Physical quantities, Units and Vectors Flashcards
100% mastery in the concepts
systematic error
errors as a result of the experimenting instrument
random error
errors as a result of the lapses in skill of the experimenter
errors consistent in magnitude and direction
systematic
human error
random error
errors that vary in magnitude and direction
random error
the SI units were a universally agreed system when
1960(adopted internationally )
1 inch to cm
2.54 cm
1 foot to meter
0.3048
1 foot to cm
30.48cm
1 ft to inches
12 inches
1 mile to meter
1609m
1 litre
10^-3 m^3
fundamental units
base units
there are _______ fundamental units
seven
the fundamental units
mass
length
Time
Electric current
Thermodynamic Temperature
Luminous Intensity
Quantity of substance
mass
Kg
Length
m
Time
s
Electric Current
A
Thermodynamic termperature
K
Luminous intensity
Candela(C)
Quantity of substance
mol
kilogram
the mass of a particular platinum-iridum cylinder, kept at the international bureau of weights and measures, servres, france
metre
the distance travelled by light in 1/299792458 sec0nd
seconds
the time interval of 9,192,631,770 periods or cycles of the duration of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom
ampere
is a constant current that will produce a force equal to 2 X 10^-7 newton per metre of length when maintained in two straight parallel conductors of infinite length of negligible circular cross section and placed one metre apart in the vaccum
kelvin
the kelvin is defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water
mole(mol)
the amount of a substance of a system which contains as many as there are carbon atoms in 0.012Kg of carbon-12
m^2
area
m^3
volume
Kgm^-3
density {mass/volume}
speed/velocity
ms^-1
ms^-2
Acceleration
Kgms^-2
force {mass X acceleration}
in terms of unit, impulse is the same as
momentum {Kgms^-1}
momentum
{Kgms^-1}
impulse
{Kgms^-1}
power/radiant flux
joules per second{watts} {force x Velocity}
energy/work done
newton meter{force X distance}
pressure
NM^-2 {Force/Area}
unit of plane angle
radian
surface tension
N/m
unit of solid angle
steridian
peta
10^15
tera
10^12
giga
10^9
mega
10^6
kilo
10^3
deci
10^-1
10^-2
centi
10^-3
milli
10^-6
micro
10^-9
nano
10^-12
pico
order of magnitude
an approximate measure of the size of a number equal to the logarithm(base 10) rounded to a whole number
the smallest power of 10 required to represent a number
the fundamental dimensions are
length mass and time
spring constant(spring stiffness constant)
unit
newton per metre
torque
force X perpendicular distance
Angular Velocity (ω)
Angular velocity represents the rate of change of angular displacement and is measured in radians per second (rad/s). Radians are dimensionless, so angular velocity has the same dimension as frequency (1/time).
The dimensional formula for angular velocity is:
\text{[ω]} = \text{T}^{-1}
Where T is time.
Viscous Force (F)
Viscous force is measured in the same units as force, typically in newtons (N).
The dimensional formula for force is:
F= MLT-2
Where:
• M is mass, • L is length, and • T is time.
Viscous Force (F)
Viscous force is measured in the same units as force, typically in newtons (N).
The dimensional formula for force is:
F= MLT-2
Where:
• M is mass, • L is length, and • T is time.
Viscous Force (F)
Viscous force is measured in the same units as force, typically in newtons (N).
The dimensional formula for force is:
\text{[F]} = \text{MLT}^{-2}
Where:
• M is mass, • L is length, and • T is time.
Viscous Force (F)
Viscous force is measured in the same units as force, typically in newtons (N).
The dimensional formula for force is:
\text{[F]} = \text{MLT}^{-2}
Where:
• M is mass, • L is length, and • T is time.
Viscosity (η)
Viscosity (dynamic viscosity) is measured in pascal-seconds (Pa·s) or equivalently in units of N·s/m² or kg/(m·s).
The dimensional formula for viscosity is:
\text{[η]} = \text{ML}^{-1}\text{T}^{-1}
Where:
• M is mass, • L is length, and • T is time.
