PHYSICS UNIT I CHAPTER 3 Length Mass And Time Measurements Class 11th Complete Solution
Physics
3
Length Mass And Time Measurements
ORDER OF MAGNITUDE OF LENGTH
Sometimes, it is sufficient to know the order of the magnitude of a quantity instead of knowing its absolute value. For this, if a number is less than 5, it is taken as 1 and if it lies between 5 and 10, it is taken as 10. For example, the velocity of light is 3 x108 ms.However, the order of magnitude of the velocity of light is regarded as 108 ms Here, 3 has been regarded as 1, as it is less than 5). Again, number of seconds in one minute is 60 or 6 x 10. The order of magnitude of seconds in a minute is taken as 102 (Here, 6 has been regarded as 10, as it is greater than 5).
In this universe, the various objects have lengths over a wide range. The following table gives order of magnitude of their lengths (from the size of the nucleus to the size of the universe):
The order of distance varies from 10¹⁴m to 10²* m The distance ranging from 10⅝m to 10² m xan be measured by direct methods.
Tho direct method involves comparison of the distance or length to be measured with chosen standard of length.
A metre rod can be used to measure distances as small as 10 m PCS upto 10* m, while a screw gauge for distances upto 10 m. L a astronomical distances cannot be measured by direct methods. For such measurements, indirect methods have to be used.
IN DIRECT METHODS FOR LARGE DISTANCE
For measuring large distances from a few hundred metres to those of astronomical objects, the following indirect methods may be used .
1.ECHO METHODS = This method may be used to find the distance of a hill. To do so, a gun is fired towards the hill and the time interval between the instant oft tiring the gun and the instant of hearing the echo of the gun shot is noted obviously, in this time interval, sound has travelled from the observer to the hill and then back to the observer. If v is the velocity of sound; S the distance of hill from the observer and t the total time taken, then
v×t = 2S
A LASER is a source of very intense, monochromatic and unidirectional beam. Ru making use of LASER beam in place of sound waves,the echo method is used measure the distance of the moon from the earth. If t is the time taken by the LASER beam in going to and returning from the moon, the distance of the moon from the earth is given by
S = c×t/2
2. To find the thickness of a matter sheet = To measure the thickness (d) of a ther sheet [Fig. 3.01I, the probe wave i.e. a signal (a pulse of radiowaves) is sent from po point A on the front surface of the sheet to the back surface. The signal gets cted from the point O on the back surface and is received at point B on the front face. If t is the time interval between the instants, the signal was sent from the surt point A and oint A and when received at B, the thickness of the sheet
d= c×t/2
Here, c is the velocity of the signal. It may be pointed out that as the thickness of the matter sheet is very small, the value of t will also be very small. Therefore, the time recorde should be such that it can measure very small time intervals.
3. Sound Navigation and Ranging = A SONAR uses ultrasonic waves to detect and locate rocks, submarines, etc submerged under water. The ultrasonics from a transmitter are sent through the ocean. If the rock or the submarine comes in the path of ultrasonics, then they are reflected back. Noting the time interval between the instants, the ultrasonic moves are sent and their receipt, the distance of the rock or the submarine can be computed. Again, the distance of a submerged rock or a submarine can be found by using the formula
S= v×t/2
where v is the velocity of ultrasound waves in water.
The principle of SONAR is also made use of in knowing the flaws in the structure of a material. It also helps in mechanical testing of the material without damaging it.
4. Radio Detection and Ranging = Radiowaves are sent into space from a transmitter. In case an aeroplane comes in their path, they are reflected back and are detected bya detector. Knowing that the velocity of radiowaves is 3x 10 m Sand measuring the time interval between the instants of transmission of the waves and their detection, the distance of the aeroplane can be computed. A sophisticated radar can tell the distance as well as the speed and elevation of the aeroplane.
5.To find height of a distant object by triangulation method = The height of a distant object, such as a tower AB may be found by measuring the angle the LOwer makes at the observation point P (say). Let 6 be the angle subtended by the Ower at point P. It is called elevation of the tower at point P [Fig. 3.02]. Let thne stance of point P from the foot of the tower i.e. PA =x. Then, in right angled triangle PAB,
h=x tan∅
To MEASURE INERTIAL MASS
The inertial mass of an object can be measured by using an inertial balance
Principle = It is based upon the principle that when a strip fixed at one end and loaded at the other end is displaced from its mean position, it executes vibratory motion.
The inertial balance consists of a pan and a block of wood joined to each other with the help of two flat metallic strips, such that the flat faces of the strips are vertical The wooden block can be firmly clamped to the top of a table.
Theory = The inertial mass of a body is measured by placing it on the pan attached to the strips and making use of the mechanics of vibratory motion. When the pan is displaced horizontally in any direction,it executes vibratory motion. It is found that the time period of the vibratory motion depends upon the following factors:
i) The distance of the pan (the point where the body is placed) from the point it leaves the wooden block.
ii) The total mass of the pan and the body placed on it.
iii) The total mass of the pan and the body placed on it.
Mathematically,the time period of the vibratory motion executed by the inertial balance is given by
T = 2π √m/k
where m is the oscillating mass (mass of the pan and the body placed on it) and k is force constant of the metallic strips.
