Q. 39

# In the expression P = E l^{2} 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.

Answer :

Given:

Equation of P = E l^{2} m^{-5} G^{-2}

The dimensional equation of energy E is [M^{2}L^{1}T^{-2}], for mass m is [M^{1}L^{0}T^{0}], for length l is [M^{0}L^{1}T^{0}] and for gravitational constant G is [M^{-1}L^{3}T^{-2}].

Now to prove that P is a dimensionless quantity we have to find, the dimensional formula for P. For this, we substitute the dimensional formulae of the above quantities in the expression for P

Hence, P is a dimensionless quantity.

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PREVIOUSA physical quantity X is related to four measurable quantities a, b, c and d as follows:X = a2 B3 c5/2 d-2The 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.NEXTIf 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.

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