The vector product is anti–commutative:
The vector product is anti–commutative: a × b = −b × a. Relations among the unit vectors for vector products are:
i×j=k j×k=i k×i=j
scalar quantities
can be described completely by their magnitudes
examples of scalar quantities
mass
area
volume
density
time
temperature
work
energy
power
vector quantities
can be described by their magnitude and direction
examples of vector quantities
displacement
velocity
acceleration
force
weight
momentum
The vector product (or cross product) of vectors a and b is a vector c whose mag- nitude is given by
c = absinφ
where φ is the smallest angle between a and b. The direction of c is perpendicular to the plane containing a and b with its orientation given by the right–hand rule. One way of using the right–hand rule is to let the fingers of the right hand bend (in their natural direction!) from a to b; the direction of the thumb is the direction of c = a × b.
1 pound =
4.448221615260 newtons (exactly)
1 kilogram is equal to approximately _____ pounds (lb).
2.20462
1 kilogram is equal to approximately _____ pounds (lb).
2.20462
1 atomic mass unit =
1u = 1.6605 × 10−27 kg
1 m is how many feet
3.28ft
1 square kilometer (km2) is how many hectares
100 hectares
1 ha =
2.471 acres
A scalar quantity can be described by a single
number.
true
A vector quantity has both a magnitude and a
direction in space.
true
unit vector
A unit vector has a magnitude
of 1 with no units.
The scalar product
(also called the “dot
product”) of two
vectors is
a * b = |a| × |b| × cos(θ)
If two vectors are perpendicular then their scalar product is
Zero
The scalar product (or dot product) of the vectors a and b is given by
a · b = ab cos φ
where a is the magnitude of a, b is the magnitude of b and φ is the angle between a and b.
The scalar product (or dot product) of the vectors a and b is given by
a · b = ab cos φ
where a is the magnitude of a, b is the magnitude of b and φ is the angle between a and b.
The scalar product is commutative: a · b = b · a. One can show that a · b is related to
the components of a and b by
a·b=axbx +ayby +azbz
The scalar product is commutative: a · b = b · a. One can show that a · b is related to
the components of a and b by
a·b=axbx +ayby +azbz
Displacement vector or scalar
Vector
Displacement vector or scalar
Vector
Area vector or scalar
Scalar
Area vector or scalar
Scalar
Work vector or scalar
Scalar
Work is not a vector quantity; it is a scalar quantity. Here’s why:
Why Work is a Scalar:
1. Dot Product: The dot product of two vectors (force and displacement) results in a scalar quantity. Since work is defined as a dot product, it is inherently scalar. 2. No Direction: Work represents a measure of energy transfer, and energy does not have direction—it’s only concerned with magnitude. Thus, work does not have components or direction associated with it, like vectors do. 3. Magnitude Only: When calculating work, we are only concerned with the amount of force applied and the distance moved in the direction of the force. There’s no need for directionality, so work is considered scalar.
Clarifying Confusion:
• Force and displacement are vector quantities, but work is the result of a dot product between these vectors, stripping the result of direction and leaving only magnitude. • You might encounter quantities like torque or momentum, which are vector quantities because they have direction, but work is specifically scalar because it measures the energy transferred, not how or in which direction.
Work is not a vector quantity; it is a scalar quantity. Here’s why:
Why Work is a Scalar:
1. Dot Product: The dot product of two vectors (force and displacement) results in a scalar quantity. Since work is defined as a dot product, it is inherently scalar. 2. No Direction: Work represents a measure of energy transfer, and energy does not have direction—it’s only concerned with magnitude. Thus, work does not have components or direction associated with it, like vectors do. 3. Magnitude Only: When calculating work, we are only concerned with the amount of force applied and the distance moved in the direction of the force. There’s no need for directionality, so work is considered scalar.
Clarifying Confusion:
• Force and displacement are vector quantities, but work is the result of a dot product between these vectors, stripping the result of direction and leaving only magnitude. • You might encounter quantities like torque or momentum, which are vector quantities because they have direction, but work is specifically scalar because it measures the energy transferred, not how or in which direction.
