A physical quantity is given by the relation z = a b 2 c where a = ( 5 . 34 ± 0 . 01 ) g , b = ( 3 . 2 ± 0 . 1 ) cm and c = ( 0 . 42 ± 0 . 01 ) s . Find the relative error in z . Which of the three quantities should be measured most precisely and why ?

Given that z = a b 2 c and a = 5 . 34 ± 0 . 01 g b = 3 . 2 ± 0 . 1 cm c = 0 . 42 ± 0 . 01 s Relative error in z , Δ z z = Δ a a + 2 Δ b b + Δ c c Δ z z = 0 . 01 5 . 34 + 2 × 0 . 1 3 . 2 + 0 . 01 0 . 42 = . 00187 + . 062 + . 0238 = 0 . 0876 ≈ 0 . 09 The largest relative error is contributed in z by the quantity b (relative error 0 . 062 ). Hence, this quantity should be measured with greatest precision.

A physical quantity is given by the relation z = a b 2 c where a = ( 5 . 34 ± 0 . 01 ) g , b = ( 3 . 2 ± 0 . 1 ) cm and c = ( 0 . 42 ± 0 . 01 ) s . Find the relative error in z . Which of the three quantities should be measured most precisely and why ?

Given that z = a b 2 c and a = 5 . 34 ± 0 . 01 g b = 3 . 2 ± 0 . 1 cm c = 0 . 42 ± 0 . 01 s Relative error in z , Δ z z = Δ a a + 2 Δ b b + Δ c c Δ z z = 0 . 01 5 . 34 + 2 × 0 . 1 3 . 2 + 0 . 01 0 . 42 = . 00187 + . 062 + . 0238 = 0 . 0876 ≈ 0 . 09 The largest relative error is contributed in z by the quantity b (relative error 0 . 062 ). Hence, this quantity should be measured with greatest precision.

(i) What are the applications of dimensional analysis ? The position ( x ) of a particle depends on its velocity ( v ) and time ( t ) as given by the relation. x = P v + Q P + t . Find the dimensions of P Q . (ii) If kinetic energy K depends on velocity v , acceleration a and density $(\rho)$ of an object, find the expression for K .

(i) Application of Dimensional Analysis Checking the dimensional consistency of physical formulae and equation. Each of the terms appearing in a mathematical relation should have the same dimensions. If it is the not so the relation is proved wrong. This is called the principle of dimensional homogeneity. Deducing relation among various physical quantities. Given equation x = P v + Q P + t Since P is added to t , therefore [ P ] = [ t ] ⇒ [ P ] = T 1 ...(i) Also, all the terms in an equation should have the same dimensions ∴ Q P + t = [ x ] ⇒ [ Q ] [ T ] = [ L ] ⇒ [ Q ] = [ L T ] ...(ii) From (i) and (ii) ∴ [ PQ ] = T 1 [ LT ] = LT 2 (ii) Let the required expression be K ∝ v x ρ y a z [ K ] = LT - 1 x ML - 3 y LT - 2 z ML 2 T - 2 = M y L x - 3 y + z T - x - 2 z Equating the dimensions of like quantities on both sides. ⇒ y = 1 ⇒ x - 3 y + z = 2 ⇒ x + z = 5 - x - 2 z = - 2 ⇒ x + 2 z = 2 or z = - 3 and x = 8 ∴ K ∝ v 8 ρ a - 3 K ∝ v 8 ρ a 3

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 54

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A physical quantity is given by the relation $z = \frac{a b^{2}}{c}$ where $a = (5.34 \pm 0.01) g, b = (3.2 \pm 0.1) cm$ and $c = (0.42 \pm 0.01) s .$ Find the relative error in $z .$ Which of the three quantities should be measured most precisely and why ?

Important Questions on Units and Measurements

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 55

(i) What are the applications of dimensional analysis ? The position $(x)$ of a particle depends on its velocity ( $v$ ) and time $(t)$ as given by the relation. $x = P v + \frac{Q}{P + t} .$ Find the dimensions of $P Q$ .

(ii) If kinetic energy $K$ depends on velocity $v,$ acceleration $a$ and density $(\rho)$ of an object, find the expression for $K .$

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 56

What points do we conclude from dimensional analysis ? Explain the principle of homogeneity of dimensions.

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 57

(i) How many significant figures are there in the following quantities ?
(a) $10.163,$
(b) $1.67 \times 10^{- 17},$
(c) $0.270,$
(d) $1.496,$
(e) $15000,$
(f) $2.4300,$
(g) $0.001040$

(ii) Round off the following numbers to three significant figures.
(a) $3.264,$
(b) $0.9462,$
(c) $1.667,$
(d) $1.285,$
(e) $45.875$

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 58

Deduce the dimensional formulae of the following physical quantities.
(i) Specific heat,
(ii) Mechanical equivalent of heat,
(iii) Coefficient of viscosity.
(iv) Time period,
(v) Relative density,
(vi) Angular speed,
(vii)