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 solve the problem of finding the volumetric strain produced in a spherical ball that contracts in radius by 2%, we can follow these steps:

### Step 1: Understand the relationship between radius and volume
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Determine the change in radius
The problem states that the radius contracts by 2%. This means:
\[ \Delta r = -0.02r \]
where \( \Delta r \) is the change in radius.

### Step 3: Calculate the change in volume
Using the formula for volume, we can express the new volume \( V' \) after the change in radius:
\[ V' = \frac{4}{3} \pi (r + \Delta r)^3 \]
Substituting \( \Delta r = -0.02r \):
\[ V' = \frac{4}{3} \pi (r - 0.02r)^3 = \frac{4}{3} \pi (0.98r)^3 \]
Calculating \( (0.98r)^3 \):
\[ (0.98)^3 = 0.941192 \]
Thus,
\[ V' = \frac{4}{3} \pi (0.941192r^3) \]

### Step 4: Find the change in volume
The change in volume \( \Delta V \) is given by:
\[ \Delta V = V' - V = \frac{4}{3} \pi (0.941192r^3) - \frac{4}{3} \pi r^3 \]
Factoring out \( \frac{4}{3} \pi \):
\[ \Delta V = \frac{4}{3} \pi (0.941192r^3 - r^3) = \frac{4}{3} \pi (-0.058808r^3) \]

### Step 5: Calculate the volumetric strain
Volumetric strain \( \epsilon_v \) is defined as the change in volume divided by the original volume:
\[ \epsilon_v = \frac{\Delta V}{V} = \frac{\frac{4}{3} \pi (-0.058808r^3)}{\frac{4}{3} \pi r^3} \]
The \( \frac{4}{3} \pi \) cancels out:
\[ \epsilon_v = -0.058808 \]
Since we are interested in the magnitude of the volumetric strain, we take the absolute value:
\[ \epsilon_v = 0.058808 \approx 0.0588 \]

### Step 6: Convert to percentage
To express this as a percentage:
\[ \epsilon_v \approx 5.88\% \]
In decimal form, this is:
\[ \epsilon_v \approx 0.0588 \]

### Final Answer
Thus, the volumetric strain produced in the ball is approximately:
\[ \epsilon_v = 6 \times 10^{-2} \text{ (unitless)} \]

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