The rate of a flow V a of liquid through a capillary under a constant pressure depends upon (i) the pressure gradient (P/l) (ii) coefficient of viscosity of the liquid `eta` (iii) the radius of the capillary tube r. Show dimesionally that the rate of volume of liquid flowing per sec V∝ Pr^4 /ηl

Explanation

To show dimensionally that the rate of volume of liquid flowing per second \( V \) is proportional to \( \frac{P r^4}{\eta l} \), we will analyze the dimensions of each variable involved in the relationship.

### Step-by-Step Solution:

1. **Identify the Variables and Their Dimensions**:
- **Rate of flow \( V \)**: This is the volume of liquid flowing per second. The dimension of volume is \( L^3 \) and time is \( T \). Therefore, the dimension of \( V \) is:
\[
[V] = L^3 T^{-1}
...