A mass is suspended from a vertica spring which is executing SHM of frequency 5 Hz.
The spring is unstretched at the highest point of oscillation. Maximum speed of the mass is (take, acceleration due to gravity, `g=10m//s^(2)` )

Options

A. `2 pi ms^(-1)`
B. ` pi ms^(-1)`
C. ` 1/(2pi) ms^(-1)`
D. ` 1/(pi) ms^(-1)`

Answer

To find the maximum speed of a mass suspended from a vertical spring executing simple harmonic motion (SHM) with a frequency of 5 Hz, we can follow these steps:

### Step 1: Understand the relationship between frequency, spring constant, and mass
The frequency \( f \) of a mass-spring system in SHM is given by the formula:
\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]
where \( k \) is the spring constant and \( m \) is the mass.

### Step 2: Rearrange the formula to find \( \frac{k}{m} \)
From the frequency equation, we can rearrange it to find \( \frac{k}{m} \):
\[ \sqrt{\frac{k}{m}} = 2\pi f \]
Squaring both sides gives:
\[ \frac{k}{m} = (2\pi f)^2 \]

### Step 3: Substitute the given frequency
Substituting \( f = 5 \, \text{Hz} \):
\[ \frac{k}{m} = (2\pi \cdot 5)^2 = (10\pi)^2 = 100\pi^2 \]

### Step 4: Find the amplitude of the oscillation
At the highest point of oscillation, the spring is unstretched, and the maximum extension (amplitude \( A \)) can be expressed as:
\[ A = \frac{mg}{k} \]
Since \( k = \frac{100\pi^2 m}{1} \) from the previous step, we can substitute this into the amplitude equation:
\[ A = \frac{mg}{\frac{100\pi^2 m}{1}} = \frac{g}{100\pi^2} \]

### Step 5: Calculate the maximum speed
The maximum speed \( V_{\text{max}} \) in SHM is given by:
\[ V_{\text{max}} = \omega A \]
where \( \omega = 2\pi f \). We already know \( f = 5 \, \text{Hz} \), so:
\[ \omega = 2\pi \cdot 5 = 10\pi \]
Now substituting for \( A \):
\[ V_{\text{max}} = 10\pi \cdot \frac{g}{100\pi^2} \]
Substituting \( g = 10 \, \text{m/s}^2 \):
\[ V_{\text{max}} = 10\pi \cdot \frac{10}{100\pi^2} = \frac{100\pi}{100\pi^2} = \frac{1}{\pi} \]

### Final Answer
Thus, the maximum speed of the mass is:
\[ V_{\text{max}} = \frac{1}{\pi} \, \text{m/s} \]

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