A vector has magnitude and direction, examples being displacement change of position, velocity, acceleration, etc.. All arrows of the same direction and magnitude denote the same vector,
Trang 1unit (AU), which is equal to the mean Earth-Sun distance(1.5 × 1011m) For stellar distances they use the light-year(1 ly = 9.461 × 1012km), which is the
distance that light travels in 1 yr(1 yr = 365.25 days = 3.156 × 107s) with a speed of 299 792 458 m/s They use also the parsec (pc), which is equal to 3.26 light-years Intergalactic distances might be described with a more appropriate unit called the megaparsec Convert the following to meters and express each with an appropriate metric prefix: (a) The astronomical unit, (b) The light-year, (c) The parsec, and (d) The megaparsecs
(6) When you observe a total solar eclipse, your view of the Sun is obstructed by the Moon Assume the distance from you to the Sun(ds) is about 400 times the
distance from you to the Moon(dm) (a) Find the ratio of the Sun’s radius to the
Moon’s radius (b) What is the ratio of their volumes? (c) Hold up a small coin
so that it would just eclipse the full Moon, and measure the distance between the coin and your eye From this experimental result and the given distance between the Moon and the Earth(3.8 × 105km), estimate the diameter of the Moon
(7) Assume a spherical atom with a spherical nucleus where the ratio of the radii is about 105 The Earth’s radius is 6.4 × 106m Suppose the ratio of the radius of the Moon’s orbit to the Earth’s radius(3.8 × 105km) were also 105.
(a) How far would the Moon be from the Earth’s surface? (b) How does this distance compare with the actual Earth-Moon distance given in exercise 6?
Time
(8) Using the day as a unit, express the following: (a) The predicted life time of proton, (b) The age of universe, (c) The age of the Earth, (d) The age of a 50-year-old tree
(9) Compare the duration of the following: (a) A microyear and a 1-minute TV commercial, and (b) A microcentury and a 60-min TV program
(10) Convert the following approximate maximum speeds from km/h to mi/h:
(a) snail(5 × 10−2km/h), (b) spider (2 km/h), (c) human (37 km/h),(d) car
(220 km/h), and (e) airplane (1,000 km/h).
(11) A 12-hour-dial clock happens to gain 0.5 min each day After setting the clock
to the correct time at 12:00 noon, how many days must one wait until it again indicates the correct time?
Trang 214 1 Dimensions and Units (12) Is a cesium clock sufficiently precise to determine your age (assuming it is exactly 19 years, not a leap year) within 10−6s ? How about within 10−3s ? (13) The slowing of the Earth’s rotation is measured by observing the occurrences
of solar eclipses during a specific period Assume that the length of a day is increasing uniformly by 0.001 s per century (a) Over a span of 10 centuries, compare the length of the last and first days, and find the average difference (b) Find the cumulative difference on the measure of a day over this period
Mass
(14) A person on a diet loses 2 kg per week Find the average rate of mass loss in milligrams every: day, hour, minute, and second
(15) Density is defined as mass per unit volume If a crude estimation of the average density of the Earth was 5.5 × 103kg/m3and the Earth is considered to be a sphere of radius 6.37 × 106m, then calculate the mass of the Earth.
(16) A carbon-12 atom(12
6C) is found to have a mass of 1.992 64 × 10−26kg How many atoms of126C are there in: (a) 1 kg? (b) 12 kg? (This number is Avogadro’s number in the SI units.)
(17) A water molecule(H2O) contains two atoms of hydrogen (1
1H), each of which has a mass of 1 u, and one atom of oxygen (16
8O), that has a mass 16 u, approx-imately (a) What is the mass of one molecule of water in units of kilograms? (b) Find how many molecules of water are there in the world’s oceans, which have an estimated mass of 1.5 × 1021kg?
