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Topic 8 of 20

Potential Energy & Conservation of Energy

Conservation of energy is one of the most powerful principles in all of physics. When only conservative forces act, total mechanical energy — the sum of kinetic and potential energy — remains constant, allowing you to relate speeds and heights without tracking the entire trajectory.

Key Concepts

Conservative Force

A force is conservative if the work it does is independent of path and depends only on start and end points. Gravity and spring force are conservative; friction is not. Conservative forces can be associated with a potential energy.

Gravitational Potential Energy

Energy stored due to height in a gravitational field:

U_g = mgh

, with

h

measured from a chosen reference level. Only differences in

U

matter — the reference level is arbitrary.

Elastic Potential Energy

Energy stored in a deformed spring:

U_s = \frac{1}{2}kx^2

, where

x

is the displacement from the natural length. Always non-negative.

Mechanical Energy

The sum of kinetic and potential energy:

E_\text{mech} = K + U

. For systems with only conservative forces,

E_\text{mech}

is conserved.

Conservation of Mechanical Energy

When only conservative forces do work:

K_i + U_i = K_f + U_f

. As an object falls,

U

decreases and

K

increases by the same amount.

Non-conservative Forces

Friction, drag, and other dissipative forces remove mechanical energy from the system, converting it to heat:

\Delta E_\text{mech} = W_\text{nc}

, where

W_\text{nc} < 0

for friction.

Key Equations

Gravitational PE

U_g = mgh

Valid near Earth's surface. h is height above the chosen reference point.

Elastic (spring) PE

U_s = \tfrac{1}{2}kx^2

x is displacement from the equilibrium (natural) length; k is the spring constant.

Conservation of mechanical energy

K_i + U_i = K_f + U_f

Valid when only conservative forces do work.

With non-conservative forces

K_i + U_i + W_\text{nc} = K_f + U_f

W_nc is work done by friction, drag, etc. (typically negative, reducing total mechanical energy).

Relation between force and PE

F_x = -\frac{dU}{dx}

Conservative force is the negative gradient of potential energy. Equilibrium where dU/dx = 0.

Worked Example

Roller Coaster Loop

Problem

A roller coaster car (mass 800 kg) starts from rest at height $h = 40$ m. Find its speed at the bottom (take $g = 10$ m/s²). Ignore friction.

Solution

Set reference level at the bottom ( $h_f = 0$ ). Apply conservation of mechanical energy:

K_i + U_i = K_f + U_f

0 + mgh = \tfrac{1}{2}mv_f^2 + 0

Solve for $v_f$ (mass cancels):

v_f = \sqrt{2gh} = \sqrt{2\times10\times40} = \sqrt{800} \approx 28.3 \text{ m/s}

Answer Speed at the bottom ≈ 28.3 m/s.

Practice

Exercises

7 problems

1 of 7

A $4 \text{ kg}$ ball is held $5 \text{ m}$ above the ground ( $g = 10 \text{ m/s}^2$ ). What is its gravitational PE relative to the ground?

Your answer

2 of 7

The same ball is released from $5 \text{ m}$ ( $g = 10 \text{ m/s}^2$ ). What is its speed just before hitting the ground?

Your answer

m/s

3 of 7

A spring with $k = 200 \text{ N/m}$ is compressed $0.30 \text{ m}$ . What is its elastic PE?

Your answer

4 of 7

A $2 \text{ kg}$ block is launched from rest by the compressed spring above ( $U_s = 9 \text{ J}$ ). What is its maximum speed?

Your answer

m/s

5 of 7

A $70 \text{ kg}$ skier starts from rest at height $h = 30 \text{ m}$ . What is their speed at the bottom? ( $g = 10 \text{ m/s}^2$ , ignore friction)

Your answer

m/s

6 of 7

A $2 \text{ kg}$ block slides $5 \text{ m}$ along a flat surface with $\mu_k = 0.30$ ( $g = 10 \text{ m/s}^2$ ). How much mechanical energy is lost to friction?

Your answer

7 of 7

A pendulum bob ( $0.5 \text{ kg}$ ) swings from rest at $0.8 \text{ m}$ above the lowest point ( $g = 10 \text{ m/s}^2$ ). What is its maximum speed at the bottom?

Your answer

m/s

Key Takeaways

Potential energy is defined only for conservative forces; friction has no associated potential energy.
The choice of reference level for $U_g = mgh$ is arbitrary — only differences in $U$ are physically meaningful.
Conservation of energy ( $K_i + U_i = K_f + U_f$ ) applies whenever only conservative forces act.
Friction converts mechanical energy to thermal energy: $\Delta E_\text{mech} = W_\text{friction} < 0$ .
Equilibrium points occur where $dU/dx = 0$ ; stable equilibrium is at a potential energy minimum.