State the law of conservation of energy, apply it to mechanical systems by setting the energy before equal to the energy after, and account for energy transformed into thermal energy (MA STE Introductory Physics, Energy, HS-PS3-1, HS-PS3-2).

What this topic is asking

This is the central law of the Energy module of the Massachusetts Introductory Physics MCAS. You must state the law of conservation of energy, that energy is never created or destroyed, only transformed from one form to another, and apply it to mechanical systems by setting the total energy before an event equal to the total energy after. You also account for energy that becomes thermal energy, usually through friction. The crosscutting idea is energy and matter: in any system, the total amount of energy is constant; it just changes form.

The law of conservation of energy

This is one of the most important ideas in all of physics, and the MCAS expects you to state it and use it. Energy appears in many forms, kinetic, gravitational potential, elastic, thermal, electrical, chemical, light, and sound, and the law says that whatever happens, the total adds up to the same amount before and after. Energy is only ever moved around or changed in form.

Mechanical energy and the before-equals-after method

This single equation solves a large family of MCAS problems. The classic case is an object falling or sliding down a frictionless slope:

• At the top, it has potential energy and (often) no kinetic energy.
• At the bottom, it has kinetic energy and (often) no potential energy.
• Setting the potential energy lost equal to the kinetic energy gained gives the speed at the bottom: $mgh = \tfrac{1}{2}mv^2$.

A pendulum and a roller coaster work the same way: energy sloshes between potential at the high points and kinetic at the low points, with the total constant if friction is ignored.

Where energy goes when there is friction

The MCAS uses friction to test whether you really understand the law. If a cart reaches the bottom of a ramp with less kinetic energy than the potential energy it started with, the difference is exactly the energy transformed into thermal energy by friction. The total, mechanical plus thermal plus sound, is still equal to the starting energy. A good answer names the destination of the energy rather than saying it is "lost," which is the most common way students lose the mark.

Worked example

Reference-sheet note

The reference sheet gives the component formulas, $KE = \tfrac{1}{2}mv^2$ and $PE = mgh$, but it does not print the law of conservation of energy. The law itself is something you recall and then apply by setting the total energy before equal to the total energy after. When friction is involved, recall that the missing mechanical energy has become thermal energy and sound.

Try this

Q1. State the law of conservation of energy. [2]

• Cue. Energy cannot be created or destroyed; it can only be transformed from one form into another or transferred, so the total energy of a system stays constant.

Q2. A ball is dropped from $0.80$ m. Using conservation of energy, find its speed just before it lands (no air resistance, $g = 10$ m/s squared). [2]

• Cue. $mgh = \tfrac{1}{2}mv^2$, so $gh = \tfrac{1}{2}v^2$, giving $v^2 = 2gh = 2(10)(0.80) = 16$ and $v = 4.0$ m/s.