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Introduction to Physics — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsIntroduction to Physics3 Marks
Question
The kinetic energy K of a rotating body depends on its moment of inertia I and its angular speed $\omega.$ Assuming the relation to be $\text{K}=\text{kI}^{\text{a}}\omega^{\text{b}}$ where k is a dimensionless constant, find a and b. Moment of inertia of a sphere about its diameter is $\frac{2}{5}\text{Mr}^2.$
✓

Answer

$\text{K = kI}^{\text{a}}\omega^{\text{b}}$ where k = Kinetic energy of rotating body and k = dimensionless constantDimensions of left side are,
$\text{K}=[\text{ML}^2\text{T}^{-2}]$
Dimensions of right side are,
$\text{I}^{\text{a}}=[\text{ML}^2]^{\text{a}},\omega^{\text{b}}=[\text{T}^{-1}]^{\text{b}}$
According to principle of homogeneity of dimension,
$[\text{ML}^2\text{T}^{-2}]=[\text{ML}^2\text{T}^{-2}][\text{T}^{-2}]^{\text{b}}$
Equating the dimension of both sides,
2 = 2a and -2 = -b ⇒ a = 1 and b = 2

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