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 48

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Earn 100

What are the base units in SI ? Define the supplementary units and the quantities they measure.

Important Questions on Units and Measurements

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 49

(i) Does a more accurate of the two values have greater precision too ? Explain with example.
(ii) What is the order of magnitude of a quantity ?

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 50

Discuss the various errors normally associated with a measurement, and also the ways to reduce them.

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 51

Define absolute, relative and percentage errors with examples.

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 52

What are significant figures ? Write the rules to round off insignificant digits in measured values.

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Gamma Question Bank for Medical Physics>Chapter 2 - Units and Measurements>Assignment>Q 53

A boy measures the diameter of a small metal ball using a vernier callipers. He takes the following five readings.

2.48 cm, 2.49 cm, 2.51 cm, 2.50 cm, 2.51 cm

Find its radius with the appropriate

%

error.

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

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 ?

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IMPORTANT

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 $P Q$ .