A body of mass 0.1 g moving with a velocity of 10 m/s hits a spring (fixed at the other end) of force constant 1000 N/m and comes to rest after compressing the spring. The compression of the spring is

Options

A. 0.01 m
B. 0.1 m
C. 0.2 m
D. 0.5 m

Explanation

To solve the problem, we will use the principle of conservation of energy. The kinetic energy of the body will be converted into the elastic potential energy of the spring when the body comes to rest after compressing the spring.

### Step-by-Step Solution:

1. **Convert Mass to Kilograms**:
The mass of the body is given as 0.1 g. We need to convert this to kilograms because the SI unit of mass is kilograms.
\[ m = 0.1 \text{ g} = \frac{0.1}{1000} \text{ kg} = 0.0001 \text{ kg} \]

**Hint**: Remember to convert grams to kilograms by dividing by 1000.

2. **Calculate Initial Kinetic Energy**:
The kinetic energy (KE) of the body can be calculated using the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \times 0.0001 \text{ kg} \times (10 \text{ m/s})^2 = \frac{1}{2} \times 0.0001 \times 100 = 0.005 \text{ J} \]

**Hint**: Use the formula for kinetic energy and make sure to square the velocity.

3. **Set Up the Equation for Elastic Potential Energy**:
The elastic potential energy (EPE) stored in the spring when it is compressed by a distance \( x \) is given by:
\[ EPE = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant (1000 N/m). Since the body comes to rest, the kinetic energy is converted entirely into elastic potential energy:
\[ KE = EPE \]
Thus,
\[ 0.005 = \frac{1}{2} \times 1000 \times x^2 \]

**Hint**: Remember that the kinetic energy is equal to the potential energy stored in the spring at maximum compression.

4. **Solve for \( x^2 \)**:
Rearranging the equation gives:
\[ 0.005 = 500 x^2 \]
Dividing both sides by 500:
\[ x^2 = \frac{0.005}{500} = 0.00001 \]

**Hint**: Make sure to isolate \( x^2 \) by dividing both sides by the spring constant.

5. **Calculate \( x \)**:
Taking the square root of both sides:
\[ x = \sqrt{0.00001} = 0.003162 \text{ m} \]

**Hint**: When you take the square root, ensure you have the correct units.

6. **Convert to Centimeters** (if necessary):
To express the compression in centimeters:
\[ x = 0.003162 \text{ m} \times 100 = 0.3162 \text{ cm} \]

**Hint**: Remember that 1 m = 100 cm for unit conversion.

### Final Answer:
The compression of the spring is approximately \( 0.003162 \) m or \( 0.3162 \) cm.