Unit & Dimensions Questions and Answers

2 The International System of Units In International System of units there are seven base quantities whose units are defined Which physical quantity has a prefix with its unit a Mass b Thermodynamic temperature c Luminous intensity d Amount of substance L fallarining in

Which of the following ratios has the dimension of mass a Volume Density b Surface tension Angular velocity c Linear momentum Force d Pressure Power AMU Med

of different time periods with suitable coefficients Example 14 2 Which of the following functions of time represent a periodic and b non periodic motion Give the period for each case of periodic motion o is any positive constant i 1 sin ot cos wt ii sin ot cos 2 wt sin 4 wt iii e t iv log at Answer sin wt cos wt is a periodic function it can also be written as 2 sin at 1 4 Now 2 sin at 4 2 sin wt 4 2

35 A river 400 m wide is flowing A boat is sailing at a velocity of 10 m s with respect to the water in a direction perpendicular to the river The time taken by the boat to reach the opposite bank is 1 80 s 2 200 s 3 40 s

The number of significant figures in the sum of 13 21 8 0 and 2 222 are a 5 b 4 9 Match the following c 3 d 2

In the following circuit what are P and Q 10 00 K A P 0 Q 0 B P 1 Q 0 C P 0 Q 1 D P 1 Q 1 op Q

What are the dimensions of the quantity g 1 being the length and g the acceleration due to gravity

Resultant of two forces P and Q acting at a point is R On doubling Q resultant also doubles The magnitude of the resultant also doubles when the direction of Q is made opposite Show that P Q R 2 3 2

28 In the cube of side a shown in the figure the vector from the central point of the face ABOD to the central point of the face BEFO will be JEE Main 2019 Online A a k C a k 1 A a i D a j k A Go a H ad la

dx 2ax x 2 n a sin 1 X 1 a 1 The value of n is a 0 b 1 c 1 d none of these You may use dimensional analysis to solve the pro

If the value of universal gravitational constant in SI is 6 6x10 1 Nm kg 2 then find its value in CGS System

175 Taking into account of the significant figures what is the value of 9 99 m 0 0099 m 9 9801 m 1 2 3 4 9 98 m 9 980 m 9 9 m 9 you 10099 9 891

3 Track OABCD as shown is figure is smooth and fixed in vertical plane What minimum speed has to be given to a particle lying at point A so that it can reach point C 0 1 60 m s 2 70 lo A B 320m 140m C D 2 100 m s 4 80 m s

F L is Y X A AL To determine the Young s modulus of a wire the formula where L length A area of cross section of the wire AL change in length of the wire when stretched with a force F The conversion factor to change it from CGS to MKS system is b 10 d 0 01 a 1 c 0 1

4 A physical quantity Q is related to four independent a ed observables a b c and d as Q The percentage error of measurement in a c and d respectively are 1 3 4 and 2 The percentage error in the quantity Q is then 1 5 2 10 3 13 TS med 2017 4 24

43 0 Find the dimensions of P in the equation where P is pressure x is distance and t is time bx ii Discuss the advantages and two limitations of using dimensional method

A particle is moving spe along the line y a in positive x direction Find magnitude of its angular velocity about origin when its position makes an angle 0 with x axis v sin 0 a v cos 0 a 3 v sin 0 None 2 cons

y 2A sin 2 ct 3 The unit of COS Which of the following statements is incorrect 1 The unit of ct is same as that of 2 The unit of x is same as that of 4 The unit of 2 2 2 X C is same as that of is same as that of X 2 X t

If the velocity of light C gravitational constant G and Planck s constant h are chason as fundame amented cuits then find the dimension of mass OR custom

1 The heart of a man beats 75 times a minute What is its a frequency and b time period 1 nec

d 4 1 a 1 2 b 2 1 c 1 4 36 The dimensions of Planck s constant is the same as the product of a force and time c force and distance b force displacement and time d time and displacement

5 The displacement of a particle is given by x t 2 where xis in metres and t in seconds The distance covered by the particle in first 4 seconds is a 4 m b 8 m c 12 m all 16

The dimensions ML T2 may correspond a work done by a force b linear momentum c pressure d energy per unit volume

