The Timpson Lyceum · University Physics I (2025)

Potential Energy and Energy Conservation – Lecture 7 (University Physics I)

Overview

This lecture introduces potential energy and the conservation of mechanical energy, building directly on the work–energy theorem from the previous chapter. It defines what makes a force conservative, derives the potential-energy functions for the two most important conservative forces — gravity and springs — and assembles the master energy-balance equation that handles non-conservative forces (like friction) as well. It closes with the deeper idea that total energy is always conserved (lost mechanical energy reappears as internal/thermal energy) and the calculus relationship that the force is the negative derivative of the potential-energy function. Mastering this material lets you solve a huge class of motion problems by bookkeeping energy at two points instead of tracking forces and accelerations the whole way.

Key Concepts

Conservative forces. A force is conservative if it has all four of these (equivalent) properties:

• The work it does moving an object from A to B is independent of the path — only the start and end points matter.
• The process is reversible: energy you put in fighting the force can be fully extracted again later.
• If the object returns to its starting point, the total work done by the force is zero.
• It has an associated potential-energy function $U$ that stores the energy.

Gravity and ideal springs are conservative; friction is non-conservative (slide a box out and back and friction does negative work both ways — it never returns the energy, it dumps it as heat).

Work, defined. Work done by a force along a path from A to B: $W = \int_A^B \vec{F}\cdot d\vec{s}$ Use the net force for total work, or a single force for the work done by that force alone.

Work–energy theorem. The total work done on an object equals its change in kinetic energy: $W_{\text{total}} = \tfrac{1}{2}mv_B^2 - \tfrac{1}{2}mv_A^2 = K_B - K_A$ with $K = \tfrac{1}{2}mv^2$.

Gravitational potential energy. Computing the work done by gravity along any path gives $W_{\text{grav}} = U_A - U_B$, where $U = mgy$ ($y$ = height; the textbook often writes $h$). Note the order: the work equals $U_A - U_B = -\Delta U$, the negative change in potential energy — when $U$ drops, $K$ rises.

Elastic (spring) potential energy. A spring obeys Hooke's law, $F = -kx$, where $x$ is the displacement from the spring's unstretched/equilibrium position (not where it started). Integrating the spring force gives $U = \tfrac{1}{2}kx^2$

Mechanical energy and its conservation. Mechanical energy is $E = U + K$. If only conservative forces act, mechanical energy is conserved: $U_1 + K_1 = U_2 + K_2$ This works for several conservative forces at once — just sum all the potential energies (gravity + spring + …).

The master equation (with non-conservative forces). In general, account for any other work done on the system: $U_1 + K_1 + W = U_2 + K_2$ Here $W$ is the work done by non-conservative/external forces — positive (a push, a batted ball) or negative (friction, air resistance). For friction, $W = -f_k \, d$, using the distance traveled $d$ (not displacement), since friction always opposes motion along the whole path.

Conservation of total energy. "Lost" mechanical energy is never destroyed — it becomes internal energy (atoms jiggling faster = higher temperature). Across any process $\Delta U + \Delta K + \Delta(\text{internal}) = 0$.

Force as the derivative of potential energy. Since $U$ is the negative integral of the force, the force is the negative derivative of $U$. In 1-D: $F_x = -\dfrac{dU}{dx}$. In 3-D, use partial derivatives (differentiate with respect to one variable while treating the others as constants): $\vec{F} = \left(-\frac{\partial U}{\partial x},\ -\frac{\partial U}{\partial y},\ -\frac{\partial U}{\partial z}\right)$ It is usually far easier to differentiate a single scalar function $U$ to get the force than to integrate a vector force to get $U$. Check: $U = mgy$ gives $\vec{F} = (0, -mg, 0)$; $U = \tfrac{1}{2}kx^2$ gives $F_x = -kx$.

Worked Example

Marble in a vertical loop. From what minimum height $h$ must a frictionless marble be released so it just completes a circular loop of radius $R$? At the top of the loop (point B, height $2R$) the minimum condition is that gravity alone supplies the centripetal acceleration: $\dfrac{v_B^2}{R} = g$, so $v_B^2 = gR$.

Apply energy conservation from release (at rest, height $h$) to the top of the loop, with gravity the only force:

The mass and $g$ cancel, so the required release height is $\tfrac{5}{2}R$, independent of mass.

Skateboarder down a quarter-circle ramp (radius $R = 3\text{ m}$, mass $25\text{ kg}$, frictionless), starting from rest. Find the speed at the bottom, with $g = 9.8\ \text{m/s}^2$:

At the bottom the normal force must also supply the centripetal acceleration, giving $N = mg + \dfrac{mv^2}{R} = 3mg = 3(25)(9.8) = 735\ \text{N}$ — three times his weight, independent of $R$.

Summary

• A conservative force is path-independent, reversible, does zero net work on a round trip, and has a potential-energy function; gravity and springs qualify, friction does not.
• The two key potential energies are gravitational $U = mgy$ and elastic $U = \tfrac{1}{2}kx^2$ (with $x$ measured from the spring's equilibrium point).
• When only conservative forces act, mechanical energy is conserved: $U_1 + K_1 = U_2 + K_2$.
• The general workhorse equation $U_1 + K_1 + W = U_2 + K_2$ folds in non-conservative work; friction contributes $W = -f_k\,d$ using distance, not displacement.
• Energy is never truly lost — missing mechanical energy shows up as internal (thermal) energy, so total energy is always conserved at everyday scales.
• Force and potential energy are linked by differentiation: $F_x = -\dfrac{dU}{dx}$ (and the partial-derivative version in three dimensions), which is usually the easier direction to compute.