A man walking briskly in rain with speed v must slant his umbrella forward making an angle  with the vertical. A student derives the following relation between θ and v: tan θ = v and checks that the relation has a correct limit : as v →0, θ → 0 as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man.) Do you think this relation can be correct?  If not, guess the correct relation. 


The relation tan straight theta = v is wrong because the dimensions on both the sides of the equation are not equal.

Dimension of right hand side is [MoL1T-1

Dimension of Left hand side quantity is [MoLoTo

Hence, the equation is wrong. 

To make the relation dimensionally correct, R.H.S should be divided by the speed of the rainfall. 

That is, 

tan space theta space equals space fraction numerator v over denominator v apostrophe end fraction, is a dimensionally correct relation. 
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The mass of a box measured by a grocerís balance is 2.300 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is
(a) the total mass of the box,
(b) the difference in the masses of the pieces to correct significant figures ? 


a) The total mass of the box is equal to the sum of the masses of box and constituents in the box.

i.e.       M = 2.3 + 0.02015 + 0.02017

               = 2.34032 kg 

Mass of the box is least accurate to one tenth of kg only. Therefore, the result must not be more accurate than one tenth of kg.


space space therefore Total mass of the box, M = 2.3 kg

b) Difference in masses of the pieces = 0.02 g 

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State the number of significant figures in the following : (a) 0.007 m (b) 2.64  kg (c) 0.2370 g cm-3 d) 6.320 J e)   6.032 N m-
f)  0.0006032 m2


The number of significant figures are: 

(a) One (b) Three (c) Four d) Three e) Four f) seven
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It is claimed that two cesium clocks, if allowed to run for 100 years, free from any disturbance, may differ by only about 0.02 s. What does this imply for the accuracy of the standard cesium clock in measuring a time-interval of 1 s ?


When the time interval of 1 sec is measured, the accuracy in measurement is given by, 

 

Therefore, accuracy is 1 part in 1011

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A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m2 sñ2. Suppose we employ a system of units in which the unit of mass equals αkg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2  in terms of the new units. 


We know that dimensional formula of heat energy is open square brackets straight M to the power of 1 straight L squared straight T to the power of negative 2 end exponent close square brackets

It is given that,

1 cal = 4.2J 

In SI system,   

M= 1 kg, L= 1 m,  T= 1 s 

In new system of units, 

straight M subscript 2 equals straight alpha space kg comma space straight L subscript 2 equals straight beta space straight m comma space straight T subscript 2 equals straight gamma space straight s

We  know,  

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space space space space straight n subscript 2 equals 4.2 open parentheses fraction numerator 1 space kg over denominator straight alpha space kg end fraction close parentheses to the power of 1 open parentheses fraction numerator 1 space straight m over denominator straight beta space straight m end fraction close parentheses squared open parentheses fraction numerator 1 space straight s over denominator straight gamma space straight s end fraction close parentheses to the power of negative 2 end exponent

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therefore space space space space space space space space 1 space cal space equals space equals 4.2 straight alpha to the power of negative 1 end exponent straight beta to the power of negative 2 end exponent straight gamma squared space new space unit 

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