chapter 1 kinematics revised autosaved

Chapter 11.1 Measurement, Standards of Length, Mass, and Time1.2-Dimensional Analysis1.3 Coordinate Systems1.4 Vectors and scalars1.5 The Displacement, Velocity, and Acceleration Vectors

General Physics I

PETROVIETNAM UNIVERSITY

FUNDAMENTAL SCIENCES DEPARTMENT

Pham Hong Quang E-mail: quangph@pvu.edu.vn

2 Credit : 03 (2 Theory and 1Tutorials)

+ Theory: 30 hour+ Tutorials: 15 hour

- Lab:

Scheme of Assessment

Assessment Type Duration Mark Weighting

[1] Cơ sở Vật lý, Tập I, II, III: Cơ học, nhiệt học, David

Halliday, Robert Resnik, Jearl Walker, bản dịch tiếng Việt, NXB Giáo dục (1999)

[2] Fundamentals of physics, 8th ed., Extended, David

Halliday, Robert Resnick and Jearl Walker, John Wiley &

Sons (2008)

[3] Vật lý đại cương, tập 1, Cơ-Nhiệt, Lương Duyên Bình

chủ biên

Course description

The course is designed to meet the needs of student majoring

in Petroleum Geology – Geophysics, Drilling and Production

Technology, and Refinery and Petrochemical It is

introductory course in Newtonian mechanics Many

concepts from General Physics I will be used in this course

such as: position, velocity, acceleration, force, Newton’s laws

of motion, work and energy The course uses algebra,

geometry and trigonometry, vectors and vector arithmetic, and

some calculus The course has lecture, homework and

Course objectives

•Provide a clear understanding of the basic concepts and

integrating their knowledge in various disciplines of

Chapter 1

1.1 Measurement, Standards of Length, Mass, and Time

1.2-Dimensional Analysis

1.3 Coordinate Systems

1.4 Vectors and scalars

1.5 The Displacement, Velocity, and Acceleration

Vectors

1.6 One dimensional motion with constant acceleration

1.7 Two dimensional motions with constant

acceleration

Kinematics of a point-like object

Learning outcome

The students should be able to:

•Identify the base quantities in the SI system.

•Identify the vectors and scalars.

•Given the components of a vector, draw the vector and

determine its magnitude and orientation.

•Identify the Dimensional Analysis

•Identify the base quantities of kinematics.

•For constant acceleration, apply the relationships between

position, displacement, velocity, acceleration, and elapsed

time.

•Apply the constant-acceleration equations to one

dimensional motion, two dimensional motion, uniform circular motion and projectile motion.

Length (1983)

Units

SI – meter, mDefined in terms of a meter – the distance traveled by light in

a vacuum during a given time(1/299792458 s)

See Table 1.1 for some examples of lengths

Length, Cont.

Mass (1887)

Units

SI – kilogram, kgDefined in terms of a kilogram, based on a specific cylinder kept at the International Bureau

of StandardsSee Table 1.2 for

1.1 Standards of Length, Mass, and Time

Time (1967)

Units SI

seconds, sDefined in terms of the oscillation of radiation from a cesium atom

(1 second has 9 192 631 770 oscillations)

See Table 1.3 for some approximate time intervals

Time, Cont.

1.1 Standards of Length, Mass, and Time

US units

Length is measured in feetTime is measured in

secondsMass is measured in slugs

often uses weight, in pounds, instead of mass

as a fundamental quantity

1.1 Standards of Length, Mass, and Time

The prefixes can be used

with any base units

They are multipliers of the

base unit

Examples:

1.2-Dimensional Analysis

•Dimension has a specific meaning – it denotes the physical nature of a quantity

•Dimensions are denoted with square brackets

Length [L]

Mass [M]

Time [T]

•Technique to check the correctness of an equation or to assist

in deriving an equation

•Dimensions (length, mass, time, combinations) can be treated

as algebraic quantities

add, subtract, multiply, divide

•Both sides of equation must have the same dimensions

•Dimensions of some common quantities are given below

•Used to describe the position of a point in space

•Coordinate system consists of

A fixed reference point called the origin

Specific axes with scales and labels

Instructions on how to label a point relative to the origin and the axes

Descartes Coordinate

System

Points are labeled (x,y,z)

Definition of Scalar and vector Scalar is a quantity that has magnitude but not

direction

For instance mass, volume, distance Vector is a directed quantity, one with both

magnitude and direction

For instance acceleration, velocity, force

Vector Example

•A particle travels from A to

B along the path shown by

the dotted red line

This is the distance

traveled and is a scalar

•The displacement is the

solid line from A to B

The displacement is

independent of the path

taken between the two

points

Displacement is a vector

Vector Notation

•Text uses bold with arrow to denote a vector:

•Also used for printing is simple bold print: A

•When dealing with just the magnitude of a vector, an

italic letter will be used: | | or A

The magnitude of the vector has physical units

The magnitude of a vector is always a positive

Adding Vectors Graphically

•Choose a scale

•Draw the first vector, , with the appropriate length and in

the direction specified, with respect to a coordinate system

•Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system

whose origin is the end of vector and parallel to the

coordinate system used for

Adding Vectors

Graphically, cont.

