Interactive module First-year engineering dynamics

Work and Energy

How fast is a roller coaster going at the bottom of a drop? How far does a car skid, how hard does a spring have to push to stop a crate, how much power does a hoist need? Energy methods answer these with one scalar equation instead of an integration of \(\sum\Fvec = m\avec\). Eight short lessons, with animated figures and questions that check your work, take you from the work of a force to the conservation of energy and the forces on curved paths.

Start Lesson 1
  • 8 lessons
  • 260 min of lessons
  • Works offline
A cart runs down a smooth ramp from three times the loop's radius and round the loop. Watch the bars: kinetic energy \(\colKE{T}\) and potential energy \(\colVg{V_g}\) trade places, and their stacked total never moves. Lesson 8 finds the force of the track at the top and bottom.

Learning outcomes

By the end of the module you can:

  • Explain how integrating \(\sum F_t = m a_t\) along the path relates speed to position, and choose energy methods for problems that link the two. (Lesson 1)
  • Calculate the work of a force, as \(\int\Fvec\cdot d\rvec\) and as the area under its force–displacement graph, and of the weight, a spring and kinetic friction. (Lessons 2–3)
  • Apply the principle of work and energy, \(T_1 + \sum U_{1\to2} = T_2\), to a particle and to bodies joined by cords, to find speeds, distances, spring compressions and forces. (Lesson 4)
  • Calculate power, \(P = \Fvec\cdot\vvec\), and mechanical efficiency for hoists, vehicles and machines in series. (Lesson 5)
  • Identify conservative forces, write the potential energies \(V_g = Wy\) and \(V_e = \tfrac12 ks^2\), and read potential energy diagrams. (Lesson 6)
  • Apply conservation of energy, \(T_1 + V_1 = T_2 + V_2\), and its general form with the work of friction and applied forces. (Lesson 7)
  • Combine energy with \(\sum F_n = mv^2/\rho\) to find normal forces and tensions on curved paths, and where a particle leaves a surface. (Lesson 8)

How to use this module

Before you start: you should be comfortable with free-body diagrams and \(\sum\Fvec = m\avec\), the tangential and normal components of acceleration (\(a_t = v\,dv/ds\), \(a_n = v^2/\rho\)), the dot product of two vectors, and integrals of simple polynomials. The module uses the notation of Hibbeler's Engineering Mechanics: Dynamics (\(T\), \(U_{1\text{-}2}\), \(V_g\), \(V_e\), \(\varepsilon\)); Beer & Johnston and Meriam & Kraige use the same ideas with small differences in notation, pointed out where they occur.

Work through the lessons in order. The times allow for working the examples on paper and doing the checks. A suggested plan in four sittings:

  1. Lessons 1–385 min

    Why energy methods, the work of a force, and the work of the weight, springs and friction.

  2. Lessons 4–570 min

    The principle of work and energy, for one particle and for connected bodies, then power and efficiency.

  3. Lessons 6–8105 min

    Potential energy, conservation of energy with and without friction, and forces on curved paths.

  4. Practice Lab and Self-Check Quiz65 min

    About 30 min of practice on the topics you found hardest, then the 35-minute quiz.

About 5½ h in total:260 min of lessons, 30 min of practice, 35 min for the quiz

Get the most out of it

Lessons

Each lesson has interactive figures, worked examples and questions with instant feedback.

Practice, tools and reference

Use these alongside the lessons, or on their own when you revise.

Requirements

Any modern browser. Works offline.

  • A current version of Chrome, Edge, Firefox or Safari, with JavaScript on. A laptop or desktop screen works best; the pages also fit tablets and phones.
  • No internet connection, installation or account. Everything is inside this folder.
  • Your progress is saved in this browser on this device. Private or incognito windows do not keep it.

Something not working? Read README.txt in the module folder for how to open the module and fix common problems.