Physics

UNIT 1 Physical Quantities, Units and Measurement TOPICS System of units − Fundamental and derived units − Measurement of Length, Mass and Time. Vernier Calipers − Micrometer principle − Screw gauge – Physical balance – mass of a body correct to centigram and milligram− sensitiveness − various types of balances. Measurement of time − various types of clocks/watches − Simple pendulum. Learning outcome: At the end of the unit the student must be able to understand • a physical quantity. • the unit in SI system. • the difference between basic (or) fundamental unit and derived unit. • the principle of Vernier and micrometer screw. • the usage of Vernier caliper and screw gauge. • the concept of physical balance. • the measurement of time • the laws of simple pendulum 1.1

Physical Quantities, Physics is a science involving measurement of many physical quantities like mass, volume, density etc.

1.1.1 Unit

In the measurement of any physical quantity some ‘reference physical standard’ is required. This reference standard of measurement is called unit. ‘The unit of a physical quantity is defined as the reference standard used to measure it’. 1.1.2 The characteristics of unit are: (i) The unit should be well defined. (ii) The unit should be neither too small nor too large in comparison with the physical quantity to be measured i.e. the unit should be of some suitable size (accessibility) FPPHY001KB

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(iii) The unit should neither change with time nor with physical conditions like pressure, temperature etc (invariability) (iv) The unit should be easily reproducible (reproducibility) The magnitude of a physical quantity is the product of unit (u) in which the quantity is measured and the number (n) of times that unit is contained in the given quantity. For example 5 × 1 metre = 500 × 1 cm = 5000 × 1 mm i.e. smaller the unit, bigger the measure and vice versa. 1.1.3 Fundamental units and derived units Units are classified into two categories, basic or fundamental units and derived units. ‘The basic or fundamental units are the units of fundamental quantities. They can neither be derived from one another nor can be further resolved into other more simpler units. The seven basic units in physics in are length, mass, time, electric current, temperature, intensity of light and amount of substance. ‘Derived units are those units which are derived from basic units’. For example, the unit of velocity = the unit of acceleration = 1.4.

dis tance m = ( unit – m s–1) time s change in velocity (unit – m s–2) . time

System of units There are four systems of units (i) centimetre, gram, second (cgs) system (ii) foot, pound and second (fps) system (iii) metre, kilogram, second (MKS) system. (iv) international system of units abbreviated as SI units. SI System of units This system is in fact an improved version of MKS system of units. In SI unit: Base physical quantity

Name of unit

Symbol for unit

length

metre

m

mass

kilogram

k

time

second

s

electric current

ampere

A

temperature

kelvin

K

amount of substance

mole

mol

candela

cd

luminous intensity

Table 1 Base physical quantity and S.I. unit FPPHY001KB

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The two supplementary units are plane angle in radian (rad) and solid angle in steradian (sr) . (i) metre (m) It is defined as the distance occupied by 1650763.73 wavelength in vacuum of the radiation (orange colour line) emitted by the krypton–86 atom. [The wavelength is unique characteristic of Kr–86 and is independent of time].

(ii) kilogram (kg) It is defined as the mass of a platinum-iridium cylinder of diameter equal to its height which is preserved in a vault at International Bureau of Weights and Measurers at Sevres near Paris. (iii) second (s) It may also be defined as the time taken by cesium–133 atoms to make 9,192,631,770 vibrations. (iv) ampere (A)

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It is that constant current which flowing in two straight parallel conductors of infinite length and of negligible area of cross section and placed one metre apart in vacuum would produce between the conductors a force equal to 2 × 10–7 newtons per metre of length. (v) kelvin (K) It is equal to

1 of the thermodynamic temperature of triple point of water. 273.16

(vi) candela (cd) It is the luminous intensity, in a direction at right angles to a surface of