What is a unit vector
A dimensionless vector that has a magnitude of 1
Describes a direction in space and has no other physical significance
Scalar product of two vectors
Yields a scalar quantity
A•B = |A||B|cos £
Vector product
Yields a vector product
|AxB| = |A||B| sin £
When is A•B = |AxB|
When A and B are perpendicular
Magnitude of the vector product will be maximum
Sin 90 = 1
Seconds
It is based on an atomic clock, which uses the energy difference between the two lowest energy states of the cesium atom (133Cs). When bombarded by microwaves of precisely the proper frequency, cesium atoms undergo a transition from one of these states to the other. One second (abbreviated s) is defined as the time required for 9,192,631,770 cycles of this microwave radiation
Seconds
It is based on an atomic clock, which uses the energy difference between the two lowest energy states of the cesium atom (133Cs). When bombarded by microwaves of precisely the proper frequency, cesium atoms undergo a transition from one of these states to the other. One second (abbreviated s) is defined as the time required for 9,192,631,770 cycles of this microwave radiation
precision is not the same as accuracy.
A cheap digital watch that gives the time as 10:35:17 a.m. is very precise (the time is given to the second), but if the watch runs several minutes slow, then this value isn’t very accurate. On the other hand, a grandfather clock might be very accurate (that is, display the correct time), but if the clock has no second hand, it isn’t very precise. A high-quality measurement is both precise and accurate.
precision is not the same as accuracy.
A cheap digital watch that gives the time as 10:35:17 a.m. is very precise (the time is given to the second), but if the watch runs several minutes slow, then this value isn’t very accurate. On the other hand, a grandfather clock might be very accurate (that is, display the correct time), but if the clock has no second hand, it isn’t very precise. A high-quality measurement is both precise and accurate.
If two vectors have the same direction, they are.
parallel
If two vectors have the same direction, they are.
parallel
If two vectors have the same direction, they are.
parallel
If two have the same magnitude and the same direction, they are.
equal, no matter where they are located in space
If two have the same magnitude and the same direction, they are.
equal, no matter where they are located in space
Negative of a vector
We define the negative of a vector as a vector having the same magnitude as the original vector but the opposite direction.
Negative of a vector
We define the negative of a vector as a vector having the same magnitude as the original vector but the opposite direction.
magnitude of a vector quantity
The magnitude of a vector quantity is a scalar quantity (a number) and is always positive. Note that a vector can never be equal to a scalar because they are different kinds of quantities. The expression ;AS = 6 m< is just as wrong as ;2 oranges = 3 apples<!
magnitude of a vector quantity
The magnitude of a vector quantity is a scalar quantity (a number) and is always positive. Note that a vector can never be equal to a scalar because they are different kinds of quantities. The expression ;AS = 6 m< is just as wrong as ;2 oranges = 3 apples<!
Given that
∣S∣=3m and
∣T∣=4m, find the possible magnitudes of the difference vector
S−T
we can apply the triangle inequality, which states that the magnitude of the difference between two vectors must fall within a specific range:
∣S−T∣≤∣S∣+∣T∣
and
∣S−T∣≥∣∣S∣−∣T∣∣.
Given that
∣S∣=3m and
∣T∣=4m, we calculate the bounds as follows:
The maximum possible magnitude occurs when the vectors are aligned in the same direction, yielding:
∣S−T∣=3+4=7m.
The minimum possible magnitude occurs when the vectors are aligned in opposite directions, yielding:
∣S−T∣=∣3−4∣=1m.
Thus, the magnitude of
S−T must lie between 1 m and 7 m.
Now, checking the options:
(i) 9 m: Not possible, as it exceeds the maximum limit of 7 m.
(ii) 7 m: Possible.
(iii) 5 m: Possible, as it lies between 1 m and 7 m.
(iv) 1 m: Possible, as it matches the minimum magnitude.
(v) 0 m: Not possible, as the vectors are not equal in magnitude.
(vi) -1 m: Not possible, as magnitude cannot be negative.
Therefore, the correct answers are: (ii) 7 m, (iii) 5 m, (iv) 1 m.
