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A physical quantity P is related to four observables a, b, c and d as follows : 
straight P space equals space straight a cubed straight b squared space divided by space square root of straight c straight d
The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%, respectively. What is the percentage error in the quantity P ? If the value ofP calculated using the above relation turns out to be 3.763, to what value should you round off the result? 

Given space comma space

straight P space equals space fraction numerator straight a cubed straight b squared over denominator open parentheses square root of straight c space straight d close parentheses end fraction space

Maximum space fractional space error space in space straight p space is space given space by comma space

fraction numerator increment straight P over denominator straight P end fraction space equals plus-or-minus open square brackets 3 fraction numerator increment straight a over denominator straight a end fraction space plus 2 fraction numerator increment straight b over denominator straight b end fraction space plus 1 half fraction numerator increment straight c over denominator straight c end fraction plus fraction numerator increment straight d over denominator straight d end fraction close square brackets space

space space space space space space space space space space equals space plus-or-minus space open square brackets 3 1 over 100 space plus 2 3 over 100 space plus 1 half 4 over 100 plus 2 over 100 close square brackets

space space space space space space space space space space equals space plus-or-minus space 13 over 100 space equals space plus-or-minus 0.13 space

That space is comma space

Percentage space error space in space straight P space equals space fraction numerator increment straight P over denominator straight P end fraction cross times 100

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals plus-or-minus 0.13 space cross times space 100 space equals plus-or-minus 13 space percent sign

Value space of space straight P space equals space 3.763

On space rounding space off space the space given space value space to space the space first space place comma space

straight P space equals space 3.8 space
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A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion:
left parenthesis straight a right parenthesis space straight y space equals space straight a space sin space 2 straight pi space straight t divided by straight T space

left parenthesis straight b right parenthesis space straight y space equals space straight a space sin space vt

space left parenthesis straight c right parenthesis space straight y space equals space left parenthesis straight a divided by straight T right parenthesis space sin space straight t divided by straight a

straight d right parenthesis space straight y space equals space open parentheses straight a square root of 2 close parentheses space left parenthesis sin space 2 space πt divided by straight T space plus space cos space 2 πt space divided by straight T space right parenthesis 
(a = maximum displacement of the particle, v = speed of the particle, T = time-period
of motion).
Rule out the wrong formulas on dimensional grounds.




The displacement y has the dimension of length.

Therefore, the formula for it should also have the dimension of length.

Trigonometric functions are dimensionless and their arguments are also dimensionless.

Based on these considerations now checking each formula dimensionally, we have

straight a right parenthesis thin space Dimensionally space correct

straight y space equals space straight a space sin space fraction numerator 2 πt over denominator straight T end fraction
Dimension space of space straight y space equals space straight M to the power of straight o straight L to the power of 1 straight T to the power of straight o space space

Dimension space of space straight a space equals space straight M to the power of straight o straight L to the power of 1 straight T to the power of straight o space

Dimsension space of space sin space fraction numerator 2 πt over denominator straight T end fraction space equals space straight M to the power of straight o straight L to the power of straight o straight T to the power of straight o space
So comma space dimension space of space straight L. straight H. straight S space equals space Dimension space of space straight R. straight H. straight S

rightwards double arrow space given space formula space is space dimensionally space correct.

straight b right parenthesis thin space Incorrect space dimensionally.

straight y space equals space straight a space sin space vt space

Dimension space of space straight y space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space space

Dimension space of space straight a space equals space straight M to the power of 0 space end exponent straight L to the power of 1 space straight T to the power of 0 space space

Dimension space of space vt space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of – 1 end exponent space cross times space straight M to the power of 0 space end exponent straight L to the power of 0 space end exponent straight T to the power of 1 space end exponent equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space

The space argument space of space the space trigonometric space function space
must space be space dimensionless comma space which space is space not space so space in space this space case. space

Hence comma space given space formula space is space dimensionally space incorrect.

straight c right parenthesis space Incorrect

Dimension space of space straight y space equals space straight M to the power of 0 space space straight L to the power of 1 space space straight T to the power of 0

Dimension space of space begin inline style straight a over straight T end style equals space straight M to the power of 0 space end exponent space straight L to the power of 1 space space straight T to the power of – 1 end exponent space

