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**Unformatted text preview: **Homework 1a: Dimensions and Units
Due: 11:59pm on Tuesday, January 24, 2017
To understand how points are awarded, read the Grading Policy for this assignment. Dimensions of Physical Quantities
Learning Goal:
To introduce the idea of physical dimensions and to learn how to find them.
Physical quantities are generally not purely numerical: They have a particular dimension or combination of dimensions
associated with them. Thus, your height is not 74, but rather 74 inches, often expressed as 6 feet 2 inches. Although feet and
inches are different units they have the same dimensionlength. Part A
In classical mechanics there are three base dimensions. Length is one of them. What are the other two? Hint 1. MKS system
The current system of units is called the International System (abbreviated SI from the French Système
International). In the past this system was called the mks system for its base units: meter, kilogram, and second.
What are the dimensions of these quantities?
ANSWER:
acceleration and mass acceleration and time acceleration and charge mass and time mass and charge time and charge Correct There are three dimensions used in mechanics: length ( l), mass ( m), and time ( t). A combination of these three dimensions
suffices to express any physical quantity, because when a new physical quantity is needed (e.g., velocity), it always obeys an
equation that permits it to be expressed in terms of the units used for these three dimensions. One then derives a unit to
measure the new physical quantity from that equation, and often its unit is given a special name. Such new dimensions are
called derived dimensions and the units they are measured in are called derived units.
For example, area A has derived dimensions [A] = l 2 . (Note that "dimensions of variable x" is symbolized as [x] .) You can find these dimensions by looking at the formula for the area of a square A
2
2
Clearly [s] = l. Plugging this into the equation gives [A] = [s] = l . Part B
Find the dimensions [ of volume. 2 = s , where s is the length of a side of the square. Find the dimensions [V ] of volume.
Express your answer as powers of length ( l), mass ( m), and time ( t). Hint 1. Equation for volume
You have likely learned many formulas for the volume of various shapes in geometry. Any of these equations will
give you the dimensions for volume. You can find the dimensions most easily from the volume of a cube V = e3 ,
where e is the length of the edge of the cube.
ANSWER:
[V ] = 3 [l] Correct Part C
Find the dimensions [v] of speed.
Express your answer as powers of length ( l), mass ( m), and time ( t). Hint 1. Equation for speed
Speed v is defined in terms of distance d and time t as
v= Therefore, [v] . d
t . = [d]/[t] Hint 2. Familiar units for speed
You are probably accustomed to hearing speeds in miles per hour (or possibly kilometers per hour). Think about the
dimensions for miles and hours. If you divide the dimensions for miles by the dimensions for hours, you will have
the dimensions for speed.
ANSWER: = [l] [v] [t] Correct The dimensions of a quantity are not changed by addition or subtraction of another quantity with the same dimensions. This
means that Δv , which comes from subtracting two speeds, has the same dimensions as speed.
It does not make physical sense to add or subtract two quanitites that have different dimensions, like length plus time. You can
add quantities that have different units, like miles per hour and kilometers per hour, as long as you convert both quantities to
the same set of units before you actually compute the sum. You can use this rule to check your answers to any physics problem you work. If the answer involves the sum or difference of two quantities with different dimensions, then it must be
incorrect.
This rule also ensures that the dimensions of any physical quantity will never involve sums or differences of the base
dimensions. (As in the preceeding example, l + t is not a valid dimension for a physical quantitiy.) A valid dimension will only
2
involve the product or ratio of powers of the base dimensions (e.g. m2/3 l t−2 ). Part D
Find the dimensions [a] of acceleration.
Express your answer as powers of length ( l), mass ( m), and time ( t). Hint 1. Equation for acceleration
In physics, acceleration a is defined as the change in velocity in a certain time. This is shown by the equation a = Δv/Δt. The Δ is a symbol that means "the change in." ANSWER: [a] = [l]
2 [t] Correct Consistency of Units
In physics, every physical quantity is measured with respect to a unit. Time is measured in seconds, length is measured in
meters, and mass is measured in kilograms. Knowing the units of physical quantities will help you solve problems in physics. Part A
Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and
causes the moon to orbit the earth. The force of gravitational attraction is represented by the equation
F = Gm 1 m 2
r 2 , where F is the magnitude of the gravitational attraction on either body, m1 and m2 are the masses of the bodies, r is the
distance between them, and G is the gravitational constant. In SI units, the units of force are kg ⋅ m/s2 , the units of mass
are kg, and the units of distance are m. For this equation to have consistent units, the units of G must be which of the
following? Hint 1. How to approach the problem
To solve this problem, we start with the equation
F = Gm 1 m 2
r 2 . For each symbol whose units we know, we replace the symbol with those units. For example, we replace m1 with kg. We now solve this equation for G.