I flows around the wire frame along the edges of a cube as shown in figure Point P is the centre of the cube Then X 1 y IX F 1 the magnetic field at P is toward y direction 2 the magnetic field at P is toward y direction 3 the unit vector of magnetic field at Pis 1 1 4 the magnitude of magnetic field at P is 4 2 1 1 3

27 Three vectors P Q and R are shown in the figure Let S be any point on the vector R The distance between the points P and S is b R The general relation among vector P Q and S is A S 1 b P b B S b 1 P bQ C S 1 b P bQ D S 1 b P bQ JEE Adv 2017 Z BIRI R Q P Q

Ans M 1 L T 2 If L and R represents inductance and resistance respectively then what of is the dimension of Ans M OLOT nd capacity respectively then give the

1 Find the dimensional formulae of the following quantities a the universal constant of gravitation G TIM b the surface tension S c the thermal conductivity k and d the coefficient of viscosity n Some equations involving these quantities are Gm m pgrh S 2 2 F Q k r 9 A 0 0 t 2 d and F A V V X X1 hola have their usual meanings

A spherical body of mass m and radius r is allowed to fall in a medium of viscosity n The time in which the velocity of the body increases from zero to 0 63 times the terminal velocity v is called time constant r Dimensionally T can be represented by AIIMS 1987 a c mr2 6 m 6 r 6mmrn 2 g d None of the above b

If two forces of equal magnitude 4 units acting at a point and the angle between them is 120 then find the magnitude and direction of the sum of the two vectors a 4 0 tan c 2 0 tan 1 1 1 73 1 73 b 4 0 tan 0 73 d 6 0 tan 1 0 73

2 14 In the figure 2 201 shown the bigger block A has a mass of 40 kg and the upper block B is of 8 kg The coefficients of friction between all surfaces of contact are 0 2 static and 0 15 sliding Find the acceleration of masses when set free

0 An object is pushed up along a plane of inclination 30 It comes down to the initial point after some time with half the initial upward velocity Find the coefficient of sliding friction between the body and the plane 10 35

direction with speed 10 m s Wind is blowing in east direction with speed 5 2 m s Find the direction in which flag will flutter 1 South 3 East 2 North 4 West

11s elocity IS 5 A car travels 6 km towards north at an angle of 45 to the east and then travels distance of 4 km towards north at an angle of 135 to the east How far is the point from the starting point What angle does the straight line joining its initial and final position makes with the east

Fill in the blanks by suitable conversion of units a 1 kg m s2 g cm s 2 b 1 m ly c 3 0 m s 2 km h d G 6 67 x 10 N m kg 2 cm s 2 g

5 The function y f x is continuous over 0 10 If the area bounded by f x and the x axis between x 0 and x 10 is S where 10 f x 10 then which of the following is are not always correct 10 a S 100 b f x dx S 0 10 c f x dx S 0 10 d S f x dx 8sff x

It A 3i 4j and B i 4j calculate the direction of 1 tan 4 with x axis in clock wise 2 tan 4 with x axis in clock wise 3 tan 4 with x axis in anti clock wise 4 tan 4 with x axis in anti clock wise

2 3 1 Measurement of Large Large distances such as the distance of a pla or a star from the earth cannot be measu m directly with a metre scale An important meth of in such cases is the parallax method ar SU lo th O When you hold a pencil in front of you again some specific point on the background a w and look at the pencil first through your lefte A closing the right eye and then look at t pencil through your right eye B closing the eye you would notice that the position of th pencil seems to change with respect to the pot of on the wall This is called parallax T distance between the two points of observati is called the basis In this example the basis the distance between the eyes To measure the distance D of a far aw planet S by the parallax method we observe from two different positions observatories Aar B on the Earth separated by distance AB at the same time as shown in Fig 2 2 measure the angle between the two direction along which the planet is viewed at these t points The ZASB in Fig 2 2 represented symbol is called the parallax angle parallactic angle As the planet is very far away therefore approximately take AB as an arc is very small of bo length circle with centre at S and the distance D b D 1 Then

