Energy transfer and dissipation

Before you start

Place the slider high on the track and release it from rest. As it moves down, it gains speed. This connects the work used to raise it with its later motion. Work here means force times the distance moved along the force. We describe the first part as gravitational potential energy of the slider–Earth system and the second as kinetic energy. We can now follow their exchange and ask where energy goes when resistance acts.

As the cart runs downhill, it is easy to notice only its increasing speed. Stay with it on the uphill part: what happens to the height as it slows? Without resistance, the kinetic energy gained downhill can turn back into gravitational potential energy. The return height gives us a check on that account. With resistance, the cart fails to reach its original height, which gives us a reason to include the surroundings. The diagram’s $\Delta E_{\text{int}}$ records their gain in internal energy. Compare it with the cart’s height and speed before deciding that some energy has gone missing.

The setupA small slider moves on a curved track whose lowest point we take as zero height. Purple shows the energy stored because of height, $U$ ; blue shows the energy of motion, $K$ . When the damping rate $\gamma$ is above zero, faint red wisps mark energy passed to the surroundings, $Q$ . The sliders set the release height and the mass, and the readout adds the three energies to give the total, $\Sigma$ .

What to do and noticeRelease the slider with no damping first. Watch purple turn into blue on the way down and back into purple on the way up, and check that $\Sigma$ keeps the same value. Then raise the damping rate and release again. The slider no longer returns to its starting height, yet $\Sigma$ still stays the same once $Q$ is counted. Use the time slider to stop at any moment and add $U + K + Q$ yourself.

01 / 04 · Define the system

Predict firstWhy can the slider have energy before it starts moving?

Preparing the model…

Purple:

U

· Blue:

K

· Ochre: internal energy gained

Guided lesson

Account for energy conversion and transfer in one release.

Prerequisites: Speed describes motion; energy is measured in joules. Choose a system and a reference level before assigning potential energy. Force is measured in newtons (N). Near Earth, each kilogram is pulled down with about 9.81 N, written g. This page writes potential energy as U, kinetic energy as K, and energy passed to the surroundings as Q. Their total is Σ, and all are in joules (J).

01 / 04

Define the system

Predict firstWhy can the slider have energy before it starts moving?

First choose the system and zero level. We consider the block and Earth, taking gravitational potential energy as zero at the track’s lowest point. Potential energy is not a colour; it records energy available when the arrangement changes. To lift mass $m$ slowly through height $h$ , an upward force must support it. That force equals the weight $mg$ , where $g$ is about $9.81\ \mathrm{N/kg}$ . The work done in lifting is force times the vertical distance.

U(h)-U(0)=mgh\quad\Rightarrow\quad U=mgh

The curved track gives the same result: gravity is vertical, so work over each small part depends only on its vertical height change. Those changes add to the total height difference. $U(h)$ is the potential energy at height $h$ , and $U(0)$ is its value at the lowest point. We set the lowest point to zero, so $U(0)=0$ and $U=mgh$ . For $m=1\ \mathrm{kg}$ and $h_0=1.4\ \mathrm{m}$ , $U=1\times 9.81\ \mathrm{N/kg}\times 1.4\ \mathrm{m}=13.734\ \mathrm{J}$ . Before release the speed and kinetic energy are zero. Purple marks this system energy, not a substance inside the block.

How to observe

Release the slider and watch purple become blue
Add damping and follow energy into the ochre environment
Scrub time and check the complete energy balance

What this model simplifies

A point slider follows a fixed parabolic track whose height grows with the square of the horizontal distance; rotational energy is omitted. Gravitational potential energy belongs to the slider–Earth system and is zero at the lowest point. The drag along the track is proportional to mass and speed and opposes the motion.

SourcesOpenStax · Conservation of energy