Potential Energy
Introduced| English |
|---|
| potential energy/pəˈtenʃl ˈenədʒi/ |
| gravitational potential energy/ˌɡrævɪˈteɪʃənl pəˈtenʃl ˈenədʒi/ |
| elastic potential energy/ɪˈlæstɪk pəˈtenʃl ˈenədʒi/ |
| conservative force/kənˈsɜːvətɪv fɔːs/ |
| stable equilibrium/ˈsteɪbl ˌiːkwɪˈlɪbrɪəm/ |
| unstable equilibrium/ʌnˈsteɪbl ˌiːkwɪˈlɪbrɪəm/ |
Energy waiting to happen
- Lift a book onto a shelf and it just sits there -- but it now holds hidden energy.
- Nudge it off and that stored energy becomes motion as it falls.
- Energy tied up in position is called potential energy.
- It is nature's rechargeable battery.
Potential energy
- Potential energy 势能 is energy stored in the arrangement of a system, not its motion.
- It depends only on where things are.
- Two common kinds power most of mechanics.
Gravitational and elastic
- Gravitational potential energy 引力势能 near the ground: $U_g = mgy$ -- lift higher, store more.
- Elastic potential energy 弹性势能 in a spring: $U_s = \tfrac{1}{2}kx^2$ -- stretch more, store more.
- Release either and it converts into kinetic energy.
Lift a $2\ \text{kg}$ book $1.5\ \text{m}$ ($g = 9.8$). How much gravitational potential energy is stored (in J)?
$U_g = mgy = 2 \times 9.8 \times 1.5 = 29.4\ \text{J}$.
A spring ($k = 200\ \tfrac{\text{N}}{\text{m}}$) is compressed $0.10\ \text{m}$. Elastic potential energy stored (in J)?
$U_s = \tfrac{1}{2}kx^2 = \tfrac{1}{2}(200)(0.10)^2 = 1\ \text{J}$.
Select all forms of potential energy covered here.
Gravitational and elastic are conservative-force energies. There is no frictional potential energy.
Force is the slope of energy
- A conservative force 保守力 (gravity, springs) is the negative slope of its potential energy:
- Steep energy hill → strong force. Flat → no force.
- This turns an energy graph into a force map.
Gravitational potential energy
Near the surface, gravitational potential energy is proportional to height.
Friction has an associated potential energy, just like gravity.
Only conservative forces have potential energy. Friction dissipates energy as heat -- it cannot be stored.
A conservative force equals the ____ slope of its potential energy, $F_x = -dU/dx$.
$F_x = -dU/dx$ -- the force points "downhill" on the energy graph.
Reading a potential-energy graph
- On a $U$-versus-position graph, valleys are points of stable equilibrium 稳定平衡 (a nudge brings it back).
- Hilltops are unstable equilibrium 不稳定平衡 (a nudge sends it away).
- An object speeds up rolling downhill on the graph and slows going up.
On a potential-energy graph, a valley (local minimum) is a point of...
At a valley a small nudge brings the object back -- stable equilibrium. A hilltop is unstable.
For a spring, $U_s = \tfrac{1}{2}kx^2$.
- The force is $F_x = -\dfrac{dU}{dx} = -kx$ -- Hooke's law falls right out of the derivative.
- The graph of $U_s$ is a parabola with its minimum at $x = 0$ -- a stable equilibrium.
Only conservative forces have a potential energy. Friction does not -- the energy it removes turns to heat and cannot be "stored" or recovered. So there is no "friction potential energy."
Potential energy is stored by position: gravitational $U_g = mgy$ and elastic $U_s = \tfrac{1}{2}kx^2$. A conservative force is the negative slope of its energy, $F_x = -dU/dx$. On a $U$-graph, valleys are stable equilibrium, hilltops unstable.