1 square metre area of 600000

a black body at a temperature of freezing platinum under a pressure of 101325 newton per square metre. (vii) mole (mol) It is the amount of substance which contains as many elementary units as there are carbon atoms in exactly 0.012 kg of C12. (viii) radian (rad) It is defined as the plane angle between the two radii of a circle which cut off on the circumference an arc of length equal to the radius of that circle. Derived quantity

physical

Relation with base and derived quantities

Symbol for unit

Special name

area

length × width

m2

volume

length × width × height

M3

density

mass ÷ volume

kg m–3

speed

distance ÷ time

ms–1

acceleration

change in velocity ÷ time

ms –2

force

mass × acceleration

kg ms–2 (N)

newton (N)

pressure

force ÷ area

kg m–1 s–2 (N m–2)

pascal (Pa)

work power

force × distance

–2

kg m s (Nm)

joule (J)

work ÷ time

kg m2 s–3 (Js–1)

watt (w)

2

Table 2 Derived physical quantity and S.I. unit 1.1.6. Advantages of SI Unit (i) This system makes use of only one unit for one physical quantity (i.e. it is a rational system of units) (ii) In this system, all the derived units can be easily obtained from basic units (i.e. it is a coherent system of units) (iii) It is metric system ie multiples and sub-multiples can be expressed as powers of 10. 1.1.7. Some practical units of distance: Microcosm (very small) units of distance are

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(i) Micron (μ) -10–6m (ii) Angstrom (A• )-10–10 m (iii) Nanometre (nm) – 10–9 m (iv) Fermi (F) – 10–15 m.

? How many light years are there in one metre? Macrocosm (very large) units of distances are (i) Light year – It is the distance travelled by light in 1 year 1 light year = (3 × 108) × 60 × 60 × 24 × 365.25 = 9.467 × 1015 m (velocity of light is 3 × 108 ms-1 ) Light year is used in astronomy. Alpha centauri, the nearest star outside the solar system is 4.3 light years away from the earth. The milky way (Akash Ganga) is nearly 10 5 light years in diameter. Prefix

Abbreviation

Power

nano

n

10–9

micro

µ

10–6

milli

m

10–3

centi

c

10–2

deci

d

10–1

kilo

k

103

mega

M

106

giga

G

109

Table 3 Some prefixes used in S.I. unit

(ii) Astronomical Unit (AU): The average distance between the sun and the earth. 1 AU = 1.496 × 1011 m Table 3 Some prefixes used in S.I. unit] (iii) parsec: It is an abbreviation of parallactic second. It is the distance at which 1 AU subtends an angle

of 1 second of an arc θ =

l l or r = ÷ r θ

where ℓ = 1 AU and θ = 1 second, r = 1 parsec ∴ 1 parsec =

π rad ÷ Q 1sec ond = 60 × 60 × 180

1.496 × 1011 × 60 × 60 × 180 π

= 3.08 × 1016 m

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Example Express the distance of Alpha Centauri in solar system in parsec. ? Distance = 4.3 light years How many = (4.3 × 365.25 × 86400) × 3 × 108 m

parsec are 4.3 × 365.25 × 864 × 3 × 1010 there in 1 = 1.319 par sec = 3.08 × 1016 light year?

[ ∵ 1 parsec = 3.08 × 1016 m] 1.1.8. Atomic to Astronomical range of variation of length 1. Radius of proton 2. Diameter of nucleus 3. Diameter of hydrogen atom 4. Diameter of red blood corpuscle 5. Length of cubit (normal person) (cubit is the length of the arm from the elbow to the tip of the middle finger) 6. Height of a child 7. Height of a double storey house 8. Length of a cricket, football and hockey field 9. Height of Mount Everest 10. Radius of the earth 11. Radius of the sun 12. Distance of sun from earth 13. Distance of Alpha Centauri (the nearest star) from earth 14. Diameter of our galaxy 15. Estimated edge of the universe distance to a distant quasar (farthest objects in our universe)

10–15 m 10–14 m 10–10 m 10–5 m 10–1m 100 m 101 m 102 m 104 m 107 m 109 m 1011 m 1017 m 1020 m 1026 m

1.1.9. Some practical units of mass: Microcosm (very small) units of mass is atomic mass unit (amu). 1 amu = 1.67 × 10 −27 kg Macrocosm (very large) are, = 1000 kg 1 Quintal = 100 kg The largest unit of mass is ‘chandra sekar limit’ (csl). 1 csl = 1.4 times the mass of the sun. Mass of the sun = 2.0 × 1030 kg. 1.1.10. Some conversions in mass unit 1 slug = 14.57 kg 1 lb (pound) = 0.4536 kg 1 kg = 2.2 lb 1.2 MEASURING INSTRUMENTS 1.2.1 Measurement of length To measure lengths rulers or measuring tapes are used. The accuracy is only 0.1 cm. For greater accuracy instruments like Vernier calipers or micrometer screw gauge must be used. FPPHY001KB

