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• Physics and MeasurementPhysics and MeasurementStandards of Length, Mass, time and Dimensional AnalysisStandards of Length, Mass, time and Dimensional Analysis

• The laws of physics are expressed in terms of basic quantities that require a clear The laws of physics are expressed in terms of basic quantities that require a clear definition.definition.

• In mechanics, the three basic quantities are length (LIn mechanics, the three basic quantities are length (L), mass (M mass (M), and time (T and time (T). All All other quantities in mechanics can be expressed in terms of these threeother quantities in mechanics can be expressed in terms of these three.

In 1960, an international committee In 1960, an international committee established a set of standards for length, established a set of standards for length, mass, and other basic quantities. The system mass, and other basic quantities. The system established is an adaptation of the metric established is an adaptation of the metric system, and it is called the SI system of units. system, and it is called the SI system of units. (The abbreviation SI comes from the (The abbreviation SI comes from the system’s French name “system’s French name “Système Inter-national.”) In this system, the ”) In this system, the units of length, mass, and time are the units of length, mass, and time are the meter, kilogram, and second, respectively meter, kilogram, and second, respectively [Kms][Kms]

As recently as 1960, the length of the meter was defined as the distance between two lines on a specific platinum – iridium bar stored under controlled conditions in France. This standard was abandoned for several reasons, a principal one being that the limited accuracy with which the separation between the lines on the bar can be determined does not meet the current requirements of science and technology. In the 1960s and 1970s, the meter was defined as 1 650 763.73 wavelengths of orange-red light emitted from a krypton-86 lamp. However, in October 1983, the meter (m) was redefined as the distance travelled by light in vacuum during a time of 1/299 792 458 second.

LengthLength

MassMassThe basic SI unit of mass, the kilogram (kg), is defined as the mass of a specific platinum –iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sèvres, France. This mass standard was established in 1887 and has not been changed since that time because platinum–iridium is an unusually stable alloy

TimeTimeBefore 1960, the standard of time was defined in terms of Before 1960, the standard of time was defined in terms of the the mean solar day for the mean solar day for the year 1900.2 The year 1900.2 The mean solar mean solar second was originally defined as of a mean second was originally defined as of a mean solar solar day. day. The rotation of the Earth is now known to vary The rotation of the Earth is now known to vary slightly with time, however, and therefore this motion is slightly with time, however, and therefore this motion is not a good one to use for defining a standard.not a good one to use for defining a standard.

24

1

60

1

60

1

Thus, in 1967 the SI unit of time, the Thus, in 1967 the SI unit of time, the second, was second, was redefined redefined using the characteristic frequency of a using the characteristic frequency of a particular kind of cesium atom as the “reference clock.” particular kind of cesium atom as the “reference clock.” The basic SI unit of time, the second (s), The basic SI unit of time, the second (s), is defined as 9 is defined as 9 192 631 770 times the period of vibration of radiation 192 631 770 times the period of vibration of radiation from the cesium-133 atomfrom the cesium-133 atom

DIMENSIONAL ANALYSISIn solving problems in physics, there is a useful In solving problems in physics, there is a useful and powerful procedure called and powerful procedure called dimensional dimensional analysis.analysis. Dimensional analysis makes use of Dimensional analysis makes use of the fact that dimensions can be treated as the fact that dimensions can be treated as algebraic quantities. algebraic quantities. • quantities can be added or subtracted only if quantities can be added or subtracted only if they have the same dimensions. they have the same dimensions. • Furthermore, the terms on both sides of an Furthermore, the terms on both sides of an equation must have the same dimensions.equation must have the same dimensions.

.

The units of time squared cancel as shown, leaving the unit of length.

Let us use dimensional analysis to check Let us use dimensional analysis to check the validity of this expressionthe validity of this expression.

