Hooke’s law &
orbital motion

Hooke’s law says the restoring force of a spring is F = −kx. The stretch x is how far the mass sits from its rest point, k is the stiffness, and the minus sign means the force points back toward rest. Gravity pulls a planet toward the sun with F = GMm/r², and a sideways speed turns that fall into an ellipse.

In the world

An automotive suspension engineer chooses the stiffness k so a car settles a comfortable amount under the passengers and still presses the tires back onto the road after a bump. A product engineer sets the stiffness of a bathroom or kitchen scale so a known stretch matches a known weight. A satellite engineer sets a satellite’s height and speed together, because a larger orbit needs a longer period. GPS satellites orbit about 20,200 km above Earth and take about 12 hours to go around once.

Common mistakes

• The minus sign in F = −kx is a direction. It means the pull points back toward rest, not that the spring is weaker.
• A heavier mass does not change k. The same spring stretches farther at rest, Δ = mg/k, and it swings with a longer period.
• Speeding a planet up does not always make a bigger closed orbit. Past escape speed, the path does not come back.
• Kepler’s third law keeps T²/a³ the same for one sun. A longer period and a larger orbit change together, and the ratio does not.