A vertical spring carries a `5kg` body and is hanging in equilibrium an additional force is applied so that the spring is further stretched. When released from this position. It performs 50 complete oscillation in 25 s, with an amplitude of 5 cm. The additional force applied is

Options

A. 80 N
B. `80pi^2N`
C. `4pi^2N`
D. `4N`

Explanation

To solve the problem step by step, we will follow the concepts of simple harmonic motion and the properties of springs.

### Step 1: Calculate the Time Period (T)
The total time for 50 oscillations is given as 25 seconds. Therefore, we can calculate the time period (T) using the formula:

\[ T = \frac{\text{Total time}}{\text{Number of oscillations}} = \frac{25 \text{ s}}{50} = 0.5 \text{ s} \]

**Hint:** Remember that the time period is the time taken for one complete oscillation.

### Step 2: Calculate the Angular Frequency (ω)
The angular frequency (ω) can be calculated using the formula:

\[ \omega = \frac{2\pi}{T} \]

Substituting the value of T:

\[ \omega = \frac{2\pi}{0.5} = 4\pi \text{ rad/s} \]

**Hint:** Angular frequency relates to the time period and is measured in radians per second.

### Step 3: Calculate the Spring Constant (K)
The spring constant (K) can be calculated using the formula:

\[ K = \omega^2 \cdot m \]

Where m is the mass (5 kg):

\[ K = (4\pi)^2 \cdot 5 = 16\pi^2 \cdot 5 = 80\pi^2 \text{ N/m} \]

**Hint:** The spring constant relates the force exerted by the spring to the displacement from its equilibrium position.

### Step 4: Calculate the Additional Force (F)
The additional force applied to the spring can be calculated using Hooke's Law:

\[ F = K \cdot x \]

Where x is the amplitude of the oscillation (5 cm = 0.05 m):

\[ F = 80\pi^2 \cdot 0.05 = 4\pi^2 \text{ N} \]

**Hint:** Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.

### Final Answer
The additional force applied is:

\[ F = 4\pi^2 \text{ N} \]

### Summary of Steps
1. Calculate the time period (T).
2. Calculate the angular frequency (ω).
3. Calculate the spring constant (K).
4. Calculate the additional force (F) using Hooke's Law.