A mass `M` is suspended from a spring of negiliglible mass the spring is pulled a little and then released so that the mass executes simple harmonic oscillation with a time period `T` If the mass is increases by `m` the time period because `((5)/(4)T)`,The ratio of `(m)/(M)` is

Options

A. `9//16`
B. `5//4`
C. `25//16`
D. `4//5`

Answer

To solve the problem, we need to analyze the time period of a mass-spring system before and after an additional mass is added.

### Step-by-Step Solution:

1. **Understanding the Time Period Formula**:
The time period \( T \) of a mass \( M \) suspended from a spring with spring constant \( K \) is given by the formula:
\[ T = 2\pi \sqrt{\frac{M}{K}} \]

2. **Initial Time Period**:
For the initial mass \( M \), the time period is:
\[ T = 2\pi \sqrt{\frac{M}{K}} \quad \text{(Equation 1)} \]

3. **New Mass After Adding \( m \)**:
When an additional mass \( m \) is added, the total mass becomes \( M + m \). The new time period \( T' \) is given as:
\[ T' = \frac{5}{4}T \]

4. **Time Period with New Mass**:
The time period with the new mass is:
\[ T' = 2\pi \sqrt{\frac{M + m}{K}} \quad \text{(Equation 2)} \]

5. **Setting Up the Equation**:
From the information given, we can equate the two expressions for \( T' \):
\[ \frac{5}{4}T = 2\pi \sqrt{\frac{M + m}{K}} \]

6. **Substituting \( T \) from Equation 1**:
Substitute \( T \) from Equation 1 into the equation:
\[ \frac{5}{4}(2\pi \sqrt{\frac{M}{K}}) = 2\pi \sqrt{\frac{M + m}{K}} \]

7. **Canceling Common Terms**:
We can cancel \( 2\pi \) from both sides:
\[ \frac{5}{4} \sqrt{\frac{M}{K}} = \sqrt{\frac{M + m}{K}} \]

8. **Squaring Both Sides**:
Squaring both sides gives:
\[ \left(\frac{5}{4}\right)^2 \frac{M}{K} = \frac{M + m}{K} \]
\[ \frac{25}{16} \frac{M}{K} = \frac{M + m}{K} \]

9. **Multiplying by \( K \)**:
Multiplying both sides by \( K \) (assuming \( K \neq 0 \)):
\[ \frac{25}{16} M = M + m \]

10. **Rearranging the Equation**:
Rearranging gives:
\[ m = \frac{25}{16} M - M \]
\[ m = \left(\frac{25}{16} - 1\right) M \]
\[ m = \left(\frac{25 - 16}{16}\right) M \]
\[ m = \frac{9}{16} M \]

11. **Finding the Ratio \( \frac{m}{M} \)**:
The ratio of \( \frac{m}{M} \) is:
\[ \frac{m}{M} = \frac{9}{16} \]

### Final Answer:
The ratio of \( \frac{m}{M} \) is \( \frac{9}{16} \).

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