Understanding vector subtraction
When subtracting vectors 𝑆−𝑇, the result is a new vector. The magnitude of this new vector can vary depending on the directions of the original vectors
𝑆 and 𝑇
Vector Magnitude: The magnitude of a vector is its “length,” and we denote the magnitude of a vector
𝐴 as
∣𝐴∣
Difference between vectors: When two vectors are subtracted, the magnitude of the result can range from the sum of their magnitudes to the absolute difference of their magnitudes, depending on the directions of the vectors.
Step 2: Applying the Triangle Inequality Theorem
The Triangle Inequality Theorem provides us with bounds for the magnitude of the difference of two vectors. This theorem tells us that:
∣𝑆−𝑇∣ ≤ ∣𝑆∣+∣𝑇∣
∣S−T∣≤∣S∣+∣T∣
and
∣S−T∣≥∣∣S∣−∣T∣∣.
The upper bound is the sum of the magnitudes of the vectors.
The lower bound is the absolute value of the difference between the magnitudes.
This theorem helps establish the range of possible values for the magnitude of
S−T.
Step 3: Finding the maximum magnitude
To find the maximum magnitude of
S−T, we consider the case when
𝑆 and 𝑇 are pointing in opposite directions. In that case, the result is just the sum of their magnitudes.
∣S∣=3m
∣T∣=4m
The maximum possible magnitude occurs when the vectors are in opposite directions (effectively aligned along the same line but pointing opposite), so:
∣S−T∣=∣S∣+∣T∣=3+4=7m.
This gives us the upper limit of the magnitude.
Step 4: Finding the minimum magnitude
The minimum magnitude occurs when the vectors are pointing in the same direction. In that case, the vectors “cancel out” part of each other, and the magnitude of the difference is the absolute value of the difference between their magnitudes.
So, we calculate the difference in magnitudes:
∣S−T∣=∣∣S∣−∣T∣∣=∣3−4∣=1m.
This gives us the lower limit of the magnitude.
Step 5: Possible Magnitude Range
From steps 3 and 4, we now know that the magnitude of
S−T must lie between 1 m and 7 m. This means that the magnitude of the difference vector can take any value between 1 m and 7 m, inclusive.
Thorough Explanation of Vector Subtraction and Magnitude Calculation
When dealing with vectors, it’s important to understand that vectors have both magnitude (size) and direction. When we subtract two vectors, the resulting vector’s magnitude depends on both the magnitude and direction of the original vectors.
Vectors
𝑆 and 𝑇
S has a magnitude of 3 meters (denoted as
∣S∣=3m).
T has a magnitude of 4 meters (denoted as
∣T∣=4m).
Vector subtraction:
The vector
S−T is the difference between the two vectors. The magnitude of this difference depends on the direction of both vectors.
The Triangle Inequality Theorem:
The magnitude of the difference vector is bounded by the triangle inequality theorem. This theorem gives us the range of possible magnitudes for
∣S−T∣≤∣S∣+∣T∣
and
∣S−T∣≥∣∣S∣−∣T∣∣.
Step-by-step breakdown:
Maximum magnitude: The largest possible magnitude of
S−T occurs when the vectors point in opposite directions. In this case, the magnitude of the difference is the sum of the magnitudes:
∣S−T∣=3+4=7m.
Minimum magnitude: The smallest possible magnitude of
S−T occurs when the vectors point in the same direction. In this case, the magnitude of the difference is the absolute value of the difference between the magnitudes:
∣S−T∣=∣3−4∣=1m.
Thus, the possible magnitudes of
S−T lie between 1 m and 7 m.
If
A has a magnitude of 6 m and
B has a magnitude of 8 m, what is the maximum possible magnitude of
A−B?
If two vectors are subtracted, the resulting magnitude is smallest when the vectors:
(a) Point in opposite directions
(b) Point in the same direction
(c) Are perpendicular
(d) Are equal in magnitude
(b) Point in the same direction
When two vectors point in the same direction and you subtract them, their magnitudes subtract directly, which results in the smallest possible magnitude (or zero if they are equal). If they point in opposite directions, the magnitudes will add up, giving a larger resultant magnitude. If they are perpendicular, the magnitude of the resultant will be between the minimum and maximum possible values.