The quantity x on the left side has the dimension of length L. We can perform a dimensional check by substituting the dimensions for acceleration, L/T2, and time, T, into the equation. That is, the dimensional form of the equation

2

2

1tax

2

2

1tax

is

CONVERSION OF UNITSSometimes it is necessary to convert units from Sometimes it is necessary to convert units from one system to another. Conversion factors one system to another. Conversion factors between the SI units and conventional units of between the SI units and conventional units of length are as followslength are as follows:

Suppose we are told that the acceleration Suppose we are told that the acceleration a of a of a particle moving a particle moving with uniform speed with uniform speed v in a v in a circle of radius r is proportional circle of radius r is proportional to some to some power of power of r, say rr, say rnn, and some power of v, say , and some power of v, say vvmm. How . How can we determine the values of can we determine the values of n and n and m?m?

EXAMPLE 1. 2 Analysis of a Power LawEXAMPLE 1. 2 Analysis of a Power Law

EXAMPLE 1.1 Analysis of an EquationEXAMPLE 1.1 Analysis of an EquationShow that the expression Show that the expression v = at is dimensionally v = at is dimensionally correct, correct, where where v represents speed, a v represents speed, a acceleration, and t a time interval.acceleration, and t a time interval.

The Laws of MotionThe Laws of Motion•The Concept of The Concept of Force•Newton’s First Law and Inertial Newton’s First Law and Inertial FramesFrames•Newton’s Second LawNewton’s Second Law•The Force of Gravity and WeightThe Force of Gravity and Weight•Newton’s Third LawNewton’s Third Law•Some Applications of Newton’s Some Applications of Newton’s LawsLaws•Forces of FrictionForces of Friction

In these examples, the word In these examples, the word force force is associated with muscular is associated with muscular activity and some change in the activity and some change in the velocity of an velocity of an object. Forces do object. Forces do not always cause motion, not always cause motion, however. For example, you can however. For example, you can push (in other words, exert a push (in other words, exert a force) on a large boulder not be force) on a large boulder not be able to move it. able to move it. If the net force If the net force exerted on an object is zero, exerted on an object is zero, then the acceleration of the then the acceleration of the object is zero and its velocity object is zero and its velocity remains constant.remains constant.

THE CONCEPT OF FORCETHE CONCEPT OF FORCE

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orce

s.

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act f

orce

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What force (if any) causes the What force (if any) causes the Moon to orbit the Earth?Moon to orbit the Earth?Newton answered this by stating Newton answered this by stating that forces are what cause any that forces are what cause any change in the velocity of an object. change in the velocity of an object. Therefore, if an object moves with Therefore, if an object moves with constant velocity, no force is constant velocity, no force is required for the motion to be required for the motion to be maintained. The Moon’s velocity is maintained. The Moon’s velocity is not constant because it moves in a not constant because it moves in a nearly circular orbit around the nearly circular orbit around the Earth. Earth. We now know that this We now know that this change in velocity is caused by the change in velocity is caused by the force exerted on the Moon by the force exerted on the Moon by the Earth.Earth.

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•Newton’s First Law and Inertial FramesNewton’s First Law and Inertial FramesBefore about 1600, scientists felt that the natural state of Before about 1600, scientists felt that the natural state of matter was the state of rest. Galileo was the first to take a matter was the state of rest. Galileo was the first to take a different approach to motion and concluded that it is not different approach to motion and concluded that it is not the nature of an object to stop once set in motion: rather, the nature of an object to stop once set in motion: rather, it is its nature to it is its nature to resist changes in its motion.resist changes in its motion. In his words, In his words, “Any velocity once imparted to a moving body will be “Any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of rigidly maintained as long as the external causes of retardation are removed.”retardation are removed.” This approach was later This approach was later formalized by Newton in formalized by Newton in Newton’s first law of motionNewton’s first law of motion::

In the absence of external forces, an object at rest In the absence of external forces, an object at rest remains at rest and an object in motion continues remains at rest and an object in motion continues in motion with a constant velocity (that is, with a in motion with a constant velocity (that is, with a constant speed in a straight line).constant speed in a straight line).

