USSC2001 Energy Lecture 1Coordinates and Kinematics

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email matwml@nus.edu.sgTel (65) 6874-2749

CONTENTS

The material in these five lectures is based on chapters 1-8 and chapter 38 of the magnificent textbook titled: Fundamentals of Physics by Halliday, Resnick, and Walker, Sixth Edition, extended, John Wiley & Sons, Inc. Singapore, 2001.

Lecture 1: physical quantities, measurement and invariance, rulers and clocks and scales, units, changing units using conversion factors, motion along a straight line, average and instantaneous velocity, acceleration, coordinate systems, vectors, change of coordinates, projectile and circular motion, relative motion

PHYSICAL QUANTITIES AND THEIR MEASUREMENTS

Physics is based on measurements of quantities: e.g. length, time, mass, temperature, pressure and electric current, each of which is measured in its own units by comparison with a standard.

Example. The length of a rod is the distance between two points chosen from opposite ends. The lengths of two rods can be compared by alignment.

A A

B

B

RULERS AND INVARIANCE

Question: Why can the lengths of two rods be compared using alignment?

Answer: Because the length of a rod does not change when the rod is translated and rotated.

The length of a rod, or more generally the set of distances between pairs of fixed points on a rigid body, provides a ruler for measuring lengths because these distances are invariant under translations and rotations

GEOMETRY

Question: How can the midpoint of a rod be found?

Answer: Use a ruler and compass

A BE

C

D

Binary rulers can be built using this bisection methodA B

TUTORIAL 11. In vufoil ‘GEOMETRY’, assume that

length AC = length BC and length AD = length BD

- prove angle XAE = angle XBE for X in {C, D}

- prove that angle AEC = 90 degrees (hint: what is the sum of the interior angles in a triangle?)

- compute length AE if length AB = 12m and length CE = 8m

- in the binary ruler if length AB = 12m locate the point F so that length AF = 4m

GEOMETRY – THE LANGUAGE OF SCIENCE

Physical objects/processes exist in space/time

The philosopher Kant argued that space/time were so fundamental that conscious experience would be impossible without them (don’t leave home without them!)

Geometry describes the structure of space\time, it is (indeed it must be) the language of science

Geometric concepts – eg points, lines, are ‘a priori synthetic’ truths - facts that can not be derived

BASIC CONCEPTS ABOUT LINES

Imagine a line L as a set of points in space

La If then the set a\Lconsists of two rays, each of which is connected

If Lb,a are distinct points then there exists

a unique continuous real-valued function

RL:f that satisfies

1)b(f,0)a(f )u(f)v(f)x(f)y(f

]y,x[interval can be translated into ]v,u[

COORDINATE SYSTEMS FOR LINES

The choice of an ordered set of two distinct points on a line yields a coordinate system for the line

Assume that (x,y) yields a coordinate system f and that (u,v) yields a coordinate system h

Define r = f(v)-f(u), then |r| is the ratio of the length of the intervals [u,v] to the length of the interval [x,y] in any coordinate system and sign r is positive if and only if the intervals have the same orientation

Furthermore

Lp),x(rh)p(rh)p(f

TUTORIAL 1

2. Two coordinate systems f, h are given for a line L such that for some pair (x,y) of distinct points in L

f(x) = 312, f(y) = 512, h(x) = 125, h(y) = 290

Compute constants (real numbers) r and c such that

f(p) = rh(p) + c, p in L

Hint: derive and solve a system of two equations for r and c by choosing carefully certain values for the point p in the equation above

TWO ASPECTS OF TIME

“When did it happen?” A point in time

“What is its duration?” Distance between time points

The duration of a process (that starts and finishes at points in time) is analogous to the length of a rod

CLOCKS AND INVARIANCE

How can we compare two time durations that start at different times?

Our duration-length analogy might provide a clue!

Clue: Find a repeatable physical process whose start and finish are points in time.

Examples: Sandclock,Pendulum, Spring

CLOCKS ARE COORDINATE SYSTEMS

An ideal clock A assigns to each point p in L a realnumber , called its reading at p (in specified units), mathematically, we say that is a real-valued function with domain L, we often illustrate as below

)p(TA

RT

LorRL:T AA

The geometry of time is that of a line in space, let us denote this line, whose points (in time), by L

AT

Here R denotes the set of real numbers

TUTORIAL 1

3. Three clocks A, B, C run at different (uniform) rates and have simultaneous readings, expressed in seconds, shown by the figure below? Express the readings of B and of C as functions of the reading of A.

Hint: consider the previous tutorial problem

A

B

C92

25

312

125 200

142

512

290

CHANGING UNITSWe can change the units in which physical quantities are expressed by using conversion factors – ratios of units that equal 1

Example How many seconds are there in 1 year ?

s31536000

)year1(year1

hours3600

dayhours24

year1days365

SCALES AND MASS MEASUREMENTS

Masses of objects can be compared using a scale.