If
C has a magnitude of 5 m and
𝐷 has a magnitude of 9 m, what is the smallest possible magnitude of
C−D?
To find the smallest possible magnitude of the vector difference C - D, we can use the fact that the magnitude of the difference between two vectors depends on their relative direction. The smallest possible magnitude occurs when the two vectors are in the same direction.
The formula for the magnitude of the difference between two vectors is:
|C-D|=||C|-|D||
Substituting the magnitudes of C and D:
|C-D|=5-9=|-4|=4m
So, the smallest possible magnitude of
C- D is 4 meters.
The magnitude of
X−Y can never be:
(a) Positive
(b) Greater than the sum of
∣X∣ and
∣Y∣
(c) Equal to zero
(d) Equal to
∣X∣
If
∣E∣=7m and
∣F∣=2m, what is the maximum possible magnitude of
E−F?
The maximum possible magnitude of ( E - F ) occurs when the vectors ( E ) and ( F ) are in opposite directions. In that case, the magnitudes add up, and the formula for the magnitude of the difference between the vectors becomes:
[
|E - F| = |E| + |F|
]
is 9 meters.
If two vectors are parallel and have the same direction, the magnitude of their difference is:
(a) Zero
(b) The sum of the two magnitudes
(c) The difference of the two magnitudes
(d) Undefined
(c) The difference of the two magnitudes
When two vectors are parallel and point in the same direction, their magnitudes simply subtract. The result of their difference will be the difference between their magnitudes. If the vectors are equal in magnitude, the result will be zero, but in general, it is just the difference between the magnitudes.
What is the maximum possible magnitude of
A−B if
∣A∣=4m and
∣B∣=4m?
(a) 4 m
(b) 0 m
(c) 8 m
(d) 2 m
The maximum possible magnitude of A - B occurs when the two vectors A and B are in opposite directions. In that case, the magnitudes add up, and the formula for the magnitude of the difference between the vectors becomes:
A-B= A + B
Given that |A| = 4m and |B |= 4m, we can
calculate:
|A-B|=4m + 4m = 8 m
Thus, the maximum possible magnitude is 8 m, and the correct answer is:
(c) 8 m.
The magnitude of
M+N is always:
(a) Less than the sum of
∣M∣ and
∣N∣
(b) Greater than
∣M∣ or
∣N∣, but not both
(c) Between the sum and the difference of
∣M∣ and
∣N∣
(d) Equal to the difference of
∣M∣ and
∣N∣
If the vectors
𝑈 and
V are perpendicular, their difference has:
(a) A magnitude greater than both
∣U∣ and
∣lV∣
(b) The same magnitude as the sum
(c) A smaller magnitude than their sum
(d) No magnitude
If the vectors
P and
𝑄point in exactly opposite directions, the magnitude of their difference is:
(a) Equal to the sum of their magnitudes
(b) Equal to zero
(c) Larger than the sum of their magnitudes
(d) The same as the magnitude of l
P
. If
∣A∣=5m and
∣B∣=3m, what is the maximum possible value of
∣A−B∣?
(a) 5 m
(b) 8 m
(c) 2 m
(d) 1 m
If two vectors have equal magnitudes but opposite directions, the magnitude of their sum is:
(a) Twice the magnitude of each vector
(b) Zero
(c) Equal to the magnitude of either vector
(d) Equal to half the magnitude of either vector
If two vectors have equal magnitudes but opposite directions, their sum results in a cancellation of their effects. In this case, the magnitude of their sum is:
|A + (-A)| = 0
Therefore, the correct answer is:
(b) Zero.
The magnitude of
R−S is equal to the difference of their magnitudes if:
(a)
R and
𝑆 are parallel
(b)
R and
𝑆 are perpendicular
(c) One vector is zero
(d) The vectors point in opposite directions
The magnitude of
T−U can be zero if:
(a)
T and
U have the same direction and magnitude
(b)
𝑇 and
U have different magnitudes
(c)
T is larger than
𝑈
(d)
T and
U are perpendicular
The magnitude of the difference vector
X−Y is minimum when the vectors:
The magnitude of the difference vector
X - Y is minimized when the vectors X and Y are in the same direction. This is because the difference in their magnitudes will be smallest when they are aligned in the same direction.
