Spring constant of a spring is calculated using formule `K=(4pi^2M)/T^2`, where T is time period of vertical oscillation when mass M is hung with the help of spring to rigid support. If time of oscillation for 10 oscillations is measured to be 5.0 s and mass M=0.20 kg, find possible error in spring constant K.

Answer

To find the possible error in the spring constant \( K \), we will follow these steps:

### Step 1: Calculate the Time Period for One Oscillation
Given that the time for 10 oscillations is 5.0 seconds, we can find the time period \( T \) for one oscillation.

\[ T = \frac{\text{Total time for 10 oscillations}}{\text{Number of oscillations}} = \frac{5.0 \, \text{s}}{10} = 0.5 \, \text{s} \]

### Step 2: Substitute Values into the Spring Constant Formula
The spring constant \( K \) is given by the formula:

\[ K = \frac{4 \pi^2 M}{T^2} \]

Substituting \( M = 0.20 \, \text{kg} \) and \( T = 0.5 \, \text{s} \):

\[ K = \frac{4 \pi^2 (0.20)}{(0.5)^2} \]

### Step 3: Calculate \( K \)
Calculating \( K \):

\[ K = \frac{4 \times (3.142)^2 \times 0.20}{0.25} \]

Calculating \( \pi^2 \):

\[ \pi^2 \approx 9.87 \quad \Rightarrow \quad K = \frac{4 \times 9.87 \times 0.20}{0.25} \]

\[ K = \frac{7.896}{0.25} = 31.584 \, \text{N/m} \approx 31.54 \, \text{N/m} \]

### Step 4: Calculate the Percentage Error in \( K \)
To find the percentage error in \( K \), we need the errors in mass \( M \) and time \( T \).

Assuming:
- The least count for mass \( \Delta m = 0.01 \, \text{kg} \)
- The least count for time \( \Delta t = 0.1 \, \text{s} \)

The formula for percentage error in \( K \) is:

\[ \frac{\Delta K}{K} \times 100 = \frac{\Delta m}{M} + 2 \frac{\Delta t}{T} \times 100 \]

### Step 5: Substitute Values into the Error Formula
Substituting the values:

\[ \frac{\Delta K}{K} \times 100 = \frac{0.01}{0.20} + 2 \frac{0.1}{5.0} \times 100 \]

Calculating each term:

\[ \frac{0.01}{0.20} = 0.05 \quad \text{and} \quad 2 \frac{0.1}{5.0} = 0.04 \]

Adding these:

\[ \frac{\Delta K}{K} \times 100 = 0.05 + 0.04 = 0.09 \times 100 = 9\% \]

### Step 6: Calculate the Absolute Error in \( K \)
Now, we can find the absolute error \( \Delta K \):

\[ \Delta K = \frac{9}{100} \times K = 0.09 \times 31.54 \]

Calculating \( \Delta K \):

\[ \Delta K \approx 2.83 \, \text{N/m} \approx 2.9 \, \text{N/m} \]

### Final Answer
The possible error in the spring constant \( K \) is approximately \( 2.9 \, \text{N/m} \).

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