Newtonian mechanics

Gravitational and elastic potential energy

Potential energy is associated with position or deformation: near Earth U_g=mgh, and an ideal spring stores U_s=½kx².

A raised object can fall and do work; a compressed spring can launch an object. Energy associated with position in a force field or with deformation is potential energy. The reference level is chosen for convenience; changes in potential energy determine work.

U_g=mgh,\qquad U_s=\frac12kx^2

Definition: Gravitational and elastic potential energy

Near Earth's surface, choose a height reference $h=0$ ; a mass $m$ at height $h$ has $U_g=mgh$ . A spring of stiffness $k$ (N/m), deformed by $x$ from its natural length, stores $U_s=\frac12kx^2$ within its elastic range. Potential energy is measured in joules and depends on the reference; in the Hooke-law range the spring force is $F=-kx$ .

Raise a mass or deform a spring; observe potential energy converting through motion, relative to the chosen reference.

Potential energy and work

For conservative forces such as gravity or an ideal spring, the force's work is $W=-\Delta U$ . With friction and other nonconservative work absent, kinetic plus potential energy is constant. Potential energy can be negative under a different reference choice; its change, not its absolute zero, predicts work.

Example: Lifting an object

A 2 kg object is raised 3 m near Earth's surface; take $g=10\,m/s^2$ . Find its increase in gravitational potential energy.

Solution

$\Delta U_g=mg\Delta h=2\times10\times3=60\,J$ .

Gravitational potential energy depends on height relative to a chosen reference, not on the path used to raise an object when only gravity is considered. Raising a 5 kg backpack by 2 m near Earth with $g=10 m/s²$ increases potential energy by 100 J. If released without drag, that energy becomes kinetic. For a spring, doubling its deformation quadruples elastic potential energy because the formula contains $x²$ .

Example: A bag on a shelf

A 3 kg bag is raised 4 m. Take g=10 m/s². How much does its potential energy increase?

Solution

ΔU=m g Δh=3×10×4=120 J.

Quick check

If height doubles, what happens to gravitational potential energy change (m and g fixed)?

What elastic potential energy does an ideal spring have when deformed by x?

References

Young, Freedman (2019). University Physics