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1.2.2 VERNIER CALIPERS Principle of Vernier Calipers A French scientist, Pierre Vernier invented a scale named after him as, vernier scale. It consists of two scales, ‘Main scale’ and another ‘vernier scale’. The vernier can slide over the main scale. The principle of vernier is that n vernier divisions are equal to (n–1) main scale divisions Construction: It consists of a steel rod graduated in cm on one side and in inches on the other sides. This is called main scale. Vernier scale slides over main scale (Figure 1) It has two set of jaws AB and PQ. AB is used to measure thickness and external diameter; PQ is used to measure internal diameter. The scale can be fixed at any position by locking screw. In Figure1. 10 vernier scale divisions coincide with 9 main scale divisions.

Figure 7 Viernier Caliper Vernier constant (or) least count Vernier constant is the least distance that can be measured by vernier calipers. It is equal to the difference between one main scale division (MSD) and one Vernier scale division (VSD). By the principle of vernier n VSD = (n –1) MSD 1 VSD =

(n − 1) MSD n

Vernier constant or least count = 1 MSD – 1 VSD = 1MSD − =

(n − 1) MSD n

1 MSD n

Example In a Vernier calipers 1MSD is 1 mm and there are 10 VSDs. Find the least count. LC =

1 MSD = 0.1 mm = 0.01 cm 10

Zero error and zero correction (i) When the two jaws are made to touch each other (without the object between them) zero mark on the vernier scale should coincide with the zero mark in the main scale. (Figure 2(a)) In this case there is no zero error.

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(a)

(b) Figure 8

(c)

(ii) Due to wear and tear to the edge of the jaws, the zeros may not coincide. This causes an error. There are two types of errors i.e. positive and negative error. If the zero of the vernier scale is to the right of the zero mark on the main scale, zero error is positive (Figure 2(b)). Also, as per figure, the third division of the vernier coincides with one of the main scale division. In this case, Zero error = 3 × 0.01 = 0.03 cm. When the zero error is positive, the zero correction is negative. Therefore, Zero correction = – zero error = – 0.03 cm. With the correction, Correct reading = (observed reading – 0.03) cm. (iii) If the zero of the vernier scale is to the left of the zero mark of the main scale, then the zero error is negative (Figure 2(c)). Also as per figure, fourth division of the vernier coincides with one of the main scale divisions. Zero error = (10 – 4) × 0.01 = – 0.06 cm. When the zero error is negative, the zero correction is positive. zero correction = – (zero error) = – (–0.06 cm) = +0.06 cm With correction Correct reading = (observed reading + 0.06) cm Determination of a length The body is put between A and B and gently gripped. The main scale reading just before zero of the vernier and the number of vernier division coinciding with any one the main scale division is noted.

Observed length = MSR + (VSC × LC) Correct length = (observed length) ± zero correction This may be repeated for many times and average is taken.

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1.2.3 SCREW GAUGE A screw gauge is used for measuring small lengths such as diameter of a wire or thickness of a thin glass plate etc. Construction It consists of a U shaped frame ABC. To one end of it is rigidly fixed a stud D or anvil (Figure 3). Second limb carries a hollow stem or sleeve S. A scale is graduated (usually in mm) on the stem and is called the pitch scale. A screw or spindle E works inside the sleeve, S. A circular cap, H is coupled with screw E and is connected to the right hand side of the screw. The circular cap carries â€˜nâ€™ equal sized division (usually n = 100).This scale is called head (circular) scale. The head scale can be rotated using a ratchet, R provided at the end of the head scale. To measure the size of the object the object is griped between D and E by rotating the head scale.

