Units of Measurement

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Units of Measurement
Units are one of those things that, like
electricity itself, are almost too obvious. We
all use them every day, without thinking about
them. Units are used to describe the size or
duration of things Units give scale to the world
around us. To describe a rope as long has no
meaning. The rope that is short to me, might
seem long to you. To describe a rope as 10
meters long has a precise meaning, conveyed by
the use of the unit of length called the meter
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Units of Measurement
The meter is a unit of length. The second is a
unit of duration, or time. Length (or
position, displacement, or distance is a
fundamental quantity. It cant be broken down
and expressed in terms of other quantities. Time
is also a fundamental quantity, as are mass,
temperature and electrical charge.
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Units of Measurement
Volume and area are derived quantities, derived
from length. The area of a square is the length
of one side, squared. If the length of a side is
measured in meters, the area is given in meters2.
Speed (or velocity, which is speed in a
particular direction) is a derived quantity,
derived from a distance traveled and the time
required to travel that distance. Suppose a
runner runs 100 meters in 10 seconds. His speed
is
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Units of Measurement
When performing a mathematical operation on
physical quantities, we can carry the units
through the operation as a sanity check. For
example, if when calculating speed, we divide
distance traveled (with units of length) by the
time required to travel the distance (units of
time)
The result has units of (length divided by time),
or length per unit time, which is the same as
units of speed. If the result does not have
units of speed, we did something wrong.
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Units of Measurement
Acceleration, the rate at which speed or velocity
changes, is also a derived unit. It is derived
from velocity and time. If the velocity of an
object changes, increasing or decreasing by an
amount DV in a time Dt, its acceleration is given
by
If an object is initially at rest, at sea level,
and is dropped, its speed increases by 9.81 m /
sec. every second. That is, if Dt is 1 second,
DV is 9.81 m / sec
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Units of Measurement
Force is yet another derived quantity, derived
from mass and acceleration
A 5 Kg. object sitting on the floor is subjected
to a gravitational acceleration of 9.81 m / sec.
It doesnt move because the floor is in the way,
but if a trap door opened under the object it
would drop with that acceleration. The force of
gravity exerted on the object (and on the floor,
by the object) is
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Units of Measurement
A force of 1 Kg m / sec2 has a special name 1
Newton. The force of gravity on our 5 Kg. object
is
The force of gravity also has a special name
weight. The Newton is therefore also the metric
unit of weight. The pound is a unit of weight,
but also of force, and NOT of mass. An object
in orbit is weightless, but not massless.
Changing its velocity (accelerating it) requires
the application of nonzero force, unless the mass
is zero. An object on the moon weighs about 1/6
what it weighs on earth, but has the same mass.
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Units of Measurement
Work and energy, which are really the same thing,
are also derived quantities. Kinetic energy is
energy of motion. An object with mass m (a
fundamental quantity) moving with velocity v has
kinetic energy
If the units of mass, length and time are
kilograms (Kg), meters (m) and seconds, then
kinetic energy has units of
An energy of 1 Kg m2/sec2 is also called 1
Joule.
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Units of Measurement
Suppose we take our 5 Kg. object, lift it 5
meters above the floor, and drop it. When it
hits the floor, it has accelerated (at an
acceleration of 9.81 m / sec2) to 1.01 m / sec.
Its kinetic energy at the moment it hits the
floor is
When the object hits the floor, assuming the
floor is strong enough to stop it, the kinetic
energy goes to zero. What happened to the
energy? It was converted from kinetic energy to
sound and heat. Energy cant be created or
destroyed, only changed. Einstein says that it
can be changed to mass, and vice versa, but we
wont have to worry about that.
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Units of Measurement
Where did this energy come from? Did we have to
work to lift the object 5 meters before dropping
it? Yes! Work is defined as a force exerted to
move an object some distance
We lifted the object 5 meters above the floor, so
d 5 meters. To lift it, we had to overcome the
force of gravity (the weight of the object) which
was 49.05 N.
Notice that the units of work are equivalent to
the units of energy, both can be expressed as
Joules. Also notice that the work required to
lift the object is equal to its kinetic energy
when dropped.
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Unit conversions
Previously, we calculated a speed of 10 m/sec.
Meters per second is a commonly used unit of
speed, but so is feet / sec. Suppose we want the
answer in ft./sec. rather than m/sec. How many
feet in one meter? 3.281
The meters in the numerator of the first quantity
cancel with the meters in the denominator of the
second, and were left with
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Unit conversions
But what if we really want the speed in miles per
hour? There are 5280 feet in one mile, and 3600
seconds in an hour, so
This time, the feet and seconds cancel, and were
left with
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Unit conversions
So 22.37 mph is the same as 10 meters per second,
and 2.237 mph is the same as 1 meter per second.
Thus, we can use
as a conversion factor for converting meters per
second to miles per hour. We can also use its
reciprocal,
for converting miles per hour to meters per
second.
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Unit conversions
Thus, 65 mph is the same as
Weve found conversion factors for converting
speed between the English and metric systems of
units. We can obtain conversion factors for any
physical quantity, and any two systems of units.
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Units
Power is the rate at which energy changes from
one form to another, or moves from one place to
another. Consider a 2000. lb. car, which
accelerates from 0 to 60 mph in 5 seconds. Its
known that an object with a mass of 1 Kg. at sea
level weighs 2.21 lb., so our car has a mass of
Its initial kinetic energy is
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Units of Measurement
At the end of the 5 second acceleration, its
moving at 60 mph. Lets convert that to meters
per second
So its final kinetic energy is
So its kinetic energy increased by 3299 J in 5
seconds. The power required to accelerate it was
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Units of Measurement
A power of 1 Joule per second as a special name
1 Watt. So the power required to accelerate the
car to 60 mph in 5 seconds is 66,000 Watts, or 66
kilowatts (66 KW). The English unit of power is
the horse power. 1 hp 746 Watts, so
Notice how the units were carried through all
these calculation, and the result is in units
that make sense units of power. If you carry the
units through your calculations and the result is
in incorrect units (like units of force or energy
in this example), you made a mistake. This can
alert you to errors you might not catch otherwise.
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Systems of Units
In the US, were all familiar with the English
system of units
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Systems of Units
Most of us are somewhat less familiar with the
metric system. There are actually two very
similar metric systems, the Meter-Kilogram-Second
(MKS) system for large physical quanities
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Systems of Units
And the Centimeter-Gram-Second (CGS) system,
which is more convenient for small physical
quanities
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Systems of Units
And there is the SI (Le Systeme International
dUnites), which is very similar to the MKS
system, and which we will use most of the time