Transcript 4.1 The Concepts of Force and Mass

Chapter 1
Introduction and
Mathematical Concepts
1.1 The Nature of Physics
Physics has developed out of the efforts
of men and women to explain our physical
environment.
Physics encompasses a remarkable
variety of phenomena:
planetary orbits
radio and TV waves
magnetism
lasers
many more!
1.1 The Nature of Physics
Physics predicts how nature will behave
in one situation based on the results of
experimental data obtained in another
situation.
Newton’s Laws → Rocketry
Maxwell’s Equations → Telecommunications
1.1.1. Which of the following individuals did not make significant contributions in
physics?
a) Galileo Galilei
b) Isaac Newton
c) James Clerk Maxwell
d) Neville Chamberlain
1.1.1. Which of the following individuals did not make significant contributions in
physics?
a) Galileo Galilei
b) Isaac Newton
c) James Clerk Maxwell
d) Neville Chamberlain
1.2 Units
Physics experiments involve the measurement
of a variety of quantities.
These measurements should be accurate and
reproducible.
The first step in ensuring accuracy and
reproducibility is defining the units in which
the measurements are made.
1.2 Units
SI units
meter (m): unit of length
kilogram (kg): unit of mass
second (s): unit of time
Let’s watch this brief clip on measurement
MEASUREMENT
1.2 Units
1.2 Units
1.2 Units
1.2 Units
The units for length, mass, and time (as
well as a few others), are regarded as
base SI units.
These units are used in combination to
define additional units for other important
physical quantities such as force and
energy.
1.1.2. Which of the following statements is not a reason that physics is a required
course for students in a wide variety of disciplines?
a) There are usually not enough courses for students to take.
b) Students can learn to think like physicists.
c) Students can learn to apply physics principles to a wide range of problems.
d) Physics is both fascinating and fundamental.
e) Physics has important things to say about our environment.
1.2.1. The text uses SI units. What do the “S” and the “I” stand for?
a) Système International
b) Science Institute
c) Swiss Institute
d) Systematic Information
e) Strong Interaction
1.2.1. The text uses SI units. What do the “S” and the “I” stand for?
a) Système International
b) Science Institute
c) Swiss Institute
d) Systematic Information
e) Strong Interaction
1.2.2. Which of the following units is not an SI base unit?
a) slug
b) meter
c) kilogram
d) second
1.2.2. Which of the following units is not an SI base unit?
a) slug
b) meter
c) kilogram
d) second
1.2.3. Complete the following statement: The standard meter is defined in terms of
the speed of light because
a) all scientists have access to sunlight.
b) no agreement could be reached on a standard meter stick.
c) the yard is defined in terms of the speed of sound in air.
d) the normal meter is defined with respect to the circumference of the earth.
e) it is a universal constant.
1.3 The Role of Units in Problem Solving
THE CONVERSION OF UNITS
1 ft = 0.3048 m
1 mi = 1.609 km
1 hp = 746 W
1 liter = 10-3 m3
1.3 The Role of Units in Problem Solving
Example 1 The World’s Highest Waterfall
The highest waterfall in the world is Angel Falls in Venezuela,
with a total drop of 979.0 m. Express this drop in feet.
Since 3.281 feet = 1 meter, it follows that
(3.281 feet)/(1 meter) = 1
 3.281feet 
Length 979.0 meters
  3212feet
 1 meter 
1.3 The Role of Units in Problem Solving
1.3 The Role of Units in Problem Solving
Reasoning Strategy: Converting Between Units
1. In all calculations, write down the units explicitly.
2. Treat all units as algebraic quantities. When
identical units are divided, they are eliminated
algebraically.
3. Use the conversion factors located on the page
facing the inside cover. Be guided by the fact that
multiplying or dividing an equation by a factor of 1
does not alter the equation.
1.3 The Role of Units in Problem Solving
Example 2 Interstate Speed Limit
Express the speed limit of 65 miles/hour in terms of meters/second.
Use 5280 feet = 1 mile and 3600 seconds = 1 hour and
3.281 feet = 1 meter.
feet
 miles
 miles 5280feet  1 hour 
Speed   65
11   65