•Continue drawing the

vectors “tip-to-tail”

•The resultant is drawn

from the origin of to

the end of the last vector

•Measure the length of

and its angle

Use the scale factor to

Component Method of Adding Vectors

•Graphical addition is not recommended when

High accuracy is required

If you have a three-dimensional problem

•Component method is an alternative method

It uses projections of vectors along coordinate axes

Components of a Vector

•A component is a projection of a

vector along an axis

Any vector can be completely

described by its components

•It is useful to use rectangular

components

These are the projections of

Components of a

Vector

•Assume you are given a

vector

•It can be expressed in

terms of two other vectors,

and

•These three vectors form a

right triangle

Unit Vectors, cont.

•The symbols

represent unit vectors

•They form a set of mutually

perpendicular vectors in a

right-handed coordinate

system

Adding Vectors Using Unit Vectors

Using

Then

and so R x = A x + B x and R y = A y + B y

Scalar product of two vectors

The dot product of two vectors is the sum of the products of

their corresponding components If a=<a1, a2> and b=<b1,

b2>, then a·b= a1b1+a2b2

Ex If a=<1,4> and b=<3,8>, then a.b=3+32=35

If θcosϕ is the angle between vectors a and b, then

Note that the dot product of two vectors produces a scalar

Therefore it is sometimes called a scalar product.

Vector product of two vectors

The cross product a x b of two

vectors a and b, unlike the dot product,

is a vector

For this reason, it is also called the vector product

Note that a x b is defined only when a and b

are three-dimensional (3-D) vectors.

If a = ‹a1, a2, a3› and b = ‹b1, b2, b3›, then

the cross product of a and b is the vector

Vector product of two vectors, cont.

In order to make the definition easier to remember, we use

the notation of determinants

A determinant of order 3 can be defined in terms of

second-order determinants as follows:

Vector product of two vectors, cont.

We see that the cross product of the vectors

a = a1i +a2j + a3k and b = b1i + b2j + b3k

is:

Vector product of two

vectors, cont.

Direction:

If the fingers of your right

hand curl in the direction of a

rotation (through an angle

less than 180°) from a to b,

then your thumb points in the

1.5 The Displacement, Velocity, and Acceleration Vectors

Position and

Displacement

•The position of an object is

described by its position

vector,

•The displacement of the

object is defined as the

change in its position

1.5 The Displacement, Velocity, and Acceleration Vectors

1.5 The Displacement, Velocity, and Acceleration

Vectors

Instantaneous velocity

•The instantaneous velocity is

the limit of the average velocity

as Δt approaches zero

As the time interval

becomes smaller, the

direction of the

displacement approaches

that of the line tangent to

the curve

1.5 The Displacement, Velocity, and Acceleration Vectors

Instantaneous Velocity, cont.

•The direction of the instantaneous velocity vector at any point in a particle’s path is

along a line tangent to the path at that point and in the direction of motion

•The magnitude of the instantaneous velocity vector is the speed

1.5 The Displacement, Velocity, and Acceleration

1.5 The Displacement, Velocity, and Acceleration Vectors

Average Acceleration,

cont.

•As a particle moves, the

direction of the change in

velocity is found by vector

subtraction

•The average acceleration is

1.5 The Displacement, Velocity, and Acceleration Vectors

Instantaneous acceleration

•The instantaneous acceleration is the limiting value of

the ratio as Δt approaches zero

The instantaneous equals the derivative of the velocity vector with respect to time

1.5 The Displacement, Velocity, and Acceleration Vectors

Producing An Acceleration

Various changes in a particle’s motion may produce an

acceleration

• The magnitude of the velocity vector may change

• The direction of the velocity vector may change

Even if the magnitude remains constant

• Both may change simultaneously

1.6 One-Dimensional Motion with Constant Acceleration

Kinematic Equation

1.6 One-Dimensional Motion with Constant Acceleration

1.7 Two dimensional motion with constant acceleration

When the two-dimensional motion has a constant

acceleration, a series of equations can be developed that

describe the motion

These equations will be similar to those of one-dimensional kinematics

Motion in two dimensions can be modeled as two

independent motions in each of the two perpendicular

directions associated with the x and y axes Any influence in

Position vector for a particle moving in the xy plane

The velocity vector can be found from the position vector.

