Identify Given Quantities and Units:
- Upward force due to surface tension ($F_T$) = 98 dyne
- Surface tension of water ($\sigma$) = $7 \times 10^{-2} \mathrm{Nm}^{-1}$
- We need to find the inner circumference ($L$) of the capillary.
Ensure Consistent Units:

The force is given in dyne (CGS unit), and surface tension is in $\mathrm{Nm}^{-1}$ (SI unit). It's best to convert the surface tension to CGS units (dyne/cm) for consistency.
- We know that $1 \mathrm{N} = 10^5 \mathrm{dyne}$ and $1 \mathrm{m} = 100 \mathrm{cm}$.
- So, $\sigma = 7 \times 10^{-2} \mathrm{Nm}^{-1} = 7 \times 10^{-2} \times \frac{10^5 \mathrm{dyne}}{100 \mathrm{cm}}$
- $\sigma = 7 \times 10^{-2} \times 10^3 \mathrm{dyne/cm}$
- $\sigma = 7 \times 10^1 \mathrm{dyne/cm}$
- $\sigma = 70 \mathrm{dyne/cm}$
Recall the Formula for Upward Force due to Surface Tension:

The upward force due to surface tension ($F_T$) acting along the inner circumference ($L$) of the capillary tube is given by:

$$F_T = \sigma \times L \times \cos\theta$$

where $\theta$ is the angle of contact between the water and the glass.
Consider the Angle of Contact:

For pure water in a clean glass capillary tube, the angle of contact ($\theta$) is approximately $0^\circ$.

Therefore, $\cos\theta = \cos(0^\circ) = 1$.
Simplify the Formula:

Substituting $\cos\theta = 1$ into the force equation:

$$F_T = \sigma \times L$$
Solve for the Inner Circumference ($L$):

We need to find $L$, so rearrange the formula:

$$L = \frac{F_T}{\sigma}$$
Substitute Values and Calculate:

Now, substitute the given force and the calculated surface tension:

$$L = \frac{98 \mathrm{dyne}}{70 \mathrm{dyne/cm}}$$

$$L = \frac{98}{70} \mathrm{cm}$$

$$L = \frac{14 \times 7}{10 \times 7} \mathrm{cm}$$

$$L = \frac{14}{10} \mathrm{cm}$$

$$L = 1.4 \mathrm{cm}$$

Therefore, the inner circumference of the capillary is $1.4 \mathrm{cm}$.

The final answer is $\boxed{\text{1.4 cm}}$.

Fluid Mechanics

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