A small metal ball of diameter `4 mm` and density `10.5 g//cm^(3)` in dropped in glycerine of density `1.5 g//cm^(3)`. The ball attains a terminal velocity of `8//cm s^(-1)`. The coefficient of viscosity of glycerine is

Options

A. `4.9` poise
B. `9.8` poise
C. `98` poise
D. `980` poise

Explanation

To find the coefficient of viscosity of glycerine, we can use the formula for terminal velocity of a sphere falling through a viscous fluid:

\[ v_t = \frac{2}{9} \cdot r^2 \cdot \frac{(ρ_b - ρ_f) \cdot g}{η} \]

Where:
- \(v_t\) = terminal velocity (8 cm/s)
- \(r\) = radius of the ball (in cm)
- \(ρ_b\) = density of the ball (10.5 g/cm³)
- \(ρ_f\) = density of the fluid (1.5 g/cm³)
- \(g\) = acceleration due to gravity (approximately 980 cm/s²)
- \(η\) = coefficient of viscosity (what we need to find)

### Step 1: Convert the diameter to radius
The diameter of the ball is given as 4 mm. To find the radius, we divide the diameter by 2.

\[ \text{Diameter} = 4 \text{ mm} = 0.4 \text{ cm} \\ r = \frac{0.4 \text{ cm}}{2} = 0.2 \text{ cm} \]

### Step 2: Substitute known values into the formula
Now we can substitute the known values into the terminal velocity formula and solve for \(η\).

\[ v_t = 8 \text{ cm/s} \\ r = 0.2 \text{ cm} \\ ρ_b = 10.5 \text{ g/cm}^3 \\ ρ_f = 1.5 \text{ g/cm}^3 \\ g = 980 \text{ cm/s}^2 \]

Substituting these into the equation:

\[ 8 = \frac{2}{9} \cdot (0.2)^2 \cdot \frac{(10.5 - 1.5) \cdot 980}{η} \]

### Step 3: Simplify the equation
First, calculate \( (0.2)^2 \):

\[ (0.2)^2 = 0.04 \]

Now, calculate \( (10.5 - 1.5) \):

\[ 10.5 - 1.5 = 9 \]

Now substitute these values back into the equation:

\[ 8 = \frac{2}{9} \cdot 0.04 \cdot \frac{9 \cdot 980}{η} \]

### Step 4: Solve for \(η\)
Multiply both sides by \(η\) to isolate it:

\[ 8η = \frac{2}{9} \cdot 0.04 \cdot 9 \cdot 980 \]

Now simplify the right side:

\[ 8η = 2 \cdot 0.04 \cdot 980 \]

Calculating \(2 \cdot 0.04 = 0.08\):

\[ 8η = 0.08 \cdot 980 \]

Now calculate \(0.08 \cdot 980\):

\[ 0.08 \cdot 980 = 78.4 \]

So we have:

\[ 8η = 78.4 \]

Now divide both sides by 8 to solve for \(η\):

\[ η = \frac{78.4}{8} = 9.8 \text{ poise} \]

### Final Answer
The coefficient of viscosity of glycerine is \(9.8 \text{ poise}\).

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