 95
second
 hour 
 hour  mile  3600s 
feet 
feet  1 meter 
meters


Speed   95
1   95

  29
second
 second
 second 3.281feet 
1.3 The Role of Units in Problem Solving
DIMENSIONAL ANALYSIS
[L] = length
[M] = mass
[T] = time
Is the following equation dimensionally correct?
x  vt
1
2
2
L 2
L   T  LT
T 
1.3 The Role of Units in Problem Solving
Is the following equation dimensionally correct?
x  vt
L
L   T  L
T 
1.3.1. Which one of the following statements concerning unit conversion is false?
a) Units can be treated as algebraic quantities.
b) Units have no numerical significance, so 1.00 kilogram = 1.00 slug.
c) Unit conversion factors are given inside the front cover of the text.
d) The fact that multiplying an equation by a factor of 1 does not change an equation
is important in unit conversion.
e) Only quantities with the same units can be added or subtracted.
1.3.1. Which one of the following statements concerning unit conversion is false?
a) Units can be treated as algebraic quantities.
b) Units have no numerical significance, so 1.00 kilogram = 1.00 slug.
c) Unit conversion factors are given inside the front cover of the text.
d) The fact that multiplying an equation by a factor of 1 does not change an equation
is important in unit conversion.
e) Only quantities with the same units can be added or subtracted.
1.3.2. Which one of the following pairs of units may not be added together, even after
the appropriate unit conversions have been made?
a) feet and centimeters
b) seconds and slugs
c) meters and miles
d) grams and kilograms
e) hours and years
1.3.2. Which one of the following pairs of units may not be added together, even after
the appropriate unit conversions have been made?
a) feet and centimeters
b) seconds and slugs
c) meters and miles
d) grams and kilograms
e) hours and years
1.3.3. Which one of the following terms is used to refer to the physical nature of a
quantity and the type of unit used to specify it?
a) scalar
b) conversion
c) dimension
d) vector
e) symmetry
1.3.3. Which one of the following terms is used to refer to the physical nature of a
quantity and the type of unit used to specify it?
a) scalar
b) conversion
c) dimension
d) vector
e) symmetry
1.3.4. In dimensional analysis, the dimensions for speed are
a)
b)
c)
d)
e)
L 2
T 
L 
T 2
L2
T 2
L 
T 
L
T
1.3.4. In dimensional analysis, the dimensions for speed are
a)
b)
c)
d)
e)
L 2
T 
L 
T 2
L2
T 2
L 
T 
L
T
1.4 Trigonometry
1.4 Trigonometry
ho
sin  
h
ha
cos  
h
ho
tan 
ha
1.4 Trigonometry
ho
tan 
ha
ho
tan 50 
67.2m