Since acceleration is constant, we can also find an expression for the velocity as a function of time:

The position vector can also be expressed as a function of time

1.6 Two dimensional motion with constant acceleration

•Uniform circular motion occurs when an object

moves in a circular path with a constant speed

•The associated analysis motion is a particle in uniform circular motion

•An acceleration exists since the direction of the

motion is changing

•This change in velocity is related to an acceleration

•The velocity vector is always tangent to the path of the object

Changing Velocity in

Uniform Circular Motion

•The change in the velocity

vector is due to the change in

direction

•The vector diagram shows

Centripetal Acceleration

•The acceleration is always perpendicular

to the path of the motion

•The acceleration always points toward the center of the circle of motion

•This acceleration is called the centripetal

acceleration

Centripetal Acceleration, cont.

•The magnitude of the centripetal acceleration vector is

given by

•The direction of the centripetal acceleration vector is

always changing, to stay directed toward the center of

•The period, T, is the time required

for one complete revolution

•The speed of the particle would be the circumference of the circle of motion divided by the period

•Therefore, the period is defined as

•The magnitude of the velocity could also be changing

•In this case, there would be a tangential acceleration

•The motion would be under the influence of both

tangential and centripetal accelerations

•An object may move in both the x and y directions simultaneously

•The form of two-dimensional motion we will deal with is called

projectile motion

Assumptions of Projectile Motion

•The free-fall acceleration is constant over the range of

motion

It is directed downward

This is the same as assuming a flat Earth over the range

of the motion

It is reasonable as long as the range is small compared

to the radius of the Earth

Projectile Motion Diagram

Analyzing Projectile Motion

•Consider the motion as the superposition of the motions in

the x- and y-directions

•The actual position at any time is given by

•The initial velocity can be expressed in terms of its

components

vxi = vi cos q and v yi = vi sin q

The x-direction has constant velocity a x = 0

Acceleration in x-direction is 0 Acceleration in y-direction is -g.

(Constant velocity) (Constant acceleration)

1.10 Projectile Motion

Range and Maximum

Height of a Projectile

•When analyzing projectile motion,

two characteristics are of special

interest

•The range, R, is the horizontal

distance of the projectile

•The maximum height the

Height of a Projectile, equation

•The maximum height of the projectile can

be found in terms of the initial velocity vector:

More About the Range of a Projectile

Range of a Projectile, final

•The maximum range occurs at q i = 45o

•Complementary angles will produce the same range

The maximum height will be different for the two angles

The times of the flight will be different for the two angles

1.10 Projectile Motion

Kinematics; Point-like object; Dimension; Coordinate

Systems; Vectors and Scalars; Displacement; Velocity; Acceleration; One and two dimensional motions;

Angular Velocity; Uniform circular motion; Centripetal

Acceleration;Projectile Motion

•Scalar: number, with appropriate units

• Vector: quantity with magnitude and direction

• Vector components: A x = A cos θ, B y = B sin θ

• Magnitude: A = (A x 2 + A y 2 ) 1/2

• Direction: θ = tan-1 (A y / A x )

•Position vector points from origin to location

• Displacement vector points from original position to final

position

• Velocity vector points in direction of motion

• Acceleration vector points in direction of change of motion

•Components of motion in the x- and y-directions can be

treated independently

•In projectile motion, the acceleration is –g

•If the launch angle is zero, the initial velocity has only an component

x-• The path followed by a projectile is a parabola

• The range is the horizontal distance the projectile travels

• The acceleration is always perpendicular to the path of the motion

•The magnitude of the centripetal acceleration vector is

given by

Check your understanding 1

The graph above shows the velocity versus time for an object moving in a straight line At what time after t = 0 does the

object again pass through its initial position?

(A) Between 0 and 1 s (B) 1 s (C) Between 1 and 2 s (D) 2 s (E) Between 2 and 3 s

(C) Area bounded by the curve is the displacement By

Check your understanding 2

A body moving in the positive x direction passes the origin at time t = 0 Between t = 0 and t = 1 second, the body has a constant speed of 24 meters per second At t = 1 second, the body is given a constant acceleration of 6 meters per second squared in the negative x direction The position x of the

body at t = 11 seconds is

(A) + 99m (B) + 36m (C) – 36 m (D) – 75 m (E) – 99 m

Check your understanding 3

A truck traveled 400 meters north in 80 seconds, and then it traveled 300 meters east in 70 seconds The magnitude of the average velocity of the truck was most nearly

(A) 1.2 m/s (B) 3.3 m/s (C) 4.6 m/s (D) 6.6 m/s (E) 9.3 m/s

Average velocity = total displacement/total time;

magnitude of total displacement = 500 m and total

time = 150 seconds Ans: B