So, the magnitude of the difference is minimized when the vectors X and Y are pointing in the same direction.
If
∣A∣=10m and
∣B∣=6m, what is the smallest possible value of
∣A−B∣?
(a) 4 m
(b) 16 m
(c) 6 m
(d) 2 m
The smallest possible value of ( |A - B| ) occurs when the vectors ( A ) and ( B ) are in the same direction. In this case, the magnitude of the difference is:
|A - B| = ||A| - |B|| = |10- 6| 4
Therefore, the correct answer is:
(a) 4 m.
components are not vectors
True components are not vectors The components Ax and Ay of a vector AS are numbers; they are not vectors themselves. This is why we print the symbols for compo- nents in lightface italic type with no arrow on top instead of in boldface italic with an arrow, which is reserved for vectors.
If A = 0 for a vector in the xy-plane, does it follow that Ax = -Ay? What can you say about Ax and Ay?
The question asks whether it is possible for both the dot product (A · B) and the cross product (A × B) of two nonzero vectors A and B to be zero.
- Dot product (A · B = 0): This occurs when the vectors are perpendicular (i.e., the angle between them is 90°).
- Cross product (A × B = 0): This occurs when the vectors are parallel (i.e., the angle between them is 0° or 180°).
It is impossible for both to be zero simultaneously for nonzero vectors. If the vectors are parallel, their dot product cannot be zero unless one of the vectors is zero, and if the vectors are perpendicular, their cross product cannot be zero.
Conclusion:
No, it is not possible for both A · B and A × B to be zero at the same time for nonzero vectors.
The question asks whether it is possible for both the dot product (A · B) and the cross product (A × B) of two nonzero vectors A and B to be zero.
- Dot product (A · B = 0): This occurs when the vectors are perpendicular (i.e., the angle between them is 90°).
- Cross product (A × B = 0): This occurs when the vectors are parallel (i.e., the angle between them is 0° or 180°).
It is impossible for both to be zero simultaneously for nonzero vectors. If the vectors are parallel, their dot product cannot be zero unless one of the vectors is zero, and if the vectors are perpendicular, their cross product cannot be zero.
Conclusion:
No, it is not possible for both A · B and A × B to be zero at the same time for nonzero vectors.
Can the magnitude of a vector be less than the magnitude of any of its components?
No, the magnitude of a vector cannot be less than the magnitude of any of its individual components. The magnitude of a vector is the resultant of all its components. Since the magnitude is based on the sum of the squares of the components, it is either equal to the largest component (in cases where one component dominates and the others are zero) or larger than each individual component (if multiple components are non-zero). Therefore, the magnitude of the vector will always be greater than or equal to the magnitude of any of its components.
Can the magnitude of a vector be less than the magnitude of any of its components?
No, the magnitude of a vector cannot be less than the magnitude of any of its individual components. The magnitude of a vector is the resultant of all its components. Since the magnitude is based on the sum of the squares of the components, it is either equal to the largest component (in cases where one component dominates and the others are zero) or larger than each individual component (if multiple components are non-zero). Therefore, the magnitude of the vector will always be greater than or equal to the magnitude of any of its components.
Can you find a vector quantity that has a magnitude of zero but components that are not zero?
Can you find a vector quantity that has a magnitude of zero but components that are not zero?
Can you find a vector quantity that has a magnitude of zero but components that are not zero?
How many correct experiments do we need to disprove a theory? How many do we need to prove a theory? Explain.
Disprove: Only one correct experiment is needed to disprove a theory, as one contradiction can invalidate the theory.
• Prove: No number of experiments can definitively prove a theory; they can only provide evidence supporting it. Scientific theories remain open to revision with new data.
How many correct experiments do we need to disprove a theory? How many do we need to prove a theory? Explain.
Disprove: Only one correct experiment is needed to disprove a theory, as one contradiction can invalidate the theory.
• Prove: No number of experiments can definitively prove a theory; they can only provide evidence supporting it. Scientific theories remain open to revision with new data.
