Derive an experession for the period of oscillation of a mass attached to a spring executing simple harmonic motion. Given that the period of oscillation depends on the mass m and the force constant k, where k is the force required to stretch the spring by unit length?

Explanation

To derive the expression for the period of oscillation \( T \) of a mass \( m \) attached to a spring executing simple harmonic motion (SHM), we can follow these steps:

### Step 1: Understand the relationship between force, mass, and acceleration
In SHM, the restoring force \( F \) exerted by the spring is proportional to the displacement \( x \) from the equilibrium position. According to Hooke's Law, this can be expressed as:
\[ F = -kx \]
where \( k \) is the spring constant.

### Step 2: Apply Newton's second law
According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration \( a \):
\[ F = ma \]
In SHM, the acceleration \( a \) can also be expressed in terms of displacement \( x \) as:
\[ a = \frac{d^2x}{dt^2} = -\omega^2 x \]
where \( \omega \) is the angular frequency.

### Step 3: Set the equations equal
Since both expressions represent the same force, we can set them equal to each other:
\[ ma = -kx \]
Substituting for acceleration gives:
\[ m(-\omega^2 x) = -kx \]
This simplifies to:
\[ m\omega^2 x = kx \]

### Step 4: Cancel \( x \) (assuming \( x \neq 0 \))
We can divide both sides by \( x \) (assuming \( x \neq 0 \)):
\[ m\omega^2 = k \]

### Step 5: Solve for angular frequency \( \omega \)
Rearranging the equation gives us:
\[ \omega^2 = \frac{k}{m} \]
Taking the square root of both sides, we find:
\[ \omega = \sqrt{\frac{k}{m}} \]

### Step 6: Relate angular frequency to the period
The angular frequency \( \omega \) is related to the period \( T \) of the oscillation by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting our expression for \( \omega \) into this equation gives:
\[ \sqrt{\frac{k}{m}} = \frac{2\pi}{T} \]

### Step 7: Solve for the period \( T \)
Rearranging this equation to solve for \( T \) yields:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

### Final Expression
Thus, the expression for the period of oscillation \( T \) of a mass attached to a spring is:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

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