A spherical ball contracts in radius by 2%, when subjected to a normal uniform force. The volumetric strain produced in ball is

Options

A. `6* 10^-2`
B. `2*10^-2`
C. `4*10^-2`
D. `10^-2`

Answer

To find the volumetric strain produced in a spherical ball that contracts in radius by 2%, we can follow these steps:

### Step 1: Define the Initial Radius
Let the initial radius of the spherical ball be denoted as \( R \).

### Step 2: Calculate the Final Radius
Since the radius contracts by 2%, the final radius \( r \) can be calculated as:
\[ r = R - 0.02R = 0.98R \]

### Step 3: Calculate the Initial Volume
The initial volume \( V_i \) of the spherical ball can be calculated using the formula for the volume of a sphere:
\[ V_i = \frac{4}{3} \pi R^3 \]

### Step 4: Calculate the Final Volume
The final volume \( V_f \) of the spherical ball after contraction can be calculated using the final radius:
\[ V_f = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (0.98R)^3 \]
Calculating \( (0.98R)^3 \):
\[ (0.98R)^3 = 0.98^3 R^3 \]
Thus,
\[ V_f = \frac{4}{3} \pi (0.98^3 R^3) \]

### Step 5: Calculate the Change in Volume
The change in volume \( \Delta V \) is given by:
\[ \Delta V = V_i - V_f = \frac{4}{3} \pi R^3 - \frac{4}{3} \pi (0.98^3 R^3) \]
Factoring out \( \frac{4}{3} \pi R^3 \):
\[ \Delta V = \frac{4}{3} \pi R^3 \left(1 - 0.98^3\right) \]

### Step 6: Calculate the Volumetric Strain
The volumetric strain \( \epsilon_v \) is defined as the change in volume divided by the initial volume:
\[ \epsilon_v = \frac{\Delta V}{V_i} = \frac{\frac{4}{3} \pi R^3 \left(1 - 0.98^3\right)}{\frac{4}{3} \pi R^3} \]
This simplifies to:
\[ \epsilon_v = 1 - 0.98^3 \]

### Step 7: Calculate \( 0.98^3 \)
Calculating \( 0.98^3 \):
\[ 0.98^3 \approx 0.941192 \]
So,
\[ \epsilon_v = 1 - 0.941192 \approx 0.058808 \]

### Step 8: Express in Scientific Notation
Converting \( 0.058808 \) into scientific notation gives:
\[ \epsilon_v \approx 5.88 \times 10^{-2} \]

### Step 9: Round to Appropriate Significant Figures
Rounding \( 5.88 \times 10^{-2} \) gives approximately \( 6 \times 10^{-2} \).

Thus, the volumetric strain produced in the ball is:
\[ \epsilon_v \approx 6 \times 10^{-2} \]

### Final Answer
The volumetric strain produced in the ball is \( 6 \times 10^{-2} \).

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