Potential Energy and Conservation of Energy: Quick Revision

A brief note on Potential Energy and Conservation of Energy. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• A conservative force (gravity, a spring) does work that depends only on the start and the end. Friction and drag are not conservative.
• Potential energy: $W = -\Delta U$. Gravity near the Earth: $U = mgh$. A spring: $U = \tfrac12 kx^2$.
• Force from $U$: $F = -\dfrac{dU}{dx}$. A minimum of $U$ is a stable equilibrium.
• Conservation: with only conservative forces working, $K + U$ stays the same.
• With friction: $K_1 + U_1 = K_2 + U_2 + E_{\text{heat}}$.

The facts at a glance

• Fall from rest through $h$: $v = \sqrt{2gh}$, whatever the path
• Spring launcher: $v = x\sqrt{\dfrac{k}{m}}$
• Pendulum from $\theta_0$: $v = \sqrt{2gL(1 - \cos\theta_0)}$ at the bottom
• Loop of radius $R$: start at least $2.5R$ high
• String in a vertical circle: at least $\sqrt{5gL}$ at the bottom, $\sqrt{gL}$ at the top

A worked example

A $1.2\ \text{kg}$ ball is thrown straight down at $3.0\ \text{m/s}$ from a window $2.2\ \text{m}$ above the ground. How fast does it land?

1. Energy at the window. $\tfrac12 mu^2 + mgh$.
2. Energy at the ground. $\tfrac12 mv^2$.
3. Set them equal. $v = \sqrt{u^2 + 2gh} = \sqrt{9.0\ \text{m}^2/\text{s}^2 + 43.1\ \text{m}^2/\text{s}^2} = 7.22\ \text{m/s}$. The mass cancels, and the answer is the same if the ball is thrown up or sideways at $3.0\ \text{m/s}$.

Practice with new numbers

A program to try

The program follows a skateboarder down a ramp, with no losses and with $15\%$ lost to friction.

Mistakes to avoid

• Height above the ground instead of distance fallen. Use the drop between the two points.
• The value of $U$ as if it were absolute. Only changes in $U$ matter; zero can be anywhere.
• Forgetting the minus in $F = -\dfrac{dU}{dx}$.
• Energy conserved with friction. Mechanical energy is not; the heat must be counted.
• $2R$ as the least height for a loop. It is $2.5R$.

Check yourself

1. A $2.0\ \text{kg}$ stone falls from $15\ \text{m}$. What is its kinetic energy $5.0\ \text{m}$ above the ground? ($g = 9.8\ \text{m/s}^2$.)

2. $U(x) = 4x^2$ joules. What is the force at $x = -0.5\ \text{m}$?

3. A spring with $k = 900\ \text{N/m}$ is squeezed $4.0\ \text{cm}$ and launches a $0.10\ \text{kg}$ ball along a smooth table. How fast does it go?

4. A $50\ \text{kg}$ child slides down a $3.0\ \text{m}$ high slide and lands at $5.0\ \text{m/s}$. How much energy became heat? ($g = 9.8\ \text{m/s}^2$.)

5. What least speed at the top keeps a stone on a $0.60\ \text{m}$ string moving in a vertical circle? ($g = 9.8\ \text{m/s}^2$.)