NEWTON’S SECOND LAWNEWTON’S SECOND LAWNewton’s second law answers the question of what Newton’s second law answers the question of what happens to an object that has a nonzero resultant force happens to an object that has a nonzero resultant force acting on it.acting on it.

From such observations, we conclude that the From such observations, we conclude that the acceleration of an object is directly proportional to the acceleration of an object is directly proportional to the resultant force acting on it.resultant force acting on it.

F1 a1

m1F2 a2 = 2a1

m1

(A)(A) 2F1 (B)(B) F1 (C)(C) 1/2 F1

F a1

m1 F a2 = 2a1

m2

(A)(A) 2m1 (B)(B) m1 (C)(C) 1/2 m1

From such observations, we conclude that the From such observations, we conclude that the acceleration of an object is inversely proportional to its acceleration of an object is inversely proportional to its mass.mass.

The acceleration of an object is directly The acceleration of an object is directly proportional to the net force acting on it and proportional to the net force acting on it and inversely pro-portional to its mass.inversely pro-portional to its mass.

These observations are summarized in These observations are summarized in Newton’s Newton’s second lawsecond law: :

The SI unit of force is the The SI unit of force is the NewtonNewton, which is , which is defined as the force that, when acting on a 1-defined as the force that, when acting on a 1-kg mass, produces an acceleration of 1 m/skg mass, produces an acceleration of 1 m/s22..

In the British engineering system, the unit of In the British engineering system, the unit of force is the force is the poundpound, which is defined as the force , which is defined as the force that, when acting on a 1-slug mass produces an that, when acting on a 1-slug mass produces an acceleration of 1 ft/sacceleration of 1 ft/s22::

A hockey puck having a mass of A hockey puck having a mass of 0.30 kg slides on the horizontal, 0.30 kg slides on the horizontal, frictionless surface of an ice rink. frictionless surface of an ice rink. Two forces act on the puck, as Two forces act on the puck, as shown in Figure. The force shown in Figure. The force FF11 has has a magnitude of 5.0 N, and the a magnitude of 5.0 N, and the force force FF22 has a magnitude of 8.0 N. has a magnitude of 8.0 N. Determine both the magnitude Determine both the magnitude and the direction of the puck’s and the direction of the puck’s acceleration.acceleration.

The resultant force The resultant force in the in the x direction isx direction is

The resultant force The resultant force in the in the y direction isy direction is

Now we use Newton’s Now we use Newton’s second law in component second law in component form to find the form to find the x and y x and y components of acceleration:components of acceleration:

and its direction relative and its direction relative to the positive to the positive x axis isx axis is

The acceleration has a The acceleration has a magnitude ofmagnitude of

NEWTON’S THIRD LAWNEWTON’S THIRD LAWThis simple experiment illustrates a general principle of This simple experiment illustrates a general principle of critical importance known as Newton’s third law:critical importance known as Newton’s third law:

If two objects interact, the If two objects interact, the force force FF1212 exerted by object 1 on exerted by object 1 on object 2 is equal in magnitude object 2 is equal in magnitude to and opposite in direction to to and opposite in direction to the force the force FF2121 exerted by object exerted by object 2 on object 1:2 on object 1:

In reality, either force can be labeled the action or In reality, either force can be labeled the action or the reaction force. the reaction force. The action force is equal in The action force is equal in magnitude to the reaction force and opposite in magnitude to the reaction force and opposite in direction.direction. In all cases, the action and reaction In all cases, the action and reaction forces act on different objectsforces act on different objects

why does the TV why does the TV not accelerate in the not accelerate in the direction of direction of FFgg ? ?

The normal force is a contact force that prevents The normal force is a contact force that prevents the TV from falling through the table and can have the TV from falling through the table and can have any magnitude needed to balance the downward any magnitude needed to balance the downward force Fforce Fgg

What is happening is that What is happening is that the table exerts on the TV the table exerts on the TV an upward force an upward force nn called called the normal force. the normal force.