(18) Density is defined as mass per unit volume The density of iron is 7.87 kg/m3,
and the mass of an iron atom is 9.27×10−26kg If atoms are cubical and tightly packed, (a) What is the volume of an iron atom, and (b) What is the distance between the centers of two adjacent atoms
Section 1.3 Dimensional Analysis
(19) A simple pendulum has periodic time T given by the relation:
T = 2πL/g where L is the length of the pendulum and g is the acceleration due to gravity
in units of length divided by the square of time Show that this equation is dimensionally correct
Trang 3(20) Suppose the displacement s of an object moving in a straight line under uniform acceleration a is giving as a function of time by the relation s = ka m t n , where
k is a dimensionless constant Use dimensional analysis to find the values of the powers m and n.
(21) Using dimensional analysis, determine if the following equations are dimen-sionally correct or incorrect: (a)v2 = v2
◦+ 2a s, (b) s = s◦+ v◦t+ 1
2a t2, (c) s = s◦cos kt , where k is a constant that has the dimension of the inverse of
time
(22) Newton’s second law states that the acceleration of an object is directly propor-tional to the force applied and inversely proporpropor-tional to the mass of the object Find the dimensions of force and show that it has units of kg·m/s2in terms of
SI units
(23) Newton’s law of universal gravitation is given by F = Gm1 m2/r2, where F is the force of attraction of one mass, m1, upon another mass, m2, at a distance r.
Find the SI units of the constant G
Trang 4Vectors 2
When a particle moves in a straight line, we can take its motion to be positive in one specific direction and negative in the other However, when this particle moves in three dimensions, plus or minus signs are no longer enough to specify the direction
of motion Instead, we must use a vector
A vector has magnitude and direction, examples being displacement (change of
position), velocity, acceleration, etc Actually, not all physical quantities involve direction, examples being temperature, mass, pressure, time, etc These physical quantities are not vectors because they do not point in any direction, and we call
them scalars.
A vector, such as a displacement vector, can be represented graphically by an
arrow denoting the magnitude and direction of the vector All arrows of the same direction and magnitude denote the same vector, as in Fig.2.1a for the case of a displacement vector
The displacement vector in Fig.2.1b tells us nothing about the actual path taken
from point A to B Thus, displacement vectors represent only the overall effect of the
motion, not the motion itself
Another way to specify a vector is to determine its magnitude and the angle it makes with a reference direction, as in Example 2.1
H A Radi and J O Rasmussen, Principles of Physics, 17
Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-642-23026-4_2,
© Springer-Verlag Berlin Heidelberg 2013
Trang 5Fig 2.1 (a) Three vectors of
the same direction and
magnitude represent the same
displacement (b) All three
paths connecting the two
points A and B correspond to
the same displacement vector
A1
B1
A
B
A2
B2
A B
Example 2.1
A person walks 3 km due east and then 2 km due north What is his displacement vector?
Solution: We first make an overhead view of the person’s movement as shown
in Fig.2.2 The magnitude of the displacement d is given by the Pythagorean
theorem as follows:
d =(3 km)2+ (2 km)2= 3.61 km
The angle that this displacement vector makes relative to east is given by:
tanθ = 2 km
3 km = 0.666
Then:θ = tan−1(0.666 ) = 33.69◦
Thus, the person’s displacement vector is 56.31◦east of north.
Fig 2.2
Start
End
3 km
2 km
d
θ
E N
W
S
Trang 62.2 Properties of Vectors 19
In text books, it is common to use boldface symbols to identify vectors, such as
A , B, etc., but in handwriting it is usual to place an arrow over the symbol, such as,
→
A, B, etc Throughout this text we shall use the handwriting style only and use the→ italic symbols A, B, etc to indicate the magnitude of vectors.
