Units and Dimensions

 Assignment 1

BASED ON UNITS 

1 What is a physical quantity? ANS Any quantity which can be measured is called a physical quantity.

2. What is a unit? ANS A certain basic, arbitrarily chosen, internationally accepted standard of reference for making measurements of a physical quantity is called a unit.

3. Which are fundamental or basic units? ANS The unit of fundamental quantities is called fundamental units

4. What are derived units? ANS The units of derived quantities that can be expressed as combination of basic units are called derived units.

5. What is meant by SI system?  ANS SI system means the international system of units, containing seven basic units.

6. Name the system of units accepted internationally. ANS SI system

7. Give the basic units of length in CGS / MKS / FPS / SI system. ANS Centimeter / meter / foot / meter respectively.

8. Given the base units of mass in CGS / MKS / FPS / SI system.  ANS gram / kilogram / pound / kilogram respectively.

9. Name the unit of time in all systems. ANS seconds

10. How many base units are there in SI system? ANS There are seven basic units in SI system.

11. Name the SI unit of current / temperature / amount of a substance / luminous intensity. ANS Ampere / Kelvin / mole / candela.

12. Name the SI unit of angle in a plane? ANS Radian

13. Name the SI unit of solid angle. ANS Steradian

14. Express the relation for angle in a plane.

15. Express the relation for solid angle.

16. Define astronomical unit (AU) ANS The average distance between earth and sun is known as astronomical unit.

17. What is meant by light year? ANS The distance travelled by light in one year of time.

18. Define parsec. ANS One parsec is the distance at which an arc of length equal to one AU subtends an angle of one second at a point.

19. Express parsec in terms of light years. ANS 1 parsec = 3.26 light year.

20. Which are the shorter units of length? Express them in meters. ANS fermi and angstrom where 1 fermi = 10-15 m and 1 angstrom = 10-10 m

21 Name the larger units of length. ANS Astronomical unit (AU) and light year (ly), parsec (pc)

22 Define unified atomic unit. ANS One unified atomic mass unit is equal to 1/12th of the mass of an atom of carbon 12 isotope including the mass of electron.

23.       Determine the number of light years in one metre.

24.       Name any two derive SI unit with the name of scientist. ANS Newton , joule, watt etc (any two)

BASED ON MEASUREMENTS 

1. Mention some direct method of measuring length? ANS Using metallic scale, vernier scale or screw gauge.

2. Name the method of measuring long distances. ANS Parallax method

3. What is parallax? ANS The change in position of object w.r.t a point when viewed with left and right eye

4. What is meant by parallax angle? ANS The angle between two directions of observation of a point (object) is called parallax angle.

5. Which is the instrument used to measure small masses like atom? ANS Mass spectrograph.

6. What is the basis of working of cesium clock or atomic clock? ANS Periodic vibration of cesium atom.

7. Name the type of clock which gives accurate time? ANS Cesium atom clock.

8. When the planet Jupiter is at a distance of 824.7 million km from the earth, its angular diameter is measured to be 35.72˝ of arc. Calculate diameter of Jupiter

9. A laser light beamed at the moon takes 2.56s and to return after reflection at the moon’s surface. What will be the radius of lunar orbit.

10. The moon is observed from two diametrically opposite points A and B on Earth. The angle θ subtended at the moon by the two directions of observation is 1o 54′. Given the diameter of the Earth to be about 1.276 × 107 m, compute the distance of the moon from the Earth.

11. The Sun.s angular diameter is measured to be 1920′′. The distance D of the Sun from the Earth is 1.496X1011 m. What is the diameter of the Sun ?

12. The nearest star to our solar system is 4.29 light years away. How much is this distance in terms of parsecs? How much parallax would this star (named Alpha Centauri) show when viewed from two locations of the Earth six months apart in its orbit around the Sun ?

13. When the planet Jupiter is at a distance of 824.7 million kilometers from the Earth, its angular diameter is measured to be 35.72. of arc. Calculate the diameter of Jupiter.

