1.4 Dimensional Analysis

Suppose we want the formula for the volume of a sphere. The two expressions commonly mentioned in elementary discussions of spheres are $4 \pi r^{2}$ and $4 \pi r^{3} / 3 .$ One is the volume of a sphere of radius r and the other is its surface area. Which one is the volume?

Is the equation v = at dimensionally consistent?

Consider the physical quantities s, v, a, and t with dimensions $\left[s\right] = \text{L} ,$ $\left[v\right] = \text{LT}^{−1} ,$ $\left[a\right] = \text{LT}^{−2} ,$ and $\left[t\right] = \text{T} .$ Determine whether each of the following equations is dimensionally consistent. (a) $v^{2} = 2 a s ;$ (b) $s = v t^{2} + 0.5 a t^{2} ;$ (c) $v = s / t ;$ (d) $a = v / t .$

Suppose quantity $s$ is a length and quantity $t$ is a time. Suppose the quantities $v$ and $a$ are defined by v = ds/dt and a = dv/dt. (a) What is the dimension of v? (b) What is the dimension of the quantity a? What are the dimensions of (c) $\int v d t ,$ (d) $\int a d t ,$ and (e) da/dt?

The arc length formula says the length $s$ of arc subtended by angle $Ɵ$ in a circle of radius $r$ is given by the equation $s = r Ɵ .$ What are the dimensions of (a) s, (b) r, and (c) $Ɵ?$