Q: A physical quantity P is related to four observations a, b, c and d as follows : P=a^3 b^2 / c d The percentage errors of measurement in a, b, c and d are 1 \%, 3 \%, 2 \% and 4 \% respectively. The p

aligned & P=a^3 b^2 c^ -1 d^ -1 / 2 \\ & ( P P 100 )= ( 3 a a + 2 b b + 1 c c + 1 2 3 d d ) 100 \\ & =3(1 \%)+2(3 \%)+1(2 \%)+ 1 2 (4 \%) \\ & =13 \% aligned

Q: A force defined by F= t^2+ t acts on a particle at a given time t. The factor which is dimensionless, if and are constants, is:

From principle of homogeneity aligned & [F]= [ t^2 ]=[ t] \\ & [ ]= [F] [t^2 ] and [ ]= [F] [t] \\ & [ ][t]=[ ] \\ & t = dimensionless aligned

Question 1

Consider that _s, k_B, b represent Stefan-Boltzmann constant, Boltzmann constant and Wien's displacement law constant, respectively. The dimension of _s k_B^ -1 b is :

Accepted Answer

The dimensional formula for the Stefan-Boltzmann constant _s is obtained from the relation E = _s A t T^4, where E is energy, A is area, t is time, and T is temperature: [ _s] = [ M L ^2 T ^ -2 ] [ L ^2][ T ][ K ^4] = [ M T ^ -3 K ^ -4 ] The dimensional formula for the Boltzmann constant k_B is obtained from E = k_B T: [k_B] = [ M L ^2 T ^ -2 ] [ K ] = [ M L ^2 T ^ -2 K ^ -1 ] The dimensional formula for Wien's displacement constant b is obtained from _ max T = b: [b] = [ L ][ K ] = [ L K ] Now,

Question 2

The speed of light in vacuum is taken as unity. If light takes 6 min 40 s to reach the Earth from the Sun, the distance between the Sun and the Earth in new unit is :

Accepted Answer

Speed of light in vacuum, c = 1 unit/s. Time taken by light to reach the Earth from the Sun, t = 6 min 40 s. Converting time into seconds: t = 6 60 + 40 = 360 + 40 = 400 s. Distance between the Sun and the Earth is given by: d = c t d = 1 400 = 400 new units. Answer: 400

Question 3

A physical quantity P is related to four observations a, b, c and d as follows : P=a^3 b^2 / c d The percentage errors of measurement in a, b, c and d are 1 \%, 3 \%, 2 \% and 4 \% respectively. The p

Accepted Answer

aligned & P=a^3 b^2 c^ -1 d^ -1 / 2 \ & ( P P 100 )= ( 3 a a + 2 b b + 1 c c + 1 2 3 d d ) 100 \ & =3(1 \%)+2(3 \%)+1(2 \%)+ 1 2 (4 \%) \ & =13 \% aligned

Question 4

The quantities which have the same dimensions as those of solid angle are:

Accepted Answer

Solid angle d = d A r^2 has dimensions [ M ^0 ~L ^0 ~T ^0 ] Strain = l l has dimensions [ M ^0 ~L ^0 ~T ^0 ] Angle measured in radians is also dimensionless [ M ^0 ~L ^ T ^0 ] = I r

Question 5

A force defined by F= t^2+ t acts on a particle at a given time t. The factor which is dimensionless, if and are constants, is:

Accepted Answer

From principle of homogeneity aligned & [F]= [ t^2 ]=[ t] \ & [ ]= [F] [t^2 ] and [ ]= [F] [t] \ & [ ][t]=[ ] \ & t = dimensionless aligned

Units and Dimensions — Previous Year Questions & Solutions