ANSWER: kg 3 m⋅s2
2 kg⋅s
m3
3 m kg⋅s2
m
kg⋅s2 Correct Part B
One consequence of Einstein's theory of special relativity is that mass is a form of energy. This massenergy relationship
is perhaps the most famous of all physics equations:
2
E = mc ,
where m is mass, c is the speed of the light, and E is the energy. In SI units, the units of speed are m/s. For the
preceding equation to have consistent units (the same units on both sides of the equation), the units of E must be which
of the following? Hint 1. How to approach the problem
To solve this problem, we start with the equation
.
For each symbol whose units we know, we replace the symbol with those units. For example, we replace m with kg. We now solve this equation for E .
E = mc 2 ANSWER: kg⋅m
s
2 kg⋅m
2 s 2 kg⋅s
2 m 2 kg⋅m
s Correct
To solve the types of problems typified by these examples, we start with the given equation. For each symbol whose
units we know, we replace the symbol with those units. For example, we replace m with kg. We now solve this
equation for the units of the unknown variable. Converting Units: The Magic of 1 Learning Goal:
To learn how to change units of physical quantities.
Quantities with physical dimensions like length or time must be measured with respect to a unit, a standard for quantities with
this dimension. For example, length can be measured in units of meters or feet, time in seconds or years, and velocity in
meters per second.
When solving problems in physics, it is necessary to use a consistent system of units such as the International System
(abbreviated SI, for the French Système International) or the more cumbersome English system. In the SI system, which is the
preferred system in physics, mass is measured in kilograms, time in seconds, and length in meters. The necessity of using
consistent units in a problem often forces you to convert some units from the given system into the system that you want to
use for the problem.
The key to unit conversion is to multiply (or divide) by a ratio of different units that equals one. This works because multiplying
any quantity by one doesn't change it. To illustrate with length, if you know that 1 inch = 2.54 cm, you can write
1= 2.54 cm
1 inch . To convert inches to centimeters, you can multiply the number of inches times this fraction (since it equals one), cancel the
inch unit in the denominator with the inch unit in the given length, and come up with a value for the length in centimeters. To
convert centimeters to inches, you can divide by this ratio and cancel the centimeters. For all parts, notice that the units are already written after the answer box; don't try to write them in your answer also. Part A
How many centimeters are there in a length 83.6 inches ?
Express your answer in centimeters to three significant figures.
ANSWER:
212 cm Correct Sometimes you will need to change units twice to get the final unit that you want. Suppose that you know how to convert from
centimeters to inches and from inches to feet. By doing both, in order, you can convert from centimeters to feet. Part B
Suppose that a particular artillery piece has a range R = 5910 yards . Find its range in miles. Use the facts that 1 mile = 5280 f t and 3 f t = 1 yard.
Express your answer in miles to three significant figures. Hint 1. Convert yards to feet
The first step in this problem is to convert from yards to feet, because you know how to then convert feet into
miles. Convert 5910 yards into feet. Use
1= 3 ft
1 yard Express your answer in feet to three significant figures. . ANSWER:
1.77×104 f t ANSWER:
5910 yards = 3.36 miles Correct Often speed is given in miles per hour (mph), but in physics you will almost always work in SI units. Therefore, you must
convert mph to meters per second (m/s). Part C
What is the speed of a car going v = 1.000 mph in SI units? Notice that you will need to change from miles to meters
and from hours to seconds. You can do each conversion separately. Use the facts that 1 mile = 1609 m and 1 hour = 3600 s.
Express your answer in meters per second to four significant figures. Hint 1. Convert miles to meters
In converting 1.000 into meters per second, you will need to multiply by mph 1= 1609 m
1 mile . When you do this, the miles will cancel to leave you with a value in meters per hour. You can then finish the
conversion. What is v = 1.000 mph in meters per hour?
Express your answer in meters per hour to four significant figures.
ANSWER:
v = 1609 m/hour Hint 2. Convert hours to seconds
Which of the following would you multiply 1609
ANSWER: 3600 s 1 hour
3600 s
1 hour
1 hour
3600 s m/hours by to convert it into meters per second (m/s)? ANSWER:
v = 0.4469 m/s Correct
Notice that by equating the two values for v , you get 1.000 mph = 0.4469 m/s. It might be valuable to remember
this, as you may frequently need to convert from miles per hour into more useful SI units. By remembering this
relationship in the future, you can reduce this task to a single conversion. Unit Conversions I
According to the label on a bottle of salad dressing, the volume of the contents is 0.473 liter (L). Part A
Using only the conversions 1 L = 1000 cm 3 and 1 , express this volume in cubic inches. in = 2.54 cm ANSWER:
28.9 in3 Correct Unit Conversions II
While driving in an exotic foreign land you see a speed limit sign on a highway that reads 1.78×105 furlongs per fortnight. Part A
1 How many miles per hour is this? (One furlong is mile, and a fortnight is 14 days. A furlong originally referred to the
length of a plowed furrow.) 8 ANSWER:
66.2 mi/h Correct
Score Summary:
Your score on this assignment is 97.6%.
You received 27.33 out of a possible total of 28 points. ...

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