If the same weight is suspended from three springs having length in the ratio 1 : 3 : 5 , the period of oscillations shall be the ratio of

Options

A. `1:3:5`
B. `1:sqrt(3):sqrt(5)`
C. `15 : 5:3`
D. `1: 1/sqrt(3) : 1/sqrt(15)`

Answer

To solve the problem of finding the ratio of the periods of oscillation of three springs with lengths in the ratio of 1:3:5, we can follow these steps:

### Step 1: Define the lengths of the springs
Let the lengths of the three springs be:
- \( L_1 = x \)
- \( L_2 = 3x \)
- \( L_3 = 5x \)

### Step 2: Understand the relationship between spring constant and length
The spring constant \( k \) is inversely proportional to the length of the spring. Therefore, we can express \( k \) as:
\[ k = \frac{C}{L} \]
where \( C \) is a constant.

### Step 3: Write the formula for the period of oscillation
The period \( T \) of a mass-spring system is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Substituting the expression for \( k \):
\[ T = 2\pi \sqrt{\frac{mL}{C}} \]

### Step 4: Calculate the periods for each spring
Using the lengths defined earlier, we can write the periods for each spring:
- For spring 1:
\[ T_1 = 2\pi \sqrt{\frac{mL_1}{C}} = 2\pi \sqrt{\frac{mx}{C}} \]
- For spring 2:
\[ T_2 = 2\pi \sqrt{\frac{mL_2}{C}} = 2\pi \sqrt{\frac{m(3x)}{C}} = 2\pi \sqrt{\frac{3mx}{C}} \]
- For spring 3:
\[ T_3 = 2\pi \sqrt{\frac{mL_3}{C}} = 2\pi \sqrt{\frac{m(5x)}{C}} = 2\pi \sqrt{\frac{5mx}{C}} \]

### Step 5: Find the ratio of the periods
Now we can find the ratio of the periods:
\[ \frac{T_1}{T_2} = \frac{2\pi \sqrt{\frac{mx}{C}}}{2\pi \sqrt{\frac{3mx}{C}}} = \frac{\sqrt{x}}{\sqrt{3x}} = \frac{1}{\sqrt{3}} \]
\[ \frac{T_2}{T_3} = \frac{2\pi \sqrt{\frac{3mx}{C}}}{2\pi \sqrt{\frac{5mx}{C}}} = \frac{\sqrt{3x}}{\sqrt{5x}} = \frac{\sqrt{3}}{\sqrt{5}} \]
Thus, the ratio of the periods \( T_1 : T_2 : T_3 \) can be expressed as:
\[ T_1 : T_2 : T_3 = 1 : \sqrt{3} : \sqrt{5} \]

### Final Answer
The ratio of the periods of oscillations of the three springs is:
\[ T_1 : T_2 : T_3 = 1 : \sqrt{3} : \sqrt{5} \]

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