Energy stored in an inductor, $$U = \frac{1}{2}L{I^2}$$
$$\eqalign{
& \Rightarrow L = \frac{{2U}}{{{I^2}}} \cr
& \therefore \left[ L \right] = \frac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{{{\left[ A \right]}^2}}} \cr
& = \left[ {M{L^2}{T^{ - 2}}{A^{ - 2}}} \right] \cr} $$
122.
The moment of inertia of a body rotating about a given axis is $$6.0\,kg\,{m^2}$$ in the SI system. What is the value of the moment of inertia in a system of units in which the unit of length is $$5\,cm$$ and the unit of mass is $$10\,g$$ ?
123.
The velocity $$v$$ of a particle at time $$t$$ is given by $$v = at + \frac{b}{{t + c}},$$ where $$a,b$$ and $$c$$ are constants. The dimensions of $$a,b$$ and $$c$$ are respectively
A
$$\left[ {L{T^{ - 2}}} \right],\left[ L \right]\,{\text{and}}\,\left[ T \right]$$
B
$$\left[ {{L^2}} \right],\left[ T \right]\,{\text{and}}\,\left[ {L{T^2}} \right]$$
C
$$\left[ {L{T^2}} \right],\left[ {LT} \right]\,{\text{and}}\,\left[ L \right]$$
D
$$\left[ L \right],\left[ {LT} \right]\,{\text{and}}\,\left[ {{T^2}} \right]$$
Answer :
$$\left[ {L{T^{ - 2}}} \right],\left[ L \right]\,{\text{and}}\,\left[ T \right]$$
$$\eqalign{
& {\text{The given expression is }}v = at + \frac{b}{{t + c}} \cr
& {\text{From principle of homogeneity}} \cr
& \left[ a \right]\left[ t \right] = \left[ v \right] \cr
& \left[ a \right] = \frac{{\left[ v \right]}}{{\left[ t \right]}} = \frac{{\left[ {L{T^{ - 1}}} \right]}}{{\left[ T \right]}} = \left[ {L{T^{ - 2}}} \right] \cr
& {\text{Similarly, }}\left[ c \right] = \left[ t \right] = \left[ T \right] \cr
& {\text{Further,}}\,\frac{{\left[ b \right]}}{{\left[ {t + c} \right]}} = \left[ v \right] \cr
& {\text{or,}}\,\left[ b \right] = \left[ v \right]\left[ {t + c} \right] \cr
& {\text{or,}}\,\left[ b \right] = \left[ {L{T^{ - 1}}} \right]\left[ T \right] = \left[ L \right] \cr} $$ NOTE
If a physical quantity depends on more than three factors, then relation among them cannot be established, because we can have only three equations by equalising the powers of $$M, L$$ and $$T.$$
124.
A physical quantity $$\zeta $$ is calculated using the formula $$\zeta = \frac{{\frac{1}{{10}}x{y^2}}}{{{z^{\frac{1}{3}}}}},$$ where $$x,y$$ and $$z$$ are experimentally measured quantities. If the fractional error in the measurement of $$x,y$$ and $$z$$ are $$2\% ,1\% $$ and $$3\% $$ respectively, then the fractional error in $$\zeta $$ will be
125.
The dimensions of the quantity $$\vec E \times \vec B$$ where $${\vec E}$$ represents the electric field and $${\vec B}$$ the magnetic field may be given as:
A
$$\left[ {M{T^3}} \right]$$
B
$$\left[ {{M^2}L{T^{ - 5}}{A^{ - 2}}} \right]$$
C
$$\left[ {{M^2}{L^{ - 3}}{T^2}{A^{ - 1}}} \right]$$