The Spring-Mass System: Why a Weight on a Spring Oscillates
Imagine hanging a weight from a spring and giving it a gentle tug downward. It bounces back up, overshoots, comes down again, and keeps going. That rhythmic up-and-down motion is simple harmonic motion, and the time it takes to complete one full bounce — down, up, and back to the start — is called the periodT.
The key question: what determines how fast or slow this bouncing happens? Intuition says two things matter: how heavy the weight is, and how stiff the spring is.
The Intuition
A heavier mass is harder to accelerate. It lumbers along, so the oscillation is slower — the period gets longer. A stiffer spring (one with a larger spring constant k) pulls back harder for the same stretch. That stronger restoring force whips the mass back faster, so the period gets shorter.
So the period T should increase with mass m and decrease with stiffness k. The exact relationship turns out to be:
T=2πkm
The 2π factor comes from the geometry of circular motion (which underlies all simple harmonic motion), and the square root tells us that doubling the mass only multiplies the period by 2≈1.4, not by 2.
Where Does This Formula Come From?
For a mass on an ideal spring, the restoring force is Hooke's law: F=−kx, where x is the displacement from equilibrium. Newton's second law F=ma gives:
mdt2d2x=−kx
This is a differential equation whose solution is a sine or cosine wave. The angular frequency ω (how fast the oscillation goes in radians per second) comes out as:
ω=mk
Since period T is the time for one complete cycle, and one cycle corresponds to 2π radians, we have T=2π/ω. Substituting ω gives the formula above.
Tip
If you ever forget which variable goes in the numerator, remember: mass in the numerator makes the period longer (heavier = slower), and stiffness in the denominator makes the period shorter (stiffer = faster). The square root just moderates the effect.
What the Formula Tells You
Mass and period: Double the mass → period increases by 2 (about 1.4 times). Quadruple the mass → period doubles.
Stiffness and period: Double the spring constant → period decreases by 2 (about 0.7 times). Quadruple the stiffness → period halves.
Independence from amplitude: The period does not depend on how far you pull the mass initially. A small bounce and a big bounce take exactly the same time. This is the hallmark of simple harmonic motion for an ideal spring. …
Step 1. Equilibrium. A massless spring of constant k, natural length L, hangs from a ceiling; attaching mass m stretches it by l to reach equilibrium, where the spring force F1=−kl balances the weight: F1+mg=0⇒mg=kl, so m/k=l/g.
Step 2. Further displacement. If the mass is pulled down a further small distance y and released, the total extension is (y+l), so the restoring force is F2=−k(y+l)=−ky−kl.
Step 3. Equation of motion. Newton's second law gives my¨=−ky−kl+mg; substituting mg=kl (Step 1) cancels the static terms, leaving my¨=−ky, i.e. y¨=−(k/m)y -- identical in form to the horizontal case. …