Period of a Mass-Spring Oscillator

For a mass $m$ attached to a spring with constant $k$, we can derive the period by comparing two expressions for acceleration.

From Hooke's law and Newton's second law, the spring force produces acceleration:

$$a = -\frac{k}{m}x$$

For any system undergoing simple harmonic motion, the acceleration relates to displacement by:

$$a = -\omega^2 x$$

Comparing these two expressions, we identify:

$$\omega^2 = \frac{k}{m}$$ $$\omega = \sqrt{\frac{k}{m}}$$

Since $T = \frac{2\pi}{\omega}$, the period equation is:

$$T = 2\pi\sqrt{\frac{m}{k}}$$

Larger mass means slower oscillation.

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