We have m= k/4π² T²
Dy measuring time period of vibration (T) with the help of a stop watch/clock and Knowing force constant (k), the value of inertial mass m of the object can be found.
However, in practice, the inertial mass of an object is found by measuring the time period of the inertial balance for an object of known mass as explained below:
An object of known inertial mass m, is placed on the pan and its time period T is measured. Then, the object of unknown inertial nass is placed on the pan and its time period T2 is measured if m2 is the inertial mass of the unknown object then from equation, we have
m2 = m1 × T²2/T²1
owing the values of m,, T, and Ta, the inertial mass of the unknown od]ect can be found from the equation.
To MEASURE GRAVITATIONAL MASS
The gravitational mass of a body can be measured by using a physical balance.
Principal = It is based on the principle of levers.
It consists of a light but rigid metallic beam balanced at the centre on a sharp knife edge" and rests on a central vertical pillar. From the two ends of the beam, two pans are suspended with the help of knife edges. To save the knife edge
From rom unnecessary wear and tear (when the balance is not in use), the beam rests or two SuPports projecting out from the pillar. By means of a lever, the beam can be raised from its position of rest. The beam at its centre carries a long pointer moving ona small ivory scale. A plumb line suspends adjacent to the vertical pillar, which can be used to level the balance by means of screws provided at the base of the balance. Two adjusting nuts are provided at the extremities of the beam fo slight adjustment. The whole arrangement is enclosed in a wooden box having glass windows to avoid air currents.
The body to be weighed is placed on the left pan and the standard weights on the right pan. The forceps are used to handle the weights. The weights are adjusted, till the beam becomes horizontal. When it is so, the gravitational force on the body to be weighed is equal to the gravitational force on the standard weights and hence the gravitational mass of the body will be equal to that of the standard weights
WEIGHT
Weight of a body is defined as the gravitational pull of the earth on it.
The weight of a body of mass M is given by
W = Mg
Weight of a body may vary rom place tO Place and it may become even zero for example of a body becomes zero at the centre of the earth. Thus, it is not an essential property of the body.
Mass Weight
1. It is the measure of 1. It is the measure measure of gravity
Inertia or gravitation
2.it is a scalar quantity 2.it is a vector quantity
3.it is a constant quantity 3.it is cannot be zero
4.it is a cannot be zero 4. It is not essential property
For any material body
5.it is not affected by the 5.its unit is Newton SI and dyne in cgs system
Presence of other bodies.
SPRING BALANCE
A spring balance is used to measure the weight of a body. It is a direct reading type balance.
Principle = it based on Hooke's law in elasticity which states that the extension reduced in a spring is directly proportional to the force applied, provided the spring is not loaded beyond the elastic limit. 
Construction = A spring balance consists of a helical steel spring S, whose upper end is fixed, and a hook(H)is connected to the lower end with the help of a metallic rod (R) as shown in The spring is enclosed in a metallic case having a slit along its length. A pointer (P) attached to the spring moves in the slit, when some weight is attached to the hook. The edge of the slit is graduated in grams or kilograms.
When the body is suspended from the hook, it experiences gravitational pull equal to the weight of the body. Depending upon the magnitude of the gravitational pull on it, the spring gets elongated. The reading of the pointer on the scale gives the weight of the body suspended from the hook.
TO MEASURE TIME INTERVALS
For measuring time intervals, a clock is used. Any phenomenon that repeats itself regularly can be regarded as a clock. Thus, the periodic motion of earth in its orbit around the sun, a pulse count, beating of heart or the measured vibrations of a cesium atom can form various types of clocks. Of these, a cesium clock is the most accurate one.Two cesium clocks may differ by only 1 second after running for 5,000 years. To measure very small time intervals, atomic clocks may be used, while for measuring long time intervals (such as age of fossils, rocks or earth), radioactive dating may be made use of. A brief account of some of the techniques for measuring ime intervals is as
1. Electrical oscillators = The The main a.c. supply is of frequency 50 Hz. The rotations of synchronous motor run on a.c. can be employed to obtain a time scale.
2. Electronic oscillators = A vacuum tube or a semiconductor transistor can be used to produce electromagnetic oscillations of very high but exact frequencies. The time period of such oscillations can be used to measure small time intervals.
3. Quartz - crystal clocks = A quartz crystal shows piezo electric effect i.e. if such a crystal is subjected to fluctuating pressure across its one pair of faces, an oscillatory e.m.f. is developed across another pair of perpendicular faces and vice-versa. The oscillations so produced can be used to measure time intervals.
4. Atomic clocks = Such clocks make use of periodic vibrations taking place within the atom and can measure a time interval with a precision of 1 s in 105 s. The first cesium atomic clock was designed in 1964.
5. Decay of elementary particles = The life times of many unstable elementary particles range from 10-b s to 10-4 s. By studying the decay of elementary particles, very small time intervals can be measured.
6.Radioactive dating = This technique is used to measure long time intervals. The basic principle of this technique is to calculate the time intervals by noting the ratio of the number of radioactive atoms which have undergone decay with the passage of time to the number of surviving atoms. Carbon dating is used to estimate the age of fossils, while uranium dating is employed for estimating the age of a rock or the earth.
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