In the adjoining vector diagram what is the angle between A and B 12 Given C A 30 C 120 B2 2 B 60 D 150 100 A

The accuracy in the measurement of the diameter of hydrogen atom as 1 06 x 10 10 m is 1 106 1 0 01 2 106 x 10 10 3 4 0 01 x 10 10 The length of a rod is measured as 31 52 cm Graduations on the scale are up to 1 1 mm 2 0 01 mm 3 0 1 mm 4 0 02 cm If L 20 0 01 m and B 10 0 02 m

2 05 56 2001mm crimp A micrometer screw gauge with pitch of 0 5 mm and 50 divisions on circular scale is used to measure the diameter of a thin wire Initially when the gap is closed the line of fourth division coincides with the reference line Three readings show 46th 48th and 44th division coinciding with the reference line which is beyond 0 5 mm of the main scale The best measured value is A 0 46 mm B 0 94 mm C 0 92 mm DIYOT 5x8 01 805mm D 1 00 mm

Example 11 a The unit of coefficient of viscosity in CGS system is poise g cm s Convert it into SI unit b The SI unit of work is joule convert it into CGS unit

ma m mass a acceleration and length The dimensional formula of is 1 MLT 2 3 MOLTO a 2 MT 4 ML

A uniform metal disc of radius R is taken and out R of it a disc of diameter is cut off from the end 2 The centre of mass of the remaining part will be R R 1 from the centre 2 from the centre 10 15 3 R from the centre 4 R from the centra

4 0 g of a gas occupies 22 4 litres at NTP The specific heat capacity of the gas at constant volume is 5 0 JK 1 mol 1 If the speed of sound in this gas at NTP is 952 ms 1 then the heat capacity at constant pressure is Take gas constant R 8 3 JK 1 mol Re AIPMT 2015 1 8 5 JK 1 mol 1 2 8 0 JK 1 mol 1 3 7 5 JK 1 mol 1 and 4 7 0 JK 1 mol 1 HOV

2 5 MEASUREMENT OF TIME To measure any time interval we need a clock We now use an atomic standard of time which is based on the periodic vibrations produced in a cesium atom This is the basis of the cesium clock sometimes called atomic clock used in the national standards Such standards are available in many laboratories In the cesium atomic clock the second is taken as the time needed for 9 192 631 770 vibrations of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium 133 atom The vibrations of the cesium atom regulate the rate of this cesium atomic clock just as the vibrations of a balance wheel regulate an ordinary wristwatch or the vibrations of a small quartz crystal regulate a quartz wristwatch The cesium atomic clocks are very accurate In principle they provide portable standard The national standard of time interval second as well as the frequency is maintained through four cesium atomic clocks A cesium atomic clock is used at the National Physical Laboratory NPL New Delhi to maintain the Indian standard of time In our country the NPL has the responsibility of maintenance and improvement of physical standards including that of time frequency etc e Indian Standard Time IST is set of atomic clocks The efficient pie plecl Note 17

The focal lengths of objective lens and eye lens of a Gallelian Telescope are respectively 30 cm and 3 0 cm Telescope produces virtual erect image of an object situated far away from it at least distance of distinct vision from the eye lens In this condition the Magnifying Power of the Gallelian Telescope should be 1 11 2 3 8 8 2 11 2 4 8 8 2014

7 Find the dimensions of linear momentum and surface tension in terms of velocity v density p and frequency v as fundamental quantities

15 A point charge q 10 1 C is placed at 4 cm abo a square plate 8 cm x8 cm having charge densi 0 5 x 10 8 C m Find the flux related with it 1 0 188 V m melinu A 3 0 288 V m 2 0 12 V m 193 0 4 0 388 V m

15 Assertion The power of an engine depends on mass angular speed torque and angular momentum then the formula of power is not derived with the help of dimensional method Reason In mechanics if a particular quantity depends on more than three quantities then we cannot derive the formula of the quantity by the help of dimensional method 2017

Let P represents radiation pressure c represents speed of light and I represents radiation energy striking a unit area per second then PX Yc will be dimensionless for 1 x 0 y z 3 x z y 2 x y z 4 x y z