Equality of Vectors
The two vectors →
A and →
B are said to be equal if they have the same magnitude,
i.e A = B, and point in the same direction; see for example the three equal vectors
AB, A1B1, and A2B2in Fig.2.1a
Addition of Vectors
Of course, all vectors involved in any addition process must have the same units The rules for vector sums can be illustrated by using a graphical method To add vector →
B to vector →
A , we first draw vector A→on graph paper with its magnitude represented by a convenient scale, and then draw vector →
B to the same scale with its tail coinciding with the arrow head of →
A , see Fig.2.3a This is known as the triangle method of addition Thus, the resultant vector →
R is the red vector drawn from the tail of →
A to the head of →
B and is shown in the vector addition equation:
→
which says that the vector →
R is the vector sum of vectors →
A and→
B The symbol +
in Eq.2.1and the words “sum” and “add” have different meanings for vectors than they do in elementary algebra of scalar numbers
B
A B
= +
R A
B
= + R
A B
B
R A B
Fig 2.3 (a) In the triangle method of addition, the resultant vector →R is the red vector that runs
from the tail of →A to the head of →B (b) In the parallelogram method of addition, the resultant
vec-tor →R is the red diagonal vector that starts from the tails of both →A and →B This method shows
that →A +→B =→B +→A
Trang 7An alternative graphical method for adding two vectors is the parallelogram rule
of addition In this method, we superpose the tails of the two vectors →
A and→
B; then
the resultant →
R will be the diagonal of the parallelogram that starts from the tail of both →
A and→
B (which form the sides of that parallelogram), as shown in Fig.2.3b Vector addition has two important properties First, the order of addition does not matter, and this is known as the commutative law of addition, i.e
→
A +B→=B→+A→ (Commutative law) (2.2) Second, if there are more than two vectors, their sum is independent of the way in which the individual vectors are grouped together This is known as the associative law of addition, i.e
→
A + (→B +C→) = ( A→+→B ) + C→ (Associative law) (2.3)
The Negative of a Vector
The negative of a vector →
B is a vector with the same magnitude which points in the opposite direction, namely−→B, see Fig.2.4a Therefore, when we add a vector and its negative we will get zero, i.e
→
Adding−B→to →
A has the same effect of subtracting →
B from →
A , see Fig.2.4b, i.e
→
S =A→+ (−→B )
-S A B
A
Fig 2.4 (a) This part of the figure shows vector →B and its corresponding negative vector−→B , both of
which have the same magnitude but are opposite in direction (b) To subtract vector →B from vector→A ,
we add the vector −→B to vector →A to get→S =→A −→B
Trang 82.2 Properties of Vectors 21
Example 2.2
A car travels 6 km due east and then 4 km in a direction 60◦north of east Find the magnitude and direction of the car’s displacement vector by using: (a) the graphical method, and (b) the analytical method
Solution: (a) Let →
A be a vector directed due east with magnitude A= 6 km
and →
B be a vector directed 60◦north of east with magnitude B = 4 km Using
graph paper with a reasonable scale and a protractor, we draw the two vectors
→
A and→
B; then we measure the length of the resultant vector R→ The
measure-ments shown in Fig.2.5indicates that R = 8.7 km Also, the angle φ that the
resultant vector→
R makes with respect to the east direction can be measured and will giveφ = 23.4◦.
Fig 2.5
Start
End
6 km
4
m
θ
E N
W
S
60°
A
B R
(b) The analytical solution for the magnitude of →
R can be obtained from
geom-etry by using the law of cosines R = √A2+ B2− 2AB cos θ as applied to an
obtuse triangle with angleθ = 180◦− 60◦= 120◦, see exercise (10b) Thus:
R=A2+ B2− 2 AB cos θ
=(6 km)2+ (4 km)2− 2(6 km)(4 km) cos 120◦
=(36 + 16 + 24) (km)2= 8.72 km
The angle that this displacement vector→
R makes relative to the east direction, see Fig.2.5, is given by:
sinφ = B sin 60◦
R = 4 km sin 60◦
8.72 km = 0.397
Then:φ = sin−1(0.397) = 23.41◦.
Trang 92.3 Vector Components and Unit Vectors
Vector Components
Adding vectors graphically is not recommended in situations where high precision
is needed or in three-dimensional problems A better way is to make use of the projections of a vector along the axes of a rectangular coordinate system
Consider a vector →
A lying in the xy-plane and making an angle θ with the positive x-axis, see Fig.2.6 This vector →
A can be expressed as the sum of two vectors→
A x and →
A ycalled the rectangular vector components of →
A along the x-axis and y-axis, respectively Thus:
→
Fig 2.6 A vector →
A in the
xy-plane can be presented by
its rectangular vector
components →A
xand →A
y , where
→A =→A x+→A y
θ
A
x
y
A y
A x
o
From the definitions of sine and cosine, the rectangular components of→
A , namely
A x and A y , will be given by:
where the sign of the components A x and A ydepends on the angleθ.
