The pendulum of a clock is replaced by a spring mass system with the spring having spring constant `0.1Nm^-1`. What mass should be attached to the spring?

Explanation

To solve the problem, we need to determine the mass that should be attached to the spring in a spring-mass system, given that the spring constant \( k \) is \( 0.1 \, \text{N/m} \).

### Step-by-Step Solution:

1. **Understand the relationship between time period and mass**:
The time period \( T \) of a spring-mass system is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where:
- \( T \) is the time period,
- \( m \) is the mass attached to the spring,
- \( k \) is the spring constant.

2. **Rearranging the formula**:
To find the mass \( m \), we can rearrange the formula:
\[ m = \frac{T^2 k}{4\pi^2} \]
Here, we need to know the time period \( T \) of the pendulum to proceed.

3. **Assuming a time period**:
For a typical pendulum clock, the time period is often around \( 1 \, \text{s} \). We will use this value for our calculations. Thus, \( T = 1 \, \text{s} \).

4. **Substituting the values**:
Now we can substitute \( T = 1 \, \text{s} \) and \( k = 0.1 \, \text{N/m} \) into the rearranged formula:
\[ m = \frac{(1)^2 \cdot 0.1}{4\pi^2} \]

5. **Calculating the mass**:
First, calculate \( 4\pi^2 \):
\[ 4\pi^2 \approx 39.478 \]
Now substituting this value back into the equation:
\[ m = \frac{0.1}{39.478} \approx 0.00254 \, \text{kg} \]

6. **Final result**:
Therefore, the mass that should be attached to the spring is approximately:
\[ m \approx 0.00254 \, \text{kg} \text{ or } 2.54 \, \text{grams} \]