harmonic oscillator

For a concrete example with which to develop the theory consider a mass oscillating on the end ofa spring. If the mass is pulled out and released from rest, its position x as a function of of timemight look something like Figure 4.1. The origin of x is chosen as the equilibrium point, where themass rests if it is not oscillating. Notice that when it is oscillating, x(t) fluctuates symmetricallyabout x = 0. The position of the mass is a cosine function of time, of the form

Figure 4.1: Position vs. time of a simple harmonic oscillator


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4.3. THE VELOCITY AND ACCELERATION OF A SIMPLE HARMONIC OSCILLATOR 19

x = A cos2t

P (4.2.1)

The argument of a cosine function is expressed in radians, and the cosine function is periodicwith period 2. By writing the argument of this function of time as 2t/P, the function has theperiodicity of a cosine function, with a time period P.

The quantity 2/P is usually written as , the Greek letter omega, and called the angularfrequency of the oscillator. Then we can rewrite equation 4.2.1 more simply as

x = A cos(t) (4.2.2)

The significance of the constant A is this: As the cosine function oscillates between +1 and 1, theposition of our oscillator varies between +A and A. A is called the amplitude of the oscillation.

4.3 The velocity and acceleration of a simple harmonic oscillator

4.3.1 Time derivatives of trigonometric functions

To discover the form of the velocity and acceleration of a simple harmonic oscillator as functionsof time, we need to be able to calculate the time derivatives of sine and cosine functions, of theform of the expression in equation 4.2.2. Well adopt a semiexperimental approach here, using thegraphical technique of drawing tangent lines to the curve in a graph of the function.

Figure 4.2 shows the position of a simple harmonic oscillator vs. time again, with time measured

in units of the period P and length measured in units of the amplitude A. Draw tangent lines tothe curve at t = 0, 0.5,...3, and estimate their slopes, making a note of the values in a table. Withthis exercise we are estimating the time derivative of x(t) for some values of t. Notice from yourtable that dx/dt is itself an oscillating function of time. Next, verify that your values are consistentwith the following expression:

dx

dt= sin(t) (4.3.1)

Now we need to repeat the exercise for a sine function of time. Figure 4.3 shows the function

x(t) = A sin(t) (4.3.2)

using the same units as in Figure 4.2. Once again, draw tangent lines to the curve at t = 0, 0.5,...3,estimate their slopes, and tabulate the values. Your values should be consistent with

dx

dt= cos(t) (4.3.3)


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-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3

xin

unitsofA

t in units of P

Figure 4.2: The simple harmonic oscillator again


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4.3. THE VELOCITY AND ACCELERATION OF A SIMPLE HARMONIC OSCILLATOR 21

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3

xin

unitsofA

t in units of P

Figure 4.3: A sine function of time


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4.3.2 The velocity

As we saw in the previous section, if the position of a simple harmonic oscillator is given by

x(t) = Acos(t) (4.3.4)

the velocity along x is

vx(t) =dx

dt= Asin(t) (4.3.5)

4.3.3 The acceleration

The acceleration in turn is the time derivative of the velocity:

ax(t) =dvxdt = A

d

dt [sin(t)] (4.3.6)

and using the result of equation (4.3.3) we find

ax(t) = 2A cos(t) (4.3.7)

4.4 Summary

Let us summarize what we have learned about the simple harmonic oscillator. If the position as afunction of time is given by

x(t) = A cos(t) (4.4.1)

the velocity along x isvx(t) = A sin(t) (4.4.2)

and the acceleration along x isax(t) =

2A cos(t) (4.4.3)

One final remark: Notice that the acceleration, like the position itself, is a cosine function of time.In fact,

ax(t) = 2x (4.4.4)

This will gain significance later when we study the forces driving the simple harmonic oscillator.


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Chapter 5

Free Fall

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24 CHAPTER 5. FREE FALL

5.1 Free fall

An important case of UAM is that of objects moving near the surface of the Earth, or any similarlarge body, like Jupiter, or the Moon. The acceleration of any object is constant; in the case of theEarth it is approximately g = 9.8 (m/s)/s1. In this case the object is said to be in free fall (eventhough it may be going up!); if y is its vertical coordinate, measured positive upwards, its positionas a function of time is

y(t) = y0 + v0yt 1

2gt2 (5.1.1)

5.2 Motion in 2 dimensions - Projectile motion

A projectile (like a shotput, for example) moves in two dimensions, both vertically and horizontally

at the same time. However, the motion along each direction is independent of the other, so wecan treat the horizontal motion and the vertical motion separately as 1-D problems. (See thisanimation). Since the acceleration is entirely in the vertical direction, the object moves withconstant velocity along the horizontal. Let x and y be the horizontal and vertica coordinatesrespectively; then the position and velocity of the projectile are given by

x(t) = x0 + v0xt (5.2.1)

y(t) = y0 + v0yt 1

2gt2 (5.2.2)

vx(t) = v0x (5.2.3)

vy(t) = v0y gt (5.2.4)

where x0, y0, v0x, and v0y are as usual the initial conditions.

