The coefficient of viscosity of a liquid of `sigma = 1.0 CGS` units as determined by measuring the terminal velocity of a spherical solid ball of density `p= 1.5 CGS` units inside the liquid. If the ball of radius 1.00cm attain terminal velocity 1cm/s, then the viscosity (in CGS units) of the liquid as calculated from the observations of the experimental is: [Take `g=10m//s^(2)` ]

Options

A. `(100)/( 9)`
B. `(1000)/( 9)`
C. `(10)/( 9)`
D. `(1)/(9)`

Answer

To calculate the coefficient of viscosity (η) of the liquid using the given parameters, we can follow these steps:

### Step 1: Understand the formula for viscosity
The coefficient of viscosity (η) can be calculated using the formula:
\[ \eta = \frac{2}{9} \cdot r^2 \cdot \frac{(p - \sigma) \cdot g}{v} \]
where:
- \( r \) = radius of the ball
- \( p \) = density of the ball
- \( \sigma \) = density of the liquid
- \( g \) = acceleration due to gravity
- \( v \) = terminal velocity of the ball

### Step 2: Substitute the known values into the formula
Given:
- \( r = 1 \, \text{cm} \)
- \( p = 1.5 \, \text{g/cm}^3 \)
- \( \sigma = 1.0 \, \text{g/cm}^3 \)
- \( g = 10 \, \text{m/s}^2 = 1000 \, \text{cm/s}^2 \)
- \( v = 1 \, \text{cm/s} \)

Now substituting these values into the formula:
\[ \eta = \frac{2}{9} \cdot (1)^2 \cdot \frac{(1.5 - 1.0) \cdot 1000}{1} \]

### Step 3: Simplify the expression
Calculating \( (1.5 - 1.0) \):
\[ 1.5 - 1.0 = 0.5 \]
Now substituting this back into the equation:
\[ \eta = \frac{2}{9} \cdot 1 \cdot \frac{0.5 \cdot 1000}{1} \]
This simplifies to:
\[ \eta = \frac{2}{9} \cdot 0.5 \cdot 1000 \]

### Step 4: Calculate the final value
Calculating \( 0.5 \cdot 1000 \):
\[ 0.5 \cdot 1000 = 500 \]
Now substituting this value:
\[ \eta = \frac{2}{9} \cdot 500 \]
Calculating \( \frac{2 \cdot 500}{9} \):
\[ \eta = \frac{1000}{9} \]

### Step 5: Final result
Thus, the coefficient of viscosity of the liquid is:
\[ \eta = \frac{1000}{9} \, \text{CGS units} \]

### Summary
The viscosity of the liquid, as calculated from the observations of the experiment, is \( \frac{1000}{9} \, \text{CGS units} \).

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