BASED ON DIMENSIONS 

1. Define dimensional formula of physical quantity? ANS The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula.

2. Write the dimensional formula of volume. ANS [M0 L3 T0]

3. Define dimensional equation of a physical quantity. ANS An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity.

4. Name a physical quantity which has no units and no dimensions? ANS strain.

5. Name a physical quantity which has units but no dimensions. ANS Angle has unit i.e. radian but it is dimensionless.

6. Name a physical quantity which has neither unit nor dimension. ANS Relative density (specific gravity)

7. Can a physical quantity have dimension but no unit? ANS No

8. State the principle of homogeneity of dimensions? ANS It states that the dimensions of all the terms on either side of an equation must be the same

9. If ‘slap’ times speed equals power, what will be the dimensional equation for ‘slap’? ANS  slap = [MLT-2]

10. If the units of force and length each are doubled, then how many times the unit of energy would be affected? ANS Energy = Work done = Force x length So when the units are doubled, then the unit of energy will increase four times

11. Can a quantity has dimensions but still has no units? ANS No, a quantity having dimension must have some units of its measurement. 

12. Justify L + L = L and L – L = L? ANS When we add or subtract a length from length we get length, 

So L + L = L and L – L = L. 

13. Can there be a physical quantity that has no unit and no dimensions?  ANS Yes, like strain. 

14. Mention two pairs of physical quantities which have the same dimensions. ANS (i) work and energy (ii) pressure and stress.

15. Mention the physical quantities whose dimensions are (i) [M1 L-1 T-2] (ii) [M1 L2 T-3]

ANS (i) pressure (ii) power

16. Mention any two constants which have dimensions. ANS Gravitational constant, Plank’s constant.

17. Mention any two applications of dimensional analysis. ANS 1) To check the correctness of a physical relation 2) to derive relationship between different physical quantities.

18. Mention any two limitations of dimensional analysis. ANS 1) Dimensionless constants in a relation cannot be determined by this method. 2) It cannot derive the exact relationship between physical quantities in any equation.

19. In Van der Wall’s equation (P + a/V2)(V – b) = RT, Determine the dimensions of a and b. ANS [a] =[ML5 T-2] and [b] = [ M0L3T0]. 

20. Give the limitations of dimensional analysis. ANS Dimensionless constants cannot be obtained by this method. The method of dimensions can only test the dimensional validity, but not the exact relationship between physical quantities in any equation. It does not distinguish between the physical quantities having same dimensions. 

21. If X= a+ bt2, where X is in meter and t is in second. Find the unit of a and b? ANS Unit of a is meter and unit of b is m/sec2. 

TYPE 1 (BASED ON RELATION BETWEEN PHYSICAL QUANTITIES)

1. Force (F) acting on a particle moving in a circle depends upon mass(m), radius of circular path(r) and speed (v). Find the relation between them using dimensional analysis ? (ANS : F=k mv2/r)

2. Time period(t) of simple pendulum depends upon mass(m),length(l) and acceleration due to gravity (g). Find the relation between them using dimensional analysis ?(ANS : t = klg )

3. Energy (E) of moving object depends upon density (ρ), speed(v) and time (t). Find the relation between them using dimensional analysis ? 

4. Energy (E) of moving object depends upon mass(m), speed(v) and time (t). Find the relation between them using dimensional analysis ? (ANS :E = kmv2)

5. Velocity of liquid flow depends upon radius(r) of tube, coefficient of viscosity (η) and acceleration due to gravity (g).Find the relation between them using dimensional analysis ? 

6. Speed of wave in a string depends upon tension(T) and mass per unit length(μ).Find the relation between them using dimensional analysis ? (ANS : v = KT )

7. The escape velocity v of a body depends on, the acceleration due to gravity ‘g’ of the planet and the radius R of the planet, Establish dimensionally for relation for the escape velocity. 

8. Bulk modulus of rigidity (B) of a substance depends upon speed(v) , density (ρ) and mass(m), Find the relation between them using dimensional analysis ? (ANS : B = K v2 ρ)

9. Velocity  (v) of water waves depends upon Wave length (λ), density (d) and acceleration due to gravity(g), find the relation using Dimensional analysis?(ANS : v = Kg)

10. Viscous force (F) of a spherical object depends upon radius (r),terminal speed (V) and , coefficient of viscosity (η).Find the relation between them using dimensional analysis ?(ANS : F =Kvrη)

11. Reynolds number (R)of liquid depends upon speed(V), coefficient of viscosity (η) and diameter of pipe, find the relation using dimensional analysis ?