Figure 11 Screw Gauge Pitch: The distance moved by the head of the screw in forward or backward direction when it is given one complete rotation is called pitch. Pitch is also the distance between two consecutive threads of the screw measured parallel to the axis of the screw. Least count =

Pitch Number of division s in the circular (head) scale

Example Pitch of the screw is 1 mm. The number of head scale divisions 100. Find the least count. Therefore, LC =

1 = 0.01 mm 100

Zero error and zero correction

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(i) If zero of the circular scale (head scale) exactly coincides with the line of graduation (datum line) when the studs D and E just touch each other, the zero error and zero correction is = 0 × LC = 0(0.01) = 0.00 mm (Fig.4a)

(ii) When D and E just touch each other, if the zero of the circular scale is below the line of graduation, the zero error is positive (Fig.4b). In this figure the third division of the head scale coincides with the line of graduation. Therefore, zero error = 3 × LC = 3 × 0.01 = 0.03 mm Zero correction = – zero error = – (0.03) = – 0.03 mm (iii) When D and E just touch each other, if the zero of the circular scale is above the line of graduation, the zero error is negative (Fig.4(c)). In this figure the ninety seventh division of the head scale coincides with the line of graduation. Therefore, zero error = (100 – 97) × LC = 3 × 0.01 = 0.04 mm Zero correction = – zero error = – (–0.03) = + 0.03 mm Determination of length (diameter of a thin wire) by screw gauge The wire is inserted between the studs D and E. The head scale is rotated so that the wire is gripped gently between the two studs. The pitch scale reading (PSR) and the head scale coincidence (HSC) are noted. The diameter = PSR + (HSC × LC) If there is any zero error, necessary zero correction is made in the observed reading. Correct reading = Observed reading ± zero correction. Example FPPHY001KB

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The pitch of the screw is 1mm. The number of head scale divisions are 100. The zero error is +2 divisions. When a wire is gripped between the studs the pitch scale reading is 3mm and the circular division that coincides with reference line is 84. What is the correct diameter of the wire? Solution Pitch = 1 mm Head scale division = 100 Least count =

1mm = 0.01 mm 100

zero error = +2 division = +2 × .01 mm = +.02 mm zero correction = – 0.02 mm Pitch scale reading = 3 mm Head scale coincidence = 84 observed diameter = 3 + (84 × .01) = 3.84 mm ∴ corrected diameter = 3.84 – 0.02 mm = 3.82 mm 1.2.4. Backlash Error When the instrument is new and the screw is well cut, the screw firmly fits in the nut. After continuous use and due to wear and tear, there is some ply or small gap left between the screw and the nut. Therefore the screw does not move backward or forward for a little motion of the head of the screw. This would cause an error called backlash error. This is avoided by turning the screw only in one direction while doing an experiment to take readings. 1.3

Mass and Weight Mass of a body is the quantity of matter contained in the body. It is constant at ordinary velocity (It changes when the body moves with velocity comparable to the velocity of light). It is measured with some standard mass with the help of a physical balance. The unit is kilogram (kg) = (1 kg = 1000 gram = 103 gram). Weight of a body is the force with which a body is pulled by the earth towards it centre. It varies from place to place. The unit is Newton (N). The practical unit is kgf (kilogram force) It is measured using spring balance. It is wrong to say that ‘my weight is 40 kg’. It is actually 40 kgf = 40 × 9.8 = 392 N. It is correct to say, ‘My mass is 40 kg’.

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Figure 13 1.3.1 PHYSICAL BALANCE Principle: It works on the principle of moments. According to principle of moments when a beam is in equilibrium under the action of a number of forces acting in the same plane, the sum of clockwise moments is equal to the sum of anticlockwise moments. The schematic diagram for a physical balance is shown in Figure 5. The construction details are given below.

Construction: It consists of a light metal beam (B) balanced at centre of gravity on a knife edge made of agate (a very hard substance) and rests on a flat agate surface fixed on the top of a vertical pillar (P) (Fig.5). The position of the pillar can be raised or lowered by means of a lever. The beam is provided with two adjustable screws S1 and S2. Two identical pans are suspended from two knife edges at the end of the beam through stirrups. A pointer (I) is fixed at the middle of the beam and can oscillate on a scale when the pillar is raised. The screws S 1 and S2 are adjusted for producing equal reading on either side of the scale. A plumb line is suspended from the support of the beam. By using the leveling screws fixed to the base, the pillar is adjusted to be vertical. The entire arrangement is placed in a box with glass doors. The standard weights of 200 gf, 100 gf, 50 gf, 20 gf, 20 gf, 10 gf, 5 gf, 2 gf, 1 gf and 500; 200; 200; 100; 50; 20; 10 mgf are provided in the weight box.

