`A` mass `M` is suspended from a spring of negligible mass. The spring is pulled a little then released, so that the mass executes simple harmonic motion of time period `T`. If the mass is increased by `m`, the time period becomes `(5T)/(3)`. Find the ratio of `m//M`.

Explanation

To solve the problem step by step, we will use the formula for the time period of a mass-spring system in simple harmonic motion (SHM).

### Step 1: Write the formula for the time period of SHM
The time period \( T \) of a mass-spring system is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( m \) is the mass attached to the spring and \( k \) is the spring constant.

### Step 2: Establish the initial condition
For the initial mass \( M \), the time period is given as \( T \):
\[ T = 2\pi \sqrt{\frac{M}{k}} \tag{1} \]

### Step 3: Establish the condition with increased mass
When the mass is increased by \( m \), the new mass becomes \( M + m \) and the new time period is given as \( \frac{5T}{3} \):
\[ \frac{5T}{3} = 2\pi \sqrt{\frac{M + m}{k}} \tag{2} \]

### Step 4: Square both equations
Now, we will square both equations (1) and (2) to eliminate the square root.

From equation (1):
\[ T^2 = (2\pi)^2 \frac{M}{k} \]
\[ T^2 = 4\pi^2 \frac{M}{k} \tag{3} \]

From equation (2):
\[ \left(\frac{5T}{3}\right)^2 = (2\pi)^2 \frac{M + m}{k} \]
\[ \frac{25T^2}{9} = 4\pi^2 \frac{M + m}{k} \tag{4} \]

### Step 5: Substitute equation (3) into equation (4)
Now we can substitute \( T^2 \) from equation (3) into equation (4):
\[ \frac{25}{9} \cdot 4\pi^2 \frac{M}{k} = 4\pi^2 \frac{M + m}{k} \]

### Step 6: Simplify the equation
We can cancel \( 4\pi^2 \) and \( k \) from both sides:
\[ \frac{25M}{9} = M + m \]

### Step 7: Rearrange to find \( m \)
Rearranging gives:
\[ m = \frac{25M}{9} - M \]
\[ m = \left(\frac{25}{9} - 1\right)M \]
\[ m = \left(\frac{25 - 9}{9}\right)M = \frac{16M}{9} \]

### Step 8: Find the ratio \( \frac{m}{M} \)
Now, we can find the ratio \( \frac{m}{M} \):
\[ \frac{m}{M} = \frac{16M/9}{M} = \frac{16}{9} \]

### Final Answer
Thus, the ratio of \( \frac{m}{M} \) is:
\[ \frac{m}{M} = \frac{16}{9} \]

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