Type 2 ( Based on correctness /accuracy of given expression )

1. Check the accuracy of the following  

1. v=B where B = bluk modulus of elasticity and ρ is the density 

2. λ = v P /g where v = speed, P = pressure , g =acceleration due to gravity and λ = wavelength of the wave 

2. Check the accuracy of the following  

1. V =u + at 

2. S=ut+at2/2

3. V2 – u2 = 2as where v = final speed, u = initial speed , s= distance, a = acceleration , and t = time 

4. Y = A sin (vt -  kx ) where A = Amplitude, v= speed , k =wave number .x =  distance and y= displacement 

5. Dn = u+a(2n-1)/2 , where Dn = Distance covered in nth second, u = speed, a = acceleration

ANSWERS 1(a) correct, 1(b) not correct),2(a) Correct 2(b) correct 2(c) Correct 2(d) correct 

Type 3 (Dimension of unknown quantities) 

1. Find the dimension of α in the (a) F = α t , where F = force and t = time and (b) E = 2αP where P = Pressure and E is the Energy

2. What are the dimensions of a and b in the relation F = ax + bt2 where F is force , x is distance and t is time.

3. Find the dimension of a and b in the following (a) F = at+bt3 (b) P=a/v + b/E where t = time,F force,P=Pressure , v = speed and E = Energy

4. Find the dimension of a/b in (P+a/V2)(V-b) RT , where V= volume, P= Pressure , R=Gas constant and T = Temp

Type – 4 (Convert one system of unit into another )

1. Convert 1 joule into CGS System?

2. Convert 1 N into CGS System?

3. Convert 1 pascal into CGS System?

4. Convert 1erg into MKS System?

5. Convert 6.7X10-11 Nm2/kg2 into CGS system?

6. Convert 6.6X 10-34 Js into CGS system?

7. Convert 4.3 ft/s2 into MKS system?

8. Convert 5.6 cm/s3 into MKS system?

Type 5 (additional)

1. Correct the equation using dimensional  method , when c  = speed of light is missing in the expression below 

m= m01- v2

Where m and m0 are the masses and v = speed of the moving particle 

2. Find the dimension of a and b in the expression v = a e-b/v where v= speed of the particle ?

3. Check the accuracy of g= GM/R2 where G = Gravitational Constant , M = Mass and R = radius of planet?

4. Volume of liquid flow per unit time depends upon pressure gradient(p/l) , coefficient of viscosity and radius of pipe, find the relation using dimensional analysis ?

5. If Energy(E), Mass (M) and Force(F) are considered as fundamental units then express Pressure (P) in terms of new Dimension.

6. If E, M, J and G respectively denote energy, mass, angular momentum and gravitational constant, Calculate the dimensions of EJ2 /M5 G2 

7. The frequency ν of vibration of stretched string depends on its length L its mass per unit length m and the tension T in the string obtain dimensionally an expression for frequency.

8. If g is the acceleration due to gravity and λ is wavelength, then which physical quantity does represented by √gλ. 

ANS Speed or velocity

9. If the unit of force is 100 N, unit of length is 10 m and unit of time is 100 s, what is the unit of mass in this system of units.

BASED ON ERRORS 

1. What is error? ANS The uncertainty in measurement is called error.

2. What is meant by accuracy? ANS The accuracy is the measure of how close the measured value is to the true value of the quantity?

3. What is meant by precision? ANS Precision means, to what resolution or limit of the instrument, the quantity is measured. It is given by least count

4. What are the types of error? ANS Systematic and random error.

5. Why does a measurement give approximate value? ANS It is due to error.

6. What is meant by systematic error? ANS Systematic errors are that tend to be in one direction and affects each measurement by same amount.

7. What are random errors? ANS The random errors are those errors which occur irregularly due to random and unpredictable fluctuations in experimental conditions.

8. What is least count? ANS The smallest value that can be measured by an instrument is called the least count.

9. What is least count error? ANS It is the error associated with the resolution of the instrument.

10. What is absolute error? ANS The magnitude of the difference between the individual measurement and the true value of the physical quantity is called absolute error. It is always positive.

11. How would you determine the true value of a quantity measured several times ? ANS By taking arithmetic mean

12. What is relative error? ANS The relative error is the ratio of the mean absolute error to the mean value of the quantity measured.

13. What is percentage error? ANS The relative error expressed in percentage is called percentage error.

14. Time period of a simple pendulum measured using different stopwatches are found to be 1.22s,1.23s,1.25s and 1.20s

Find (a) mean observation (b) absolute errors (c) mean absolute error (d) relative error and (e) percentage error.

15. Length of cylinder measured by Vernier scale as followed 3.33cm,3.35cm, 3.39cm, 3.32cm and 3.36cm. Find (a) mean observation (b) absolute errors (c) mean absolute error (d) relative error and (e) percentage error. 

16. The length of a rod measured in an experiment was found to be 2.48m, 2.46, 2.50m and 2.48m and 2.49m, Find the average length , the absolute error in each observation and % error.