Notice that it becomes important now to distinguish between motion alongx and alongy, so the subscripts are important! We must keep separate, for instance, vx from vy.

1g varies slightly with latitude, from about 9.79 (m/s)/s at the equator to 9.81 (m/s)/s at the poles.
http://www.phas.ucalgary.ca/physlets/proj.htmhttp://www.phas.ucalgary.ca/physlets/proj.htm


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Chapter 6

Vectors

25


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26 CHAPTER 6. VECTORS

6.1 Vector algebra

By now we have seen that in describing how something moves, the position, velocity, and accelera-tion are measured along independent directions. We speak of the velocity along x and velocityalong y, for instance. It is convenient to combine the information, and speak simply of thevelocity. Similarly, we would say position and acceleration.

However, it is clear that these are not like other physical quantities like length or time, whichcan be specified by a single number. Instead, to specify a velocity we would have to give twonumbers, if the object is moving on a two-dimensional surface1. Such a quantity is called a vector,as opposed to a scalar, like length or time.

6.1.1 Representing a 2D vector

Cartesian representation

Well use velocity as an example. A velocity vector would be written

v = (vx, vy) (6.1.1)

Notice the arrow on top, indicating that this is a vector quantity. Here, x and y are two perpen-dicular directions. The numbers vx and vy are called the x and y components ofv.

Polar representation

One can also specify a vector by giving its size, or magnitude, and its direction. (The magnitudeof a velocity is the speed). The direction may be given as an angle between the direction of thevector and some reference axis. Conventionally, a positive angle means that the direction of thevector is counterclockwise from the reference axis.

The magnitude of a vector v is written |v|.

Representing a vector graphically

For the visually oriented, a very good way of representing a vector graphically is by means of anarrow. If the drawing is made to scale, the length of the arrow may be made proportional to themagnitude of the vector. The direction of the arrow is of course the same as the direction of thevector.

Passing between the cartesian and the polar representation

1. Given the components vx and vy of a vector, what are its magnitude and direction? Refer tofigure 6.1. From the shaded triangle in the figure, Phythagoras theorem gives right away

1Throughout this course we will only consider at most 2D situations.


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6.1. VECTOR ALGEBRA 27

x

y

v_x

v_y|v|

Figure 6.1: Anatomy of a vector

|v| =

v2x + v2y

and the angle between the vector and the x axis is

= arctan

vyvx

A word of caution about calculators: An equation of the type = arctan(z) has two possiblesolutions (like a quadratic equation has two possible solutions). If you use a calculator to findthe direction of a vector, you will get one of the two possible values, but you have to checkthe signs of the components vx and vy to see if your calculator gave you the right answer; youmight have to add 180o.

2. Given the magnitude and direction, what are the components vx and vy? Again, refer to

Figure 6.1. The x and y components are given by

vx = |v| cos

vy = |v| sin


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6.2 Operations with vectors

6.2.1 Vector addition

Vectors may be added by adding each component separately:

a + b = (ax + bx, ay + by)

Graphically, two vectors may be added by the parallelogram method illustrated in Figure6.2, or by the tip-to-tail method illustrated in Figure 6.3.

x

y

a_y

b_y

a_xb_x

b

a

a+b

Figure 6.2: Parallelogram method of addition

6.2.2 Multiplication by a scalar

Scalar multiplication involves multiplying each component of a vector by the same scalar:

Ca = (Cax, Cay)

This operation is equivalent to stretching the vector, and if C is negative, flipping it.

6.2.3 Time derivative of a vector

If the components of a vector change over time, it is legitimate to find the time derivative ofthe vector, by taking the time derivatives of each component. Thus for example, the velocity


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6.3. USING VECTOR NOTATION 29

x

y

a

b

a+b

Figure 6.3: Tip-to-tail method of addition

vector is the time derivative of the position vector:

v =dr

dt

which means simply that

vx =dx

dt

vy =dydt

6.3 Using vector notation

Consider for example, the equations for the x and y components of the position of an objectin free fall:

x = x0 + v0xt

y = y0 + v0yt 1

2gt2

(As usual, x is horizontal, and y is vertical, positive upwards). We can convey the sameinformation with a single vector equation:

r = r0 + v0t +1

2at2


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where r is the position vector at a time t, r0 is the initial position vector, and a = (0, g)

is the acceleration vector.It is comforting to see that the time derivative dr/dt does indeed give the velocity vectorv0 gt. (Verify this!)

dr. a louro - physics 211 notes (1) - kinematics

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