Oscillations and SHM: Real-World Scenarios (8)

Question

A 200 g block hangs from a spring and stretches it by 5 cm at rest. The block is then pulled down 3 cm and released. Find the period of oscillation and the maximum speed. Take $g = 10\text{ m/s}^2$.

Solution — Step by Step

Final answer: $T \approx 0.44\text{ s}$, $v_{max} \approx 0.42\text{ m/s}$

Why This Works

The neat trick of the vertical spring: gravity just shifts the equilibrium position. Once we measure the oscillation from this new equilibrium, the SHM equations are identical to a horizontal spring. That is why $T = 2\pi\sqrt{m/k}$ has no $g$ in it, even though gravity is acting.

The static stretch $x_0 = mg/k$ is a free measurement of $k$ — no need for force gauges or any other setup.

Alternative Method

We can find $v_{max}$ via energy conservation: at the extreme, all energy is potential ($\frac{1}{2}kA^2$); at equilibrium, all is kinetic ($\frac{1}{2}mv_{max}^2$).

$v_{max} = A\sqrt{k/m}$

Same answer, sometimes faster if $\omega$ is messy.

Common Mistake