17. A physical quantity X is given by X = A2B3/C1/3D , If the percentage errors of measurement in A,B,C and D are 4%,2%,3% and 1% respectively, then calculate the % error in X. 

18. If two resistors of resistance R1=(4 ± 0.5)Ω and R2=(16 ± 0.5)Ω are connected (1) In series (2) In parallel. Find the resistances with limits of error?

19. What are significant figures? ANS The reliable digits plus the first uncertain digit are known as significant figures.

20. Does the number of significant figures depend on the choice of unit? ANS No

21. State the number of significant figures in the following a) 0.006 m2 (b) 2.65 x 103 kg (c) 0.2309 m-3 (d) 6.320 J (e) 0.006032 m2 ? ANS (a) 1 (b) 3 (c) 3 (d) 4 (e) 4

22. Round off the following result to three significant figures (a) 2.746 (b) 2.744 (3) 2.745 (4) 2.735

ANS (a) 2.75 (b) 2.744 (c) 2.74 (d) 2.74

23. Mention the base quantities in SI system

24. What are sources of systematic error? ANS (i) Instrumental error (ii) imperfection in experimental procedure (iii) personal error

25. Explain the method of reducing systematic error/ ANS It can be minimized by improving experimental techniques, selecting better instruments and removing personal bias.

26. Give any two methods of reducing least count error. ANS Least count error can be reduced by using instruments of higher precision, improving experimental techniques and taking mean of all observations.

27. The period of oscillation of a simple pendulum is T = 2πLg. the measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a watch of 1s resolution. Find accuracy in % error. ANS 2.7%

28. In an experiment, on the measurement of g using a simple pendulum the time period was measured with an accuracy of 0.2 % while the length was measured with accuracy of 0.5%. Calculate the percentage error in the value of g. 

29. Given relative error in the measurement of length is 0.02, what is the percentage error? ANS  Percentage error = 2 % 

30. If heat dissipated in a resistance can be determined from the relation H = I2Rt joule. If the maximum error in the measurement of current, resistance and time are 2%, 1% and 1% respectively, what would be the maximum error in the dissipated heat? ANS % error in heat dissipated is ± 6 %.

31. Name any three physical quantities having the same dimensions and also give their dimensions.  ANS Any group of physical quantities, like work, energy and torque and their dimensions [ML2 T-2]. 

32. A physical quantity X is given by X = A2B3/CD , If the percentage errors of measurement in A,B,C and D are 4%,2%,3% and 1% respectively, then calculate the % error in X. 

33. If two resistors of resistance R1=(4 ± 0.5) Ω and R2=(16 ± 0.5) Ω are connected (1) In series and (2) Parallel . Find the equivalent resistance in each case with limits of % error. 

34. The length of a rod measured in an experiment was found to be 2.48m, 2.46, 2.50m and 2.48m and 2.49m, Find the average length , the absolute error in each observation and % error. 

35. A calorie is a unit of heat energy and it equals about 4.2 J, where 1 J = 4.2 kgm2s-2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the units of time is ϒ sec. show that a calorie has a magnitude 4.2 α-1 β-2 ϒ2 in terms of the new units. 

RULES OF SIGNIFICANT FIGURES 

(1) All non-zero digits are significant.

(2) A zero becomes significant figure if it appears between to non-zero digits.

(3) Leading zeros or the zeros placed to the left of the number are never significant.

Example: 0.543 has three significant figures and 0.006 has one significant figures.

(4) Trailing zeros or the zeros placed to the right of the number are significant.

Example: 4.330 has four significant figures and 343.000 has six significant figures.

(5) In exponential notation, the numerical portion gives the number of significant figures. Example: 1.32 × 10–2 has three significant figures.

RULES OF ROUNDING OFF

(1) If the digit to be dropped is less than 5, then the preceding digit is left unchanged.

Example: x = 7.82 is rounded off to 7.8, again x = 3.94 is rounded off to 3.9.

(2) If the digit to be dropped is more than 5, then the preceding digit is raised by one.

Example: x = 6.87 is rounded off to 6.9, again x = 12.78 is rounded off to 12.8.

(3) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.

Example: x = 16.351 is rounded off to 16.4, again x = 6.758 is rounded off to 6.8.

(4) If digit to be dropped is 5 or 5 followed by zeros, then preceding digit is left unchanged, if it is even.

Example: x = 3.250 becomes 3.2 on rounding off, again x = 12.650 becomes 12.6 on rounding off.

(5) If digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one,

if it is odd. Example: x = 3.750 is rounded off to 3.8