A block of mass `m` is suspended from a spring fixed to the ceiling of a room. The spring is stretched slightly and released so that the block executes S.H.M. of frequency v. IF the mass is increased by `m'` the frequency becomes `3v//5`. If `(m')/(m) = (q)/(p)`, then find the value of `(q-p)` .

Explanation

To solve the problem step by step, we will use the concepts of simple harmonic motion (SHM) and the relationship between mass, spring constant, and frequency.

### Step 1: Understand the frequency of SHM
The frequency \( v \) of a mass \( m \) attached to a spring with spring constant \( k \) is given by the formula:
\[ v = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]
This is our first equation.

### Step 2: Write the frequency after increasing the mass
When the mass is increased by \( m' \), the new mass becomes \( m + m' \), and the new frequency is given as \( \frac{3v}{5} \). Thus, we can write:
\[ \frac{3v}{5} = \frac{1}{2\pi} \sqrt{\frac{k}{m + m'}} \]
This is our second equation.

### Step 3: Set up the equations
From the first equation, we can express \( v \) as:
\[ v = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]
Now, substituting this expression for \( v \) into the second equation gives:
\[ \frac{3}{5} \left(\frac{1}{2\pi} \sqrt{\frac{k}{m}}\right) = \frac{1}{2\pi} \sqrt{\frac{k}{m + m'}} \]

### Step 4: Simplify the equation
We can cancel \( \frac{1}{2\pi} \) from both sides:
\[ \frac{3}{5} \sqrt{\frac{k}{m}} = \sqrt{\frac{k}{m + m'}} \]
Now, squaring both sides results in:
\[ \left(\frac{3}{5}\right)^2 \frac{k}{m} = \frac{k}{m + m'} \]
This simplifies to:
\[ \frac{9}{25} \frac{k}{m} = \frac{k}{m + m'} \]

### Step 5: Cross-multiply to eliminate \( k \)
Cross-multiplying gives:
\[ 9(m + m') = 25m \]
Expanding this results in:
\[ 9m + 9m' = 25m \]
Rearranging gives:
\[ 9m' = 25m - 9m \]
Thus:
\[ 9m' = 16m \]

### Step 6: Find the ratio \( \frac{m'}{m} \)
Dividing both sides by \( 9m \) gives:
\[ \frac{m'}{m} = \frac{16}{9} \]

### Step 7: Relate \( \frac{m'}{m} \) to \( \frac{q}{p} \)
From the problem, we know:
\[ \frac{m'}{m} = \frac{q}{p} \]
Thus, we can identify \( q = 16 \) and \( p = 9 \).

### Step 8: Calculate \( q - p \)
Finally, we need to find \( q - p \):
\[ q - p = 16 - 9 = 7 \]

### Final Answer
The value of \( q - p \) is \( 7 \).

---