In a spring block system if length of the spring is reduced by `1%`, then time period

Options

A. increase by `2%`
B. increase by `0.5%`
C. decrease by `2%`
D. decrease by `0.5%`

Explanation

To solve the problem of how the time period of a spring block system changes when the length of the spring is reduced by 1%, we can follow these steps:

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

### Step 2: Determine the effect of spring length on spring constant
The spring constant \( k \) is inversely proportional to the length of the spring \( l \). This means:
\[ k \propto \frac{1}{l} \]
If the length \( l \) decreases, the spring constant \( k \) increases.

### Step 3: Calculate the change in length
Given that the length of the spring is reduced by 1%, we can express this mathematically as:
\[ \frac{dl}{l} = -0.01 \]
This indicates that the length is decreasing.

### Step 4: Relate the change in spring constant to the change in length
Since \( k \) is inversely proportional to \( l \), we can express the change in \( k \) as:
\[ \frac{dk}{k} = -\frac{dl}{l} \]
Substituting the value of \( \frac{dl}{l} \):
\[ \frac{dk}{k} = 0.01 \]
This indicates that \( k \) increases by 1%.

### Step 5: Differentiate the time period formula
To find the change in the time period \( T \), we differentiate the time period equation:
\[ dT = \frac{d}{dk} \left( 2\pi \sqrt{\frac{m}{k}} \right) \]
Using the chain rule, we find:
\[ dT = -\frac{\pi m}{k^{3/2}} dk \]

### Step 6: Relate the change in time period to the change in spring constant
Now, we can express the relative change in time period:
\[ \frac{dT}{T} = -\frac{1}{2} \frac{dk}{k} \]

### Step 7: Substitute the change in spring constant into the time period equation
Substituting \( \frac{dk}{k} = 0.01 \):
\[ \frac{dT}{T} = -\frac{1}{2} \times 0.01 = -0.005 \]

### Step 8: Convert to percentage change
To find the percentage change in time period:
\[ \frac{dT}{T} \times 100 = -0.5\% \]
This indicates that if the length of the spring decreases by 1%, the time period decreases by 0.5%.

### Final Answer
If the length of the spring is reduced by 1%, then the time period decreases by 0.5%.

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