Period of a Mass on a Spring

A mass on a spring obeys T = 2 π √(m/k), where m is the mass and k is the spring constant. The square root is not decoration. Force from the spring scales with k, and acceleration from that force scales with 1/m, so the restoring push and the object's resistance to being pushed both shape the motion together. That combined effect only shows up cleanly once you take a square root of the ratio m/k. Double the mass alone and the period does not double — it grows by √2, about 1.41 times. Quadruple the mass and the period exactly doubles, since √4 = 2.

Mass and spring constant sit on opposite sides of that ratio. More mass means more inertia, so the system swings slower and the period grows. A stiffer spring (larger k) pulls back harder, speeds up every cycle, and shrinks the period. Because both live under one square root, you can isolate their relationship by squaring the formula: T² = 4 π² (m/k). That equation is linear in m, so plotting T² against m gives a straight line through the origin, with slope 4 π²/k. That is exactly how a spring constant gets measured in a lab without ever touching gravity.

Notice what is missing from T = 2 π √(m/k): amplitude. However far you stretch or compress the spring, the period stays the same, which is the signature of simple harmonic motion and the idea the next topic on pendulums will echo in a different form.