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$ F = A_0 ( 1 -e ^ { -Bxt^2} )$ where F is force and x is displacement. write the dimension formula of B
$ F = A ( 1 - e^{-Bxt^2 } )$ $ Bxt^2 = dimensional less $ $ B = { M^0 L^0 T^0 \over xt^2 } = M^0 L^{-1} T^{-2} $
Equation of physical quantity $ \nu = at + bt^2 $ where v = velocity t = time so write the dimensional formula of a in this equation
Density of substance in CGS system is $ 3.125 gm / cm^3$ what is its magnitude is SI system ?
$ Density = 3.125 \times { gm \over cm^3 } $ $ = { 3.125 \times 10 ^ {-3} kg \over 10 ^ {-6} m^3 } $ $ = 3125 kg / m^3 $
The resistivity of resistive wire is $ \rho = { AR \over L } $ where L = length of wire A = Area of wire and R is resistance of wire find dimension formula of $ \rho $
A cube has numerically equal volume and surface area calculate the volume of such a cube.
$$ volume of cube V = a^ 3$$ $$ total surface area of cube A = 6a^2$$ $ \therefore V = A $ $ a^3 = 6a^ 2$ $a = 6$ $ \therefore V = ( 6) ^3 = 216 unit $
Which out of the following is dimensionally correct.
If energy $ E = G^p h^q c^r $ where G is the universal gravitational constant. h is the plank’s constant and c is the velocity of light, then the values of p, q and r are respectively
$ E = G^p h^q c^r $ $ E = M^1 L^2 T^{-2} $ $ G = M^ {-1} L^3 T^{-2} $ $ h = M^1 L^2 T^{-1} $ $ c = M^0 L^1 T^{-1} $ take it $ ( M^1 L^2 T^{-2} ) = ( M^{-1} L^3 T^{-2} )^p = ( M^1 L^2 T^{-1} )^q (M^0 L^1 T^{-1} )^c $ $ = M ^ { -p +q } L^{3p + 2q+r} T^{-2p -q-r } $ $ \therefore P = {1 \over 2 } , q = {1 \over 2 } , r = { 5 \over 2 } $
If the centripetel force is of the form $m^av^br^c$ find the values of a, b and c
$ F \alpha m^a \nu^b r^c $ $ F = M^1 L^1 T^{-2} $ $ \nu = M^0 L^1 T^{-1} $ $ r = M^0 L^1 T^0 $ $ m = M^1 L^0 T^0 $ take it $ ( M^1 L^1 T^{-2} ) = (M^1)^a (L^1 T^{-1} )^b (L^1)^c $ $ = M^a L^{b+c } T^{-b} $ $ \therefore a =1 , b =2 , c = -1 $
Test if the following equation are dimensionally correct (S = surface tension $ \rho $ = density P = pressure v = volume n = coefficient of viscocity r = radius)
Match list - I with list - II List - I (1) Joule (2) Walt (3) volt (4) Resistivity List - II (a) $ henry \times ampere/sec$ (b) $coulomb \times volt $ (c) $ metre \times ohm$ (d) $ (ampere)^2 \times ohm $
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