1.3.2. Requisites of a good balance (i) Truth (ii) Sensitivity and (iii) Stability (i) Truth: A balance is said to be true if the beam remains horizontal when the pans are empty or when loaded equally.

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(ii) Sensitivity: A balance is said to be sensitive if small difference in mass in the two pans causes a large deflection in the beam. Sensitivity is measured by the deflection produced in the beam when a difference of mass in two pans is 1 mg. It is measured in radian/mg. (iii) Stability: A balance is said to be stable if it returns quickly to its position of equilibrium after it has been disturbed. Turning point: When the balance is released, the pointer swings to and fro in front of the scale. The point at which the pointer changes its direction is called the turning point. Resting point: A sensitive balance does not quickly come to rest after being disturbed. It takes a long time to come to rest. To find the resting point, five consecutive turning points (3 on the left and 2 on the right) on the scale starting from left are noted. 1 L1 + L2 + L 3 R1 + R 2 + Then the resting point (RP) = 2 3 2

÷,

where, L1, L2 and L3 are the readings on the left side and R 1 and R2 are the reading on the right side. When the pans are empty, the resting point is called zero resting point (ZRP). Determination of mass of a body When the pans are empty, by noting down three readings on the left and two on the right, the resting point is determined as explained earlier. Let the zero resting point be a. The given body is placed on the left pan and the weights are added to the right pan in the order as in the weight box until the pointer oscillates on the scale almost equally on both sides. Again the resting point is found and let it be b. If b is greater than the zero resting point a,10 mg is added and again the resting point is determined as c. It should be less than the zero resting point. By trial one has to repeat this procedure, till the one of the resting point is above the zero resting point and the other is below the zero resting point,

} {

}

Mass of the body = Mass of the body corresponding + 10 (b − a) correct to milligram to higher resting point,b (b − c)

Example In a physical balance experiment, the zero resting point is 10.2 with the body on the left pan. After adding 50, 10, 5, 2, 1g and 100, 50, 20 mg, in the right pan the resting point is 10.8. On adding 10 mg the resting point is 9.8. What is the mass of the body in centigram? Find the sensibility and determine the mass of the body correct to milligram. Solution Zero resting point = 10.2 = a With mass 68.17 g the resting point is 10.8. = b The mass added is less and hence 10 mg is added. With mass 68.18 g the resting point is 9.8 = c. Out of these two resting points, 9.8 is closer to 10.2 ∴ The mass of the body correct to centrigram is 68.18 g Sensibility is defined as the small mass required to produce the deflection of the pointer, through one division. FPPHY001KB

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Sensibility =

10 mg 10 mg 10 = 10.8 − 9.8 = 1 b−c

Mass correct to milligram

= 10 mg

{ Mass of the body }

10

× (b − a) = at higher restingpoint + b−c

= 68.170 +

10 × (10.8 − 10.2) 10.8 − 9.8

= 68.170 + 0.006 = 68.176 g Precautions for using a balance

(i) Pans should be clean and dry (ii) The stirrups should rest properly on the knife edges (iii) Plumb line must be vertical (iv) Standard weights should be added or removed from the pan only when the beam is in arrested position. (v) While taking the resting points, the glass doors of the balance should be closed. (vi) Hot bodies should not be weighed (vii) Weights from the weight box should be handled with the help of forceps.

1.4

Type of Clock/Watch

Accuracy

Atomic clock

10–10 s

Digital stop watch

± 0.01 s

Analogue stop watch (electronic oscillator)

± 0.1 s

Ticker tape timer (electrical oscillation)

0.02 s

Quartz watch

1s

Pendulum clock (mechanical vibration)

1s

Measurement of time The concept of time is a very familiar concept. We talk of yesterday, today and tomorrow. Time is measured in years, months, days, hours, minutes and seconds. The SI unit for time is second(s).

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Due to the wide range of time intervals to be measured, there are different kinds of clocks and watches. The measurement of time is essentially a process of counting.