Suppose you are asked to compute the tangent of 5.00 meters. Is this possible? Why or why not?
No, it is not possible to compute the tangent of 5.00 meters because the tangent function applies to angles, not lengths. A trigonometric function like tangent requires an angular input (in radians or degrees), not a distance.
What is your height in centimeters? What is your weight in newtons?
• Height: This is a personal measurement; to convert height to centimeters, multiply meters by 100.
• Weight: Weight in newtons can be calculated by multiplying mass (in kilograms) by the gravitational constant (9.81 m/s²).
Does the apparent mass gain of the NIST standard kilograms have any importance? Explain.
• Yes, it is important because these mass standards are used to define and calibrate weights worldwide. Even small deviations could affect high-precision measurements in science, engineering, and industry.
Does the apparent mass gain of the NIST standard kilograms have any importance? Explain.
• Yes, it is important because these mass standards are used to define and calibrate weights worldwide. Even small deviations could affect high-precision measurements in science, engineering, and industry.
What physical phenomena (other than a pendulum or cesium clock) could you use to define a time standard?
• Radioactive decay, vibrations of quartz crystals (used in quartz clocks), or the orbital period of celestial bodies (like Earth’s rotation around the Sun) could define a time standard.
What physical phenomena (other than a pendulum or cesium clock) could you use to define a time standard?
• Radioactive decay, vibrations of quartz crystals (used in quartz clocks), or the orbital period of celestial bodies (like Earth’s rotation around the Sun) could define a time standard.
Describe how you could measure the thickness of a sheet of paper with an ordinary ruler.
• Stack a large number of sheets (e.g., 100), measure the total thickness with the ruler, and divide the measurement by the number of sheets to estimate the thickness of one sheet.
Describe two or three other geometrical or physical quantities that are dimensionless.
• Reynolds number (ratio of inertial forces to viscous forces), Strain (ratio of deformation to original length), and Coefficient of friction (ratio of frictional force to normal force) are examples of dimensionless quantities.
Describe two or three other geometrical or physical quantities that are dimensionless.
• Reynolds number (ratio of inertial forces to viscous forces), Strain (ratio of deformation to original length), and Coefficient of friction (ratio of frictional force to normal force) are examples of dimensionless quantities.
What must be true about the directions and magnitudes of A and B if C = A + B and C = 0?
• If C = A + B = 0, A and B must have equal magnitudes but opposite directions, meaning they cancel each other out.
Is it possible for both A · B and A × B to be zero for nonzero vectors A and B?
• No, it is not possible. If A · B = 0, the vectors are perpendicular. If A × B = 0, the vectors are parallel. They can’t be both at the same time unless one of the vectors is zero.
Is it possible for both A · B and A × B to be zero for nonzero vectors A and B?
• No, it is not possible. If A · B = 0, the vectors are perpendicular. If A × B = 0, the vectors are parallel. They can’t be both at the same time unless one of the vectors is zero.
What does A · A give? What about A × A?
• A · A gives the magnitude squared of A: |A|^2.
• A × A is always zero because the cross product of a vector with itself is zero.
What does A · A give? What about A × A?
• A · A gives the magnitude squared of A: |A|^2.
• A × A is always zero because the cross product of a vector with itself is zero.
Why is A / |A| a unit vector, and what is its direction?
• A / |A| is a unit vector because its magnitude is 1. Its direction is the same as the direction of vector A.
Why is A / |A| a unit vector, and what is its direction?
• A / |A| is a unit vector because its magnitude is 1. Its direction is the same as the direction of vector A.
If a train overshoots by 10.0 m after traveling 890 km, what is the percent error in the distance covered?
• The percent error is calculated as (10.0 \, \text{m} / 890,000 \, \text{m}) \times 100 = 0.00112\%.
Can the vector products A × (B × C) and (A × B) × C be equal?
• Generally, A × (B × C) and (A × B) × C do not have the same magnitude or direction. They follow different vector product rules and are not equal unless specific conditions hold (e.g., parallel vectors).
Prove that A · (A × B) = 0.
• The cross product A × B is perpendicular to A by definition. The dot product of any vector with a perpendicular vector is zero.
