Q40 of 44 Page 9

If velocity of light c, Planck’s constant h and gravitational constant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

The dimensional formula for velocity of light c is [M0L1T-1], for plank’s constant h is [ML2T-1] and for gravitational constant G is [M-1L3T-2].


1. Let the dimensional formula for mass m = cahbGc, then substituting the dimensional formula for c, h and G we have





Equating the exponents of like quantities on both sides we have





Adding the three equations and solving for b we have



Using this value of b and substituting in the above equations, we have




Now, substituting these equations in the expression for mass m we have




2. Let the dimensional formula for length l = cahbGc, then substituting the dimensional formula for c, h and G we have





Equating the exponents of like quantities on both sides, we have





Adding the three equations and solving for b we have



Using this value of b and substituting in the above equations, we have




Now, substituting these equations in the expression for length l we have




3. Let the dimensional formula for time t = cahbGc, then substituting the dimensional formula for c, h and G we have





Equating the exponents of like quantities on both sides, we have





Adding the three equations and solving for b we have



Using this value of b and substituting in the above equations, we have




Now, substituting these equations in the expression for length l we have




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38

A physical quantity X is related to four measurable quantities a, b, c and d as follows:

X = a2 B3 c5/2 d-2


The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4%, respectively. What is the percentage error in quantity X ? If the value of X calculated on the basis of the above relation is 2.763, to what value should you round off the result.


 39

In the expression P = E l2 m-5 G-2, E, m, l and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity.

 41

An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s Third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that

T =


where k is a dimensionless constant and g is acceleration due to gravity.


 42

In an experiment to estimate the size of a molecule of oleic acid 1 mL of oleic acid is dissolved in 19 mL of alcohol. Then 1 mL of this solution is diluted to 20 mL by adding alcohol. Now 1 drop of this diluted solution is placed on water in a shallow trough. The solution spreads over the surface of water forming one molecule thick layer. Now, lycopodium powder is sprinkled evenly over the film and its diameter is measured. Knowing the volume of the drop and area of the film we can calculate the thickness of the film which will give us the size of oleic acid molecule.

Read the passage carefully and answer the following questions:


(a) Why do we dissolve oleic acid in alcohol?


(b) What is the role of lycopodium powder?


(c) What would be the volume of oleic acid in each mL of solution prepared?


(d) How will you calculate the volume of n drops of this solution of oleic acid?


(e) What will be the volume of oleic acid in one drop of this solution?