Topic 3.4 Conservation of Energy: apply conservation of mechanical energy to systems with conservative forces, and account for energy dissipated by nonconservative forces such as friction.

What this topic is asking

The College Board (Topic 3.4) wants you to apply conservation of energy: in a system with only conservative forces, the total mechanical energy (kinetic plus potential) stays constant, and energy simply trades between forms. When nonconservative forces such as friction act, they dissipate mechanical energy into thermal energy, and you account for that loss. Energy conservation lets you solve for speeds and heights without tracking the forces through time.

Conservation of mechanical energy

This is the workhorse of Unit 3. A falling object converts gravitational potential energy into kinetic energy; a mass on a spring trades elastic potential energy for kinetic energy and back. Because only the start and end states appear, you avoid solving the motion step by step.

The kinetic-potential trade

This trade-off is why the energy method is so powerful. You do not need to know the shape of the path, only the heights and speeds at the start and end. A roller coaster, a swinging pendulum and a ball rolling in a frictionless bowl are all the same problem: $K + U$ is constant.

When friction dissipates energy

Friction and other nonconservative forces convert mechanical energy into thermal energy, so mechanical energy is no longer conserved. You account for this with an energy balance:

where $E_{dissipated}$ is the magnitude of the (negative) work done by friction, equal to the friction force times the distance over which it acts, $f\,d$. The total energy of the wider system, including thermal energy, is still conserved; it is only the mechanical part that decreases. This is the bridge to the friction topic from Unit 2: the negative work done by kinetic friction is exactly the mechanical energy lost to heat.

Energy bookkeeping as a problem-solving tool

The practical skill this topic builds is energy bookkeeping: list the energy at the start, the energy at the end, and any energy dissipated, then set the total in equal to the total out. This converts many force-and-motion problems into a single algebraic equation. A block launched up a rough incline, a ball dropped onto a spring, a cart on a looping track: in each case you identify the kinetic and potential energies at two instants and the energy lost to friction in between, and solve. The method shines when the force varies or the path is complicated, because the kinematic equations would be hard to apply but energy only cares about the endpoints. The discipline is to choose a clear reference level for potential energy, decide whether the system is conservative, and not double-count: the work done by gravity is already captured in the potential-energy term, so do not also include it as a separate work. Master this and a large fraction of Unit 3 (and much of the exam) reduces to writing one energy equation and solving it.

Try this

Q1. A $1.0$ kg ball is dropped from rest at a height of $5.0$ m. Calculate its speed just before it lands ($g = 9.8$ m/s squared, no air resistance). [2 points]

• Cue. $mgh = \tfrac{1}{2}mv^2$, so $v = \sqrt{2gh} = \sqrt{2(9.8)(5.0)} = \sqrt{98} = 9.9$ m/s.

Q2. A block with $40$ J of kinetic energy slides to a stop over $4.0$ m on a rough floor. Calculate the friction force. [2 points]

• Cue. $E_{dissipated} = f\,d$, so $f = 40/4.0 = 10$ N.