ho  tan50 67.2m  80.0m

1.4 Trigonometry
 ho 
  sin  
h
1
 ha 
  cos  
h
1
 ho
  tan 
 ha
1



1.4 Trigonometry
 ho
  tan 
 ha
1



 2.25m 

  tan 
  9.13
 14.0m 
1
1.4 Trigonometry
Pythagorean theorem:
h  h h
2
2
o
2
a
1.4.1. Which one of the following terms is not a trigonometric function?
a) cosine
b) tangent
c) sine
d) hypotenuse
e) arc tangent
1.4.1. Which one of the following terms is not a trigonometric function?
a) cosine
b) tangent
c) sine
d) hypotenuse
e) arc tangent
1.4.2. For a given angle , which one of the following is equal to the ratio of sin /cos
?
a) one
b) zero
c) sin1 
d) arc cos 
e) tan 
1.4.2. For a given angle , which one of the following is equal to the ratio of sin /cos
?
a) one
b) zero
c) sin1 
d) arc cos 
e) tan 
1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the
following ratios is equal to the sine of the angle ?
a)
b)
c)
d)
e)
A
B
A
C
B
C
B
A
C
B
1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the
following ratios is equal to the sine of the angle ?
a)
b)
c)
d)
e)
A
B
A
C
B
C
B
A
C
B
1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the
following ratios is equal to the tangent of the angle  ?
a)
b)
c)
d)
e)
A
B
A
C
B
C
B
A
C
B
1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the
following ratios is equal to the tangent of the angle  ?
a)
b)
c)
d)
e)
A
B
A
C
B
C
B
A
C
B
1.4.5. Which law, postulate, or theorem states the following: “The square of the length
of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths
of the other two sides.”
a) Snell’s law
b) Pythagorean theorem
c) Square postulate
d) Newton’s first law
e) Triangle theorem
1.4.5. Which law, postulate, or theorem states the following: “The square of the length
of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths
of the other two sides.”
a) Snell’s law
b) Pythagorean theorem
c) Square postulate
d) Newton’s first law
e) Triangle theorem
1.5 Scalars and Vectors
A scalar quantity is one that can be described
by a single number:
temperature, speed, mass
A vector quantity deals inherently with both
magnitude and direction:
velocity, force, displacement
1.5 Scalars and Vectors
Arrows are used to represent vectors. The
direction of the arrow gives the direction of
the vector.
By convention, the length of a vector
arrow is proportional to the magnitude
of the vector.
8 lb
4 lb
1.5 Scalars and Vectors
1.5.1. Which one of the following statements is true concerning scalar quantities?
a) Scalar quantities have both magnitude and direction.
b) Scalar quantities must be represented by base units.
c) Scalar quantities can be added to vector quantities using rules of trigonometry.
d) Scalar quantities can be added to other scalar quantities using rules of ordinary
addition.
e) Scalar quantities can be added to other scalar quantities using rules of
trigonometry.
1.5.1. Which one of the following statements is true concerning scalar quantities?
a) Scalar quantities have both magnitude and direction.
b) Scalar quantities must be represented by base units.
c) Scalar quantities can be added to vector quantities using rules of trigonometry.
d) Scalar quantities can be added to other scalar quantities using rules of ordinary
addition.
e) Scalar quantities can be added to other scalar quantities using rules of
trigonometry.
1.5.2. Which one of the following quantities is a vector quantity?
a) the age of the pyramids in Egypt
b) the mass of a watermelon
c) the sun's pull on the earth
d) the number of people on board an airplane
e) the temperature of molten lava
1.5.2. Which one of the following quantities is a vector quantity?
a) the age of the pyramids in Egypt
b) the mass of a watermelon
c) the sun's pull on the earth
d) the number of people on board an airplane
e) the temperature of molten lava
1.5.3. A vector is represented by an arrow. What is the significance of the length of
the arrow?
a) Long arrows represent velocities and short arrows represent forces.
b) The length of the arrow is proportional to the magnitude of the vector.
c) Short arrows represent accelerations and long arrows represent velocities.
d) The length of the arrow indicates its direction.
e) There is no significance to the length of the arrow.
1.5.3. A vector is represented by an arrow. What is the significance of the length of
the arrow?
a) Long arrows represent velocities and short arrows represent forces.
b) The length of the arrow is proportional to the magnitude of the vector.
c) Short arrows represent accelerations and long arrows represent velocities.
d) The length of the arrow indicates its direction.
e) There is no significance to the length of the arrow.
1.5.4. Which one of the following situations involves a vector quantity?
a) The mass of the Martian soil probe was 250 kg.
b) The overnight low temperature in Toronto was 4.0 C.
c) The volume of the soft drink can is 0.360 liters.
d) The velocity of the rocket was 325 m/s, due east.
e) The light took approximately 500 s to travel from the sun to the earth.
1.5.4. Which one of the following situations involves a vector quantity?
a) The mass of the Martian soil probe was 250 kg.
b) The overnight low temperature in Toronto was 4.0 C.
c) The volume of the soft drink can is 0.360 liters.
d) The velocity of the rocket was 325 m/s, due east.
e) The light took approximately 500 s to travel from the sun to the earth.
1.6 Vector Addition and Subtraction
Often it is necessary to add one vector to another.
1.6 Vector Addition and Subtraction
3m
5m
8m
1.6 Vector Addition and Subtraction
1.6 Vector Addition and Subtraction
2.00 m
6.00 m
1.6 Vector Addition and Subtraction
R  2.00 m  6.00 m
2
R
2
2
2.00 m  6.00 m
2
2
 6.32 m
R
2.00 m
6.00 m
1.6 Vector Addition and Subtraction
tan  2.00 6.00
  tan 2.00 6.00  18.4
1

6.32 m
2.00 m

6.00 m
1.6 Vector Addition and Subtraction
When a vector is multiplied
by -1, the magnitude of the
vector remains the same, but
the direction of the vector is
reversed.
1.6 Vector Addition and Subtraction

B
 
AB

A

A
 
AB

B
1.6.1. A and B are vectors. Vector A is directed due west and vector B is directed
due north. Which of the following choices correctly indicates the directions of vectors
A and B?
a) A is directed due west and B is directed due north
b) A is directed due west and B is directed due south
c) A is directed due east and B is directed due south
d) A is directed due east and B is directed due north
e) A is directed due north and B is directed due west
1.6.1. A and B are vectors. Vector A is directed due west and vector B is directed
due north. Which of the following choices correctly indicates the directions of vectors
A and B?
a) A is directed due west and B is directed due north
b) A is directed due west and B is directed due south
c) A is directed due east and B is directed due south
d) A is directed due east and B is directed due north
e) A is directed due north and B is directed due west
1.6.2. Which one of the following statements concerning vectors and scalars is false?
a) In calculations, the vector components of a vector may be used in place of the
vector itself.
b) It is possible to use vector components that are not perpendicular.
c) A scalar component may be either positive or negative.
d) A vector that is zero may have components other than zero.
e) Two vectors are equal only if they have the same magnitude and direction.
1.6.2. Which one of the following statements concerning vectors and scalars is false?
a) In calculations, the vector components of a vector may be used in place of the
vector itself.
b) It is possible to use vector components that are not perpendicular.
c) A scalar component may be either positive or negative.
d) A vector that is zero may have components other than zero.
e) Two vectors are equal only if they have the same magnitude and direction.
1.7 The Components of a Vector


x and y are called the x vectorcomponent

and the y vectorcomponentof r.
1.7 The Components of a Vector

T hevectorcomponentsof A are two perpendicular


vectorsA x and A y thatare parallelto the x and y axes,
 

and add togethervectorially so thatA  A x  A y .
1.7 The Components of a Vector
It is often easier to work with the scalar components
rather than the vector components.
Ax and Ay are thescalar components

of A.
xˆ and yˆ are unit vectors with magnitude1.