The clocks and watches have a common characteristics. They are based on some regular event or process such as repetitive motion (as in swinging pendulum) or vibration (as in the quartz crystals in watches or stop watches). Such repetitive motion or vibrations are called oscillations. The time taken to make one complete oscillation is known as the period of the oscillation. There are many repetitive phenomena taking place in nature. One such repetitive phenomenon is the spinning of earth about its axis. A solar day is the time interval between two consecutive passage of sun across the plane of the meridian at a place in question. The standard of time for all purposes was the mean solar second. It is defined as the

1 part of the mean solar day. Now86400

a-days, the unit is based on the periodic vibration produced in cesium atom. A few examples of range of variation of time

1.5

1.

Period of atomic vibration

10–15 s

2.

Time taken by atom to emit visible light

10–9 s

3.

Time for electron beam to go from source to screen in TV tube

10–7 s

4.

Period of sound signal

10–4 s

5.

Time for electric fan to complete one revolution

10–2 s

6.

Time between heart beats

100s= 1s

7.

Time for light from the sun to reach the earth

103 s

8.

Time for earth to rotate once on its axis (day)

105 s

9.

Time for earth to revolve around the sun (year)

107 s

10.

Human life span

109 s

SIMPLE PENDULUM A simple pendulum is a metallic sphere (usually called as the bob of the pendulum) suspended by a light torsion less thread. Refer Figure 6. The length of the simple pendulum is the distance from the point of suspension of the string to the centre of gravity of the bob. When the bob of the pendulum is pulled to one side and released, the bob goes on swinging to and fro about its equilibrium position.

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Figure 14

Physics

The amplitude of the simple pendulum is the maximum extent of swing to either side measured from the equilibrium position. In Figure 6, the length of the arc BA or BC gives the amplitude. When the bob swings to and fro, one complete to and fro motion is called an oscillation. (That is the bob swings from B to C, then to A and back to B). The period of the simple pendulum is the time taken for one complete oscillation.

1.5.1. Laws of the simple pendulum For small amplitudes, the period of oscillation of a simple pendulum is (i)

independent of amplitude (length being constant),

(ii)

independent of the size, material or mass of the bob (length being constant),

(iii)

directly proportional to the square root of its length and

(iv)

indirectly proportional to the square root of the acceleration due to gravity at that place.

If l the length of the simple pendulum and g the acceleration due to gravity then the the period of oscillation T is given by,

T =2π

l g

The period of the simple pendulum can be easily determined using a stop clock or stop watch for

l 2 ÷ will be found constant.. T

different lengths of the simple pendulum and the quantity

If l1, l2, l3 are different lengths of the simple pendulum and T1, T2, T3 are the respective periods then,

l1 l2 l3 = = T12 T22 T32 The acceleration due to gravity can be calculated using the relation given below.

l T 2 =4π2 g

or

l g = 4 π2 2 T

Example A simple pendulum, 1 m length has a period of 2 s. Calculate the value of g. Solution π =3.14, π2 =9.8approximately l T 2 = 4 π2 ÷ g 1 T 2 = 4 ×9.8 ÷=4 g g = 9.8 ms–2

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Example A simple pendulum whose length is 64 cm takes 32 s for 20 oscillations. What will be the period of the pendulum if the length is increased to 80 cm? Solution

l1 = 0.64m,

l2 = 0.8m,

l1 l2 = T12 T22

;

32 = 1.6s T2 = ? 20 0.64 0.8 = 2 2 ( 1.6 ) T2 T1 =

T22 = 3.2 T2 = 3.2 = 1.79s

1.5.2. Seconds pendulum A seconds pendulum is one for which the period of oscillation is 2 seconds. The length of the seconds pendulum is almost equal to 1m. Pendulum clocks are usually fitted with seconds pendulums.

?

SOLUTION

1 light year = 9.467 × 1015 m ∴1m=

1 light year = 0.1056 × 10–15 light years. 9.467 × 1015

1 m = 1.056 × 10–16 light years Q PG: 4

1 parsec = 3.26 light years ∴ 1 light year =

1 parsec = 0.3067 parsec 3.26 Q PG: 5

atomic mass unit (amu)

Q PG: 12

The time taken by moon to complete one revolution around earth is called lunar month [1 lunar month = 27.3 days] Q PG: 14

It is the time taken by earth to complete one revolution around the sun. Q PG: 14

Tropical year is that year in which there is total solar eclipse. Q PG: 14

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