# Conservation of Energy

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/conservation-of-energy

## Energy Accounting & Conservation

Conservation laws are physics’ accounting tricks: instead of following every push and pull through time, you compare two snapshots—energy before and energy after. In an **isolated system** where only **conservative forces** act, total mechanical energy is conserved: $\Delta E = \Delta KE + \Delta PE = 0$.

| Force | Potential Energy |
| --- | --- |
| Gravity (near surface) | $PE = mgh$ |
| Gravity (universal) | $PE = -\dfrac{GMm}{r}$ |
| Spring (Hooke's law) | $PE = \dfrac{1}{2}kx^{2}$ |

**Energy conservation equation**: $\frac{1}{2}mv_i^2 + PE_i = \frac{1}{2}mv_f^2 + PE_f$.

> **Problem-solving tip**: Energy methods bypass acceleration and time. If a problem asks for speed at a given position, try energy conservation before $F=ma$.

## Conservative vs Non-Conservative

A force is **conservative** if the work it does is **path-independent**, depending only on start and end points. Equivalently, the work around any closed loop is zero. Gravity and springs are conservative; friction and air resistance are not.

| Force Type | Properties & Behavior |
| --- | --- |
| **Conservative** | Path-independent work; zero net work in closed loop (e.g., Gravity, Springs). |
| **Non-Conservative** | Path-dependent work; energy leaves mechanical system (e.g., Friction). |

When **non-conservative forces** are present: $E_f = E_i + W_{\text{nc}}$, where $W_{\text{nc}} < 0$ for friction as energy turns to heat.

> **Common pitfall**: "Energy is conserved" does not mean "mechanical energy is conserved." Friction quietly converts $KE$ into thermal energy. Total energy survives, but mechanical books stop balancing when sliding friction appears.

## Practice questions

3 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Sort these forces into Conservative or Non-conservative.

**Answer:**

- Conservative: Gravity, Spring force, Electrostatic force
- Non-conservative: Kinetic friction, Air resistance, Tension (with deformation)

**Why:** Conservative forces have path-independent work; friction and drag do not.

Page: https://tryals.app/practice/physics-i/conservation-of-energy/sort-these-forces-into-conservative-or-non-conservative

### 2. A block sliding to a halt by friction seems to destroy energy. Why is energy still conserved?

A. Into sound, which carries away all of the kinetic energy
B. Into the potential energy of the surface’s deformation
C. It is genuinely destroyed, which is why friction is called non-conservative
D. Into thermal energy of the block and surface

**Answer:** D. Into thermal energy of the block and surface

**Why:** Friction converts organised kinetic energy into random molecular motion. Nothing is lost; what changes is the form, and that one-way conversion is where the second law enters.

Page: https://tryals.app/practice/physics-i/conservation-of-energy/a-block-sliding-to-a-halt-by-friction-seems-to-destroy-energy-why-is

### 3. A pendulum bob is pulled up until it is **20 cm above** its lowest point, then released. Estimate its speed at the bottom of the swing.

**Answer:** 2 (within ±0.3)

**Why:** $v = \sqrt{2 \times 9.8 \times 0.2} \approx 1.98$ m/s. Notice the string tension does no work, it always pulls perpendicular to the motion, so only gravity matters.

Page: https://tryals.app/practice/physics-i/conservation-of-energy/a-pendulum-bob-is-pulled-up-until-it-is-20-cm-above-its-lowest-point