A mass $$m = 1.0$$ kg is put on a flat pan attached to a vertical spring fixed on the ground. The mass of the spring and the pan is negligible. When pressed slightly and released, the mass executes simple harmonic motion. The spring constant is 500 N/m. What is the amplitude A of the motion, so that the mass m tends to get detached from the pan? (Take $$g = 10$$ m/s$$^2$$). The spring is stiff enough so that it does not get distorted during the motion.

The mass $$m$$ tends to get detached from the pan when the normal force between them becomes zero. In simple harmonic motion (SHM), this occurs at the highest point of the oscillation where the downward acceleration of the pan exceeds the acceleration due to gravity $$g$$.

$$a_{max} = \omega^2 A$$

$$a_{max} \geq g \implies \omega^2 A \geq g$$

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

$$\omega^2 = \frac{500}{1.0} = 500 \text{ rad}^2/\text{s}^2$$

$$500 \times A \geq 10$$

$$A \geq \frac{10}{500} \text{ m}$$

$$A \geq 0.02 \text{ m} = 2.0 \text{ cm}$$