Topic 3.4 Conservation of Energy: apply conservation of mechanical energy for conservative systems, and extend the energy balance to include the work done by non-conservative forces.

What this topic is asking

The College Board (Topic 3.4) wants you to apply conservation of mechanical energy in systems where only conservative forces act, and to extend the energy balance to account for non-conservative forces such as friction that dissipate mechanical energy. Energy conservation is one of the most powerful problem-solving tools in the course, letting you connect speeds and positions at different points without tracking the detailed forces in between.

Conservation of mechanical energy

When gravity and springs are the only forces doing work, energy simply shifts between kinetic and potential forms while the total stays fixed. A ball dropped from height $h$ trades all its gravitational PE ($mgh$) for kinetic energy ($\tfrac{1}{2}mv^2$), giving $v = \sqrt{2gh}$, the same result as a frictionless ramp or a swinging pendulum through the same height. The strategy is to pick two instants, write the total energy at each, and set them equal. The mass often cancels, and you never need the acceleration or the time.

Including non-conservative forces

Real situations usually involve non-conservative forces, chiefly friction, which dissipate mechanical energy as heat. The energy balance then has an extra term:

where $W_{nc}$ is the work done by non-conservative forces. For friction acting over a distance $d$, this work is negative, $W_{friction} = -f_k d$, and represents mechanical energy lost from the system (it becomes thermal energy). An applied force can also do positive non-conservative work, adding energy. The accounting is the same: start with the initial mechanical energy, add the non-conservative work (signed), and you get the final mechanical energy.

Choosing a system and a reference

Two setup choices make energy problems clean. First, choose the system so that as many forces as possible become internal conservative forces; including the spring and the Earth in the system lets you use $U$ instead of tracking their work. Second, choose a reference level for potential energy (often the lowest point of the motion), since only changes in $U$ matter. With those fixed, you write the total mechanical energy at the start and end, include any friction or applied work, and solve. This approach shines in multi-stage problems, a ramp leading to a rough floor leading to a spring, where forces change but energy bookkeeping stays simple.

Try this

Q1. A $0.20$ kg ball is thrown straight up at $12$ m/s. Ignoring air resistance, calculate the maximum height ($g = 9.8$ m/s squared). [2 points]

• Cue. $\tfrac{1}{2}mv^2 = mgh \Rightarrow h = v^2/(2g) = 144/19.6 = 7.3$ m.

Q2. A $4.0$ kg crate slides $5.0$ m across a floor with $\mu_k = 0.20$ before stopping. Calculate the mechanical energy dissipated ($g = 9.8$ m/s squared). [2 points]

• Cue. $|W_{friction}| = \mu_k mg\,d = (0.20)(4.0)(9.8)(5.0) = 39$ J.