A · B = 0, does it necessarily follow that A = 0 or B = 0?
No, it only means A and B are perpendicular. They can both be nonzero.
A · B = 0, does it necessarily follow that A = 0 or B = 0?
No, it only means A and B are perpendicular. They can both be nonzero.
If A × B = 0, does it necessarily follow that A = 0 or B = 0?
• No, it means A and B are parallel or one of them is zero.
If A × B = 0, does it necessarily follow that A = 0 or B = 0?
• No, it means A and B are parallel or one of them is zero.
If A = 0 for a vector in the xy-plane, does it follow that Ax = -Ay?
No, it does not follow. If A = 0, then both Ax and Ay must be zero.
What is the displacement of a bicyclist when she travels around a circular racetrack from the north side to the south side? When she makes one complete circle around the track? Explain.
• From north to south: The displacement is the diameter of the circle, because displacement is the straight-line distance between starting and ending points, so it’s 1000 m (diameter = 2 × 500 m).
• One complete circle: The displacement is zero because the bicyclist returns to the starting point.
What is the displacement of a bicyclist when she travels around a circular racetrack from the north side to the south side? When she makes one complete circle around the track? Explain.
• From north to south: The displacement is the diameter of the circle, because displacement is the straight-line distance between starting and ending points, so it’s 1000 m (diameter = 2 × 500 m).
• One complete circle: The displacement is zero because the bicyclist returns to the starting point.
Can you find two vectors with different lengths that have a vector sum of zero? What length restrictions are required for three vectors to have a vector sum of zero?
Two vectors with different lengths cannot sum to zero; they must have equal magnitudes and opposite directions to sum to zero.
• For three vectors, the sum can be zero if they form a closed triangle, meaning the vectors must have lengths that allow them to satisfy the triangle inequality theorem.
Can you find two vectors with different lengths that have a vector sum of zero? What length restrictions are required for three vectors to have a vector sum of zero?
Two vectors with different lengths cannot sum to zero; they must have equal magnitudes and opposite directions to sum to zero.
• For three vectors, the sum can be zero if they form a closed triangle, meaning the vectors must have lengths that allow them to satisfy the triangle inequality theorem.
The “direction of time” is said to proceed from past to future. Does this mean that time is a vector quantity? Explain.
No, time is not a vector quantity. Even though it has a direction (from past to future), it lacks the other essential feature of a vector: multiple components and the ability to operate in different directions. Time is considered a scalar.
The “direction of time” is said to proceed from past to future. Does this mean that time is a vector quantity? Explain.
No, time is not a vector quantity. Even though it has a direction (from past to future), it lacks the other essential feature of a vector: multiple components and the ability to operate in different directions. Time is considered a scalar.
Are air traffic controller instructions called “vectors” correctly named?
• Yes, the name is correctly used because the instructions involve both direction and magnitude (such as speed or velocity), which are characteristics of vectors.
Does it make sense to say that a vector is negative? Why?
• No, a vector itself cannot be “negative.” However, the direction of a vector can be reversed, and we can describe one vector as the negative of another if they have the same magnitude but opposite directions.
Does it make sense to say that one vector is the negative of another?
• Yes, if two vectors have the same magnitude but opposite directions, one can be called the negative of the other.
Would you like me to continue answering further questions in this series?
Does it make sense to say that one vector is the negative of another?
• Yes, if two vectors have the same magnitude but opposite directions, one can be called the negative of the other.
Would you like me to continue answering further questions in this series?
Can you find a vector quantity that has a magnitude of zero but components that are not zero? Can the magnitude of a vector be less than the magnitude of any of its components?
• No, if the magnitude of a vector is zero, all of its components must be zero.
• No, the magnitude of a vector is the result of the combination of its components, so it cannot be less than the magnitude of any individual component.
Can you find a vector quantity that has a magnitude of zero but components that are not zero? Can the magnitude of a vector be less than the magnitude of any of its components?
• No, if the magnitude of a vector is zero, all of its components must be zero.
• No, the magnitude of a vector is the result of the combination of its components, so it cannot be less than the magnitude of any individual component.
