Topic 3.3 Potential Energy: define potential energy as stored energy of a system's configuration, and calculate gravitational potential energy (mgh) and elastic potential energy (1/2 kx^2).

What this topic is asking

The College Board (Topic 3.3) wants you to define potential energy as the energy stored in the configuration of a system, to calculate gravitational potential energy near Earth as $U_g = mgh$, to calculate elastic potential energy in a spring as $U_s = \tfrac{1}{2}kx^2$, and to understand that potential energy is associated with conservative forces and is measured relative to a chosen reference point.

What potential energy is

A raised book, a stretched spring, and two masses held apart all store potential energy: do work to set up the configuration and that energy is recoverable. Potential energy is associated only with conservative forces (gravity and ideal springs in this course), for which the work done depends only on the start and end positions, not the path taken.

Gravitational potential energy

The reference level (where $h = 0$) is yours to choose: the floor, a tabletop, or any convenient height. The absolute value of $U_g$ changes with that choice, but the change $\Delta U_g$ between two heights does not, which is why only differences in potential energy carry physical meaning. The work done by gravity as an object falls a height $h$ is $+mgh$, exactly the loss in gravitational potential energy.

Elastic potential energy

A stretched or compressed spring stores elastic potential energy:

where $k$ is the spring constant and $x$ is the displacement from the natural length. This formula comes straight from the work done against the spring force: because the spring force $F = kx$ grows linearly with displacement, the work to stretch it from $0$ to $x$ is the triangular area under the force-displacement graph, $\tfrac{1}{2}(kx)(x) = \tfrac{1}{2}kx^2$. Like kinetic energy, elastic potential energy depends on the square of the displacement, so doubling the stretch quadruples the stored energy, while the spring force only doubles. This is the energy counterpart to the spring force from Topic 2.8, and it is what drives the oscillation of a mass on a spring: energy trades back and forth between elastic potential energy at the extremes and kinetic energy at the center. The deeper point is that both potential-energy formulas are statements about stored work: lift an object and you store $mgh$; compress a spring and you store $\tfrac{1}{2}kx^2$. In each case the energy was supplied by a force acting through a displacement, and it can be recovered as kinetic energy when the object is released, which is exactly what the conservation of energy (Topic 3.4) formalises.

Try this

Q1. A $3.0$ kg mass is raised $4.0$ m. Calculate its gain in gravitational potential energy ($g = 9.8$ m/s squared). [2 points]

• Cue. $\Delta U_g = mg\,\Delta h = (3.0)(9.8)(4.0) = 117.6$ J (about $118$ J).

Q2. A spring of constant $300$ N/m is stretched $0.20$ m. Calculate the elastic potential energy stored. [2 points]

• Cue. $U_s = \tfrac{1}{2}kx^2 = \tfrac{1}{2}(300)(0.20)^2 = \tfrac{1}{2}(300)(0.04) = 6.0$ J.