The magnitudes A x and A yform two sides of a right triangle that has a hypotenuse
of magnitude A Thus, from A x and A ywe get:
A=A2
x + A2
y and θ = tan−1
A y
A x
(2.8)
The inverse tan obtained from your calculator is from −90◦< θ < 90◦ This may
lead to incorrect answer when 90◦< θ ≤ 360◦ A method used to achieve the correct
answer is to calculate the angleφ such as:
Trang 102.3 Vector Components and Unit Vectors 23
φ = tan−1|Ay|/|Ax| (2.9)
Then, depending on the signs of A x and A y , we identify the quadrant where the vector
→
A lies, as shown in Fig.2.7
x y
|Ax|
A
o
x y
o
x y
o
x y
o
|Ay|
|Ax|
A
|Ax|
A
|Ay|
|Ax|
A
|Ay|
θ θ
Ax positive
Ax negative
Ay negative
Ax positive
Ay positive
Ax negative
Ay negative
Ay positive
φ
φ
φ
θ
Quadrant I Quadrant II
θ ≡ φ
Fig 2.7 The signs of A x and A ydepend on the quadrant where the vector →A is located
Once we determine the quadrant, we calculateθ using Table2.1
Table 2.1 Calculatingθ from
φ according to the signs of A x
and A y
Sign of A x Sign of A y Quadrant Angleθ
Unit Vectors
A unit vector is a dimensionless vector that has a magnitude of exactly one and points
in a particular direction, and has no other physical significance The unit vectors in
Trang 11the positive direction of the x, y, and z axes of a right-handed coordinate system
are often labeled→
i,→j, and→k, respectively; see Fig.2.8 The magnitude of each unit vector equals unity; that is:
x
y
o
x y
o
OR
Z
Z
i k
j
i
j k
Fig 2.8 Unit vectors →
system
Consider a vector →
A lying in the xy-plane as shown in Fig.2.9 The product
of the component A xand the unit vector→
i is the vector→
A x = Ax→i, which is paral-lel to the x-axis and has a magnitude A x Similarly, A→y = Ay→j is a vector parallel
to the y-axis and has a magnitude A y Thus, in terms of unit vectors we write A→
as follows:
→
Fig 2.9 A vector →A in the
xy-plane can be represented by
its rectangular components A x
and A yand the unit vectors
→
i and →
as →A = A x→i + A y→j
A
x y
A xi o
A y j
This method can be generalized to three-dimensional vectors as:
→
Trang 122.3 Vector Components and Unit Vectors 25
We can define a unit vector→n along any vector, say,A→, as follows:
→
n =
→
A
Adding Vectors by Components
Suppose we wish to add the two vectors →
A = Ax→i + Ay→j and→
B = Bx→i + By→j using the components method, such as:
→
R =→A+B→
= (Ax→i + Ay→j ) + (B x→
i + By→j )
If the vector sum →
R is denoted by→
R = Rx→i + Ry→j, then the components of the
resultant vector will be given by:
R x = Ax + Bx
The magnitude of →
R can then be obtained from its components or the components
of →
A and →
B using the following relationships:
R=R2
x + R2
y =(A x + Bx )2+ (Ay + By )2 (2.16) and the angle that →
R makes with the x-axis can also be obtained by using the
fol-lowing relationships:
θ = tan−1
R y
R x
= tan−1
A y + By
A x + Bx
(2.17)
The components method can be verified using the geometrical method, as shown in Fig.2.10
If →
A = Ax→i +Ay→j +Az→k and→
B = Bx→i +By→j +Bz→k, then we can generalize
the previous case to three dimensions as follows:
→
R =A→+→B = (Ax + Bx )→i + (Ay + By )→j + (Az + Bz )→k