A  Ax xˆ  Ay yˆ
1.7 The Components of a Vector
Example
A displacement vector has a magnitude of 175 m and points at
an angle of 50.0 degrees relative to the x axis. Find the x and y
components of this vector.
sin   y r




y  r sin   175m sin 50.0  134m
cos  x r
x  r cos  175m cos50.0  112m

r  112mxˆ  134myˆ
1.7.1. A, B, and, C are three vectors. Vectors B and C when
added together equal the vector A. In mathematical form, A = B +
C. Which one of the following statements concerning the
components of vectors B and C must be true if Ay = 0?
a) The y components of vectors B and C are both equal to zero.
b) The y components of vectors B and C when added together
equal zero.
c) By  Cy = 0 or Cy  By = 0
d) Either answer a or answer b is correct, but never both.
e) Either answer a or answer b is correct. It is also possible that
both are correct.
1.7.1. A, B, and, C are three vectors. Vectors B and C when
added together equal the vector A. In mathematical form, A = B +
C. Which one of the following statements concerning the
components of vectors B and C must be true if Ay = 0?
a) The y components of vectors B and C are both equal to zero.
b) The y components of vectors B and C when added together
equal zero.
c) By  Cy = 0 or Cy  By = 0
d) Either answer a or answer b is correct, but never both.
e) Either answer a or answer b is correct. It is also possible that
both are correct.
1.7.2. Vector r has a magnitude of 88 km/h and is directed at 25 relative to the x
axis. Which of the following choices indicates the horizontal and vertical components
of vector r?
a)
rx
+22 km/h
ry
+66 km/h
b)
+39 km/h
+79 km/h
c)
+79 km/h
+39 km/h
d)
+66 km/h
+22 km/h
e)
+72 km/h
+48 km/h
1.7.2. Vector r has a magnitude of 88 km/h and is directed at 25 relative to the x
axis. Which of the following choices indicates the horizontal and vertical components
of vector r?
a)
rx
+22 km/h
ry
+66 km/h
b)
+39 km/h
+79 km/h
c)
+79 km/h
+39 km/h
d)
+66 km/h
+22 km/h
e)
+72 km/h
+48 km/h
1.7.3. A, B, and, C are three vectors. Vectors B and C when added together equal
the vector A. Vector A has a magnitude of 88 units and it is directed at an angle of
44 relative to the x axis as shown. Find the scalar components of vectors B and C.
a)
Bx
63
By
0
Cx
0
Cy
61
b)
0
61
63
0
c)
63
0
61
0
d)
0
63
0
61
e)
61
0
63
0
1.7.3. A, B, and, C are three vectors. Vectors B and C when added together equal
the vector A. Vector A has a magnitude of 88 units and it is directed at an angle of
44 relative to the x axis as shown. Find the scalar components of vectors B and C.
a)
Bx
63
By
0
Cx
0
Cy
61
b)
0
61
63
0
c)
63
0
61
0
d)
0
63
0
61
e)
61
0
63
0
1.8 Addition of Vectors by Means of Components
  
C AB

A  Ax xˆ  Ay yˆ

B  Bx xˆ  By yˆ
1.8 Addition of Vectors by Means of Components

C  Ax xˆ  Ay yˆ  Bx xˆ  B y yˆ
  Ax  Bx xˆ  Ay  B y yˆ
Cx  Ax  Bx
Cy  Ay  By
1.8.1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s. Vector B has
scalar components Bx = 22 m/s and By = 18 m/s. Determine the scalar components
of vector C = A  B.
a)
Cx
13 m/s
Cy
3 m/s
b)
57 m/s
33 m/s
c)
13 m/s
33 m/s
d)
57 m/s
3 m/s
e)
57 m/s 3 m/s
1.8.1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s. Vector B has
scalar components Bx = 22 m/s and By = 18 m/s. Determine the scalar components
of vector C = A  B.
a)
Cx
13 m/s
Cy
3 m/s
b)
57 m/s
33 m/s
c)
13 m/s
33 m/s
d)
57 m/s
3 m/s
e)
57 m/s 3 m/s