Question: 27. The quantities which have the same dimensions as those of solid angle are :
(1) strain and angle
(2) stress and angle
(3) strain and arc
(4) angular speed and stress
Answer: Option (1)
Explanation:
Solid angle is defined as the ratio of area subtended on a sphere to the square of the radius.
\Omega = \frac{\text{Area}}{r^{2}}Since both area and r^{2} have dimensions of L^{2},
the solid angle is dimensionless.
Strain is defined as the ratio of change in length to original length.
\text{Strain} = \frac{\Delta l}{l}Hence, strain is also dimensionless.
Plane angle is defined as the ratio of arc length to radius.
\theta = \frac{s}{r}Since both arc length and radius have the same dimension L,
angle is dimensionless.
Therefore, strain and angle have the same dimensions as solid angle.
Stress has dimensions of force per unit area and angular speed has dimensions of T^{-1},
so they are not dimensionless.
Hence, the correct answer is Option (1).