Dimension space of space straight t over straight a equals space equals space straight M 0 space straight L to the power of – 1 space end exponent straight T to the power of 1 space
Argument space of space the space trignometric space function space is
space dimensionless. space Hence comma space the space formula space is space dimensionally space
incorrect.

straight d right parenthesis thin space Correct

Dimension space of space straight y space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0

Dimension space of space straight a space equals straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space

Dimension space of space straight t over straight T space equals space straight M to the power of 0 space end exponent straight L to the power of 0 space end exponent straight T to the power of 0 space
Since space the space srgument space of space the space trignometric space
function space is space dimensionless comma space the space dimensions
space of space straight y space and space straight a space are space the space same. space Hence comma space the
given space formula space is space dimensionally space correct. space


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One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratio so large?


Diameter space of space hydrogen space atom comma space straight D space equals space 0.5 space straight A with straight o on top space equals space 0.5 cross times 10 to the power of negative 10 space end exponent straight m

Volume comma space straight V space equals space 4 over 3 straight pi space open parentheses straight D over 2 close parentheses cubed space space

equals space 5.238 space cross times space 10 to the power of negative 31 end exponent space straight m cubed space

Ratio space of space molar space volume space to space volume space of space one space atom space is comma space

fraction numerator 22.4 over denominator 5.238 cross times 10 to the power of negative 28 end exponent end fraction space equals space 4.276 space cross times space 10 to the power of 28 space

Ratio space os space large space because semicolon space

straight i right parenthesis thin space No. space of space atoms space in space straight a space mlole space of space gas space is space large. space

ii right parenthesis thin space Distance space between space hydrogen space molecules space almost equal to space 10 space straight A with 0 on top
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A famous relation in physics relates ‘moving mass’ m to the ‘rest mass’ moof a  particle in terms of its speed v and the speed of light, c. (This relation first arose as
a consequence of special relativity due to Albert Einstein). A boy recalls the relation
almost correctly but forgets where to put the constant c. He writes :
straight m space equals space fraction numerator straight m subscript straight o over denominator left parenthesis square root of 1 minus straight v squared end root right parenthesis to the power of bevelled 1 half end exponent end fraction

Guess where to put the missing c.


Given the relation, 

straight m space equals space fraction numerator straight m subscript straight o over denominator left parenthesis square root of 1 minus straight v squared end root right parenthesis to the power of bevelled 1 half end exponent end fraction 

Dimension of m = M1 L0 T0

Dimension of mo = M1 L0 T

Dimension of v = M0 L1 T–1

Dimension of v2 = M0 L2 T–2

Dimension of c = M0 L1 T–1

The given formula will be dimensionally correct only when the dimension of L.H.S is the same as that of R.H.S.

This is possible when the factor in the denominator is dimensionless. This is only possible if straight nu squared space is space divided space by space straight c squared. 
Hence the correct relation is 

straight m space equals space straight m subscript straight o over open parentheses 1 minus begin display style straight v squared over straight c squared end style close parentheses to the power of bevelled 1 half end exponent

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The unit of length convenient on the atomic scale is known as an angstrom and is
denoted by Å: 1 Å = 10–10 m. The size of a hydrogen atom is about 0.5 Å. What is the
total atomic volume in m 3 of a mole of hydrogen atoms ?

Radius of hydrogen atom, r = 0.5 = 0.5 × 10–10 m

Volume space of space hydrogena space atom space equals space 4 over 3 πr cubed space

equals 4 over 3 cross times 22 over 7 cross times left parenthesis 0.5 cross times 10 to the power of negative 10 end exponent right parenthesis cubed space equals space 0.524 space cross times space 10 to the power of negative 30 end exponent space straight m cubed space

1 space mole space of space hydrogen space equals space 6.023 space cross times space 10 to the power of 23 space atoms space of space hydrogen

therefore space Volume space of space 1 space mole space of space hydrogen space atom

space equals space space 6.023 space cross times space 10 to the power of 23 space cross times 0.524 space cross times 10 to the power of negative 30 end exponent

equals space 3.16 space cross times space 10 to the power of negative 7 end exponent space straight m cubed

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