Question: in which direction will the right side move ?

A B

Answer: up, down iff mass B < , > mass A

TUTORIAL 1

4. The speed limit on a road in the United States is posted at 55 miles per hour. Compute the speed limit in units of meters per second?

5. Eight of nine coins have the same mass and one coin has more mass than the other coins. Describe how to determine the more massive coin by making two weighings using a perfect scale balance.

SYSTEMS OF UNITS

SI (International System) of seven base units include: meter (m)– length or distance traveled by light in a vacuum during 1/299792458 of a secondsecond (s) – time taken by 9192631770 oscillations of the light (of a specified frequency) emitted by a cesium-133 atomkilogram (kg) – mass of a certain platinum-iridium cylinder

These base units can be used to define derived units

watt (W) - energy unit = (1 N) x (1 m)

newton (N) – force unit =

)s/m1()kg1(2

TUTORIAL 1

6. We explained a geometric method to divide a ruler’s length by 2. Explain an approach to divide a clock’s duration by 2. Hint: what happens to the oscillation frequency of a pendulum if its length changes? What happens to an oscillator if its mass is changed? Devise a method to compute the ratio of the masses of two objects using a binary ruler and a scale with a movable top arm?

PLANAR AND SPATIAL COORDINATES

Orthonormal coordinates can be constructed to represent points in a plane or in space by ordered pairs or ordered triplets of real numbers called coordinates

Likewise, displacement vectors between ordered pairs of points can be represented by such pairs or triplets and the norm or magnitude of a vector defined as the distance between the points

The Pythagorean theorem implies that the squared vector norm equals the sum of the squares of its coordinates

KINEMATICS

A particle is an idealized object that has a location in space at each point in a specified time interval

This means that a particles position in space is a function of its point in time

Velocity vectors are the derivatives of these functions if time and space are given unit coordinate systems

MOTION IN ONE DIMENSION

Consider a particle that moves along a line in space

We can describe the time and the position of the particle using coordinates (clock and ruler)Then the particles position coordinate is a function of its time coordinate

We can use analytic geometry to construct a plot of this function using orthonormal planar coordinates

MOTION IN ONE DIMENSION

Consider a particle thrown upward from the ground

We can use analytic geometry to construct a graph of this function using orthonormal planar coordinates

t

)t(h

)t(h 4

t

4t

2t

1t

3t

AVERAGE AND INSTANTANEOUS VELOCITIES

The average velocity over the interval is the slope of the dotted line

t

)t(h

4t

2t

1t

3t

]tt[43

,

34

34

tt

)t(h)t(havV

The instantaneous velocity at is the slope of the solid line and equals the derivative dh/dt evaluated at

4tt

4tt

Tangent

Secant

COMPUTING INSTANTANEOUS VELOCITIES

1t

3t

The instantaneous velocity v(t) is computed using differential calculus

2t

4t

t

)t(v

)t(h)t(h

0limtt|dt

dh)t(v 44

44

If the graph of v is shown above2

gtt)0(v)t(h2

1

COMPUTING DISTANCE FROM VELOCITY

The fundamental theorem of calculus says that h equals the signed area under the graph of v

tt

)t(v

ts

0sds)s(vdssu|

ts

0s du

)u(dh)t(h

ACCELERATION, VELOCITY, AND SPEED

Acceleration is the derivative of velocity, therefore velocity is the integral of the acceleration, in the example on the preceeding page the accleration equals –g if the positive direction of height is measured away from the Earth, speed is the absolute value or magnitude of the velocity

t

)t(a

g

TUTORIAL 1

7. Consider the preceeding example of motion in one dimension and

- compute the value of time when the object is at a maximum height

- explain why the average speed over the time interval is larger than the magnitude of the average velocity over the same interval

- graph the possible velocity given air friction(to show the qualitative effect of the air friction)

]tt[42

,

MOTION IN TWO AND THREE DIMENSIONS

Motion can be described coordinatewise, for Example: if the position of a particle moving in a plane is expressed as function of time using an orthonormal coordinate system

2t)t(y,t)t(x,)t(y

)t(x)t(p

Then its velocity, acceleration and speed are

2t41)t(s,2

0)t(a,

t2

1)t(v

TUTORIAL 18. A particle moves with constant speed 5 m/s along a path in a plane described in terms of orthonormal coordinates x,y by the equation9/4

2xy Compute the two possible velocity vectors for the particle when its x coordinate equals 3.

9. A gun is fired directly at an object thrown directly upward when its height is maximum height. Show that the bullet will strikes the target unless it hits the ground before the object hits the ground (ignore air friction and assume that the ground is flat).

10. Compute the acceleration and speed of a particle moving in a circle of radius R and angle ).t(θ