Units and Dimensions

PARAGRAPH FOR QUESTION NO. 10 TO 12 A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement. The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, may represent length in the x direction as $L_{x}$ , and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar or vector) must be dimensionally consistent. As an example, suppose wish to calculate the drift S of a swimmer crossing a river flowing with velocity $V_{x}$ and of width D and he is swimming in direction perpendicular to the river flow with velocity $V_{y}$ relative to river, assuming no use of directed lengths, the quantities of interest are then $V_{x}, V_{y}$ both dimensioned as $\frac{L}{T}$ , S the drift and D width of river both having dimension L. With these four quantities, may conclude that the equation for the drift S may be written: $S = V_{x}^{a} V_{y}^{b} D^{c}$ Or dimensionally $L = \left(\frac{L}{T}\right)^{a + b} \times (L)^{c}$ from which may deduce that $a + b + c = 1$ and $a + b = 0$ , which leaves one of these exponents undetermined. If, however, use directed length dimensions, then $V_{x}$ will be dimensioned as $\frac{L_{x}}{T}$ , $V_{y}$ as $\frac{L_{y}}{T}$ , S as $L_{x}$ and D as $L_{y}$ . The dimensional equation becomes: $L_{x} = \left(\frac{L_{x}}{T}\right)^{a} \left(\frac{L_{y}}{T}\right)^{b} \left(L_{y}\right)^{c}$ and may solve completely as a = 1, b = -1 and c = 1. The increase in deductive power gained by the use of directed length dimensions is apparent. From the concept of directed dimension what is the formula for a range (R) of a cannon ball when it is fired with vertical velocity component $V_{y}$ and a horizontal velocity component $V_{x}$ assuming it is fired on a flat surface. [Range also depends upon acceleration due to gravity, g and k is numerical constant]

PARAGRAPH FOR QUESTION NO. 10 TO 12 A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement. The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, may represent length in the x direction as $L_{x}$ , and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar or vector) must be dimensionally consistent. As an example, suppose wish to calculate the drift S of a swimmer crossing a river flowing with velocity $V_{x}$ and of width D and he is swimming in direction perpendicular to the river flow with velocity $V_{y}$ relative to river, assuming no use of directed lengths, the quantities of interest are then $V_{x}, V_{y}$ both dimensioned as $\frac{L}{T}$ , S the drift and D width of river both having dimension L. With these four quantities, may conclude that the equation for the drift S may be written: $S = V_{x}^{a} V_{y}^{b} D^{c}$ Or dimensionally $L = \left(\frac{L}{T}\right)^{a + b} \times (L)^{c}$ from which may deduce that $a + b + c = 1$ and $a + b = 0$ , which leaves one of these exponents undetermined. If, however, use directed length dimensions, then $V_{x}$ will be dimensioned as $\frac{L_{x}}{T}$ , $V_{y}$ as $\frac{L_{y}}{T}$ , S as $L_{x}$ and D as $L_{y}$ . The dimensional equation becomes: $L_{x} = \left(\frac{L_{x}}{T}\right)^{a} \left(\frac{L_{y}}{T}\right)^{b} \left(L_{y}\right)^{c}$ and may solve completely as a = 1, b = -1 and c = 1. The increase in deductive power gained by the use of directed length dimensions is apparent. A conveyor belt of width D is moving along x-axis with velocity V. A man moving with velocity U on the belt in the direction perpendicular to the belt's velocity with respect to belt wants to cross the belt. The correct expression for the drift (S) suffered by man is given by (k is numerical constant)

PARAGRAPH FOR QUESTION NO. 13 TO 15
L, M and T are units of length, Mass and Time respectively in a system of units.
Column-I
(I) $M = 1 0 0 \mathrm{gm}$
(II) $M = 10kg$
(III) $M = 10gm$
(IV) M = 1 tonne
Column-III
(P) $T = 0.1\mathrm{sec}.$
(Q) $T = 10ms$
(R) $T = 10$ sec
(S) $T = 0.01\mathrm{sec}.$ In which of the following combinations unit of force is $10^{6}$ dyne. [F = ma]

PARAGRAPH FOR QUESTION NO. 13 TO 15
L, M and T are units of length, Mass and Time respectively in a system of units.
Column-I
(I) $M = 1 0 0 \mathrm{gm}$
(II) $M = 10kg$
(III) $M = 10gm$
(IV) M = 1 tonne
Column-III
(P) $T = 0.1\mathrm{sec}.$
(Q) $T = 10ms$
(R) $T = 10$ sec
(S) $T = 0.01\mathrm{sec}.$ In which of the following system, unit of energy is $10^{9}$ erg? $[E = F \times S]$

PARAGRAPH FOR QUESTION NO. 13 TO 15
L, M and T are units of length, Mass and Time respectively in a system of units.
Column-I
(I) $M = 1 0 0 \mathrm{gm}$
(II) $M = 10kg$
(III) $M = 10gm$
(IV) M = 1 tonne
Column-III
(P) $T = 0.1\mathrm{sec}.$
(Q) $T = 10ms$
(R) $T = 10$ sec
(S) $T = 0.01\mathrm{sec}.$ In which of the following system, unit for coefficient of viscosity is 100 poiseuille? $[\eta = \frac{F}{6\pi rv}]$

In a new system of units known as RMP, length is mea sured in 'retem', mass is measured in 'marg' and time is measured in 'pal'. 100 retem = 1.0 meter 1.0 marg = $10^{-3}$ kilogram 10 pal = 1.0 second In the given table some unit conversion factors are given. Suggest suitable match.

A physical quantity $\vec{S}$ is defined as $\vec{S} = (\vec{E} \times \vec{B}) / \mu_{0}$ , where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and $\mu_{0}$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity/quantities?