Here are questions that often come up while learning this chapter — with short, clear answers. Questions are ordered from foundational to advanced. If you have a question not listed, reach us via the contact page.
Potential energy: the basics
1. What exactly is potential energy?
Potential energy is energy stored due to an object's position. A book on a high shelf has gravitational PE — when it falls, this energy converts to kinetic energy. A compressed spring has elastic PE — when released, this energy converts to motion. Potential energy is a function of position and is only defined for conservative forces.
2. Why is gravitational PE \( -GMm/r \), not positive?
Convention on where to set the zero point. Potential energy is always defined relative to a reference point. For gravity, we choose the zero at infinity. Any finite distance \( r \) is therefore below this reference, giving \( PE < 0 \). This negative energy is a sign that the object is bound — trapped by gravity. To completely escape, you'd need to add energy to reach \( U = 0 \) (at infinity).
3. Why are \( U = mgh \) and \( U = -GMm/r \) two different formulas?
One is an approximation; one is exact. The formula \( mgh \) is valid for small heights (near Earth's surface). For \( h \ll R_E \) (height small compared to Earth's radius), you can use \( mgh \) — simple and accurate. For large distances (satellites, planets), you must use the exact formula \( -GMm/r \). The key difference: \( mgh \) is linear (uniform field), while \( -1/r \) is curved (varying field).
4. How can potential energy be negative?
PE is always relative. Whether it's negative or positive depends on where you put your zero. For gravity, we put zero at infinity, so all nearby locations are negative. For springs, we put zero at equilibrium, so all other positions are positive (\( PE = \frac{1}{2}kx^2 \geq 0 \)). What matters physically is the difference in PE, not the absolute value. You can choose your zero however makes the problem easiest.
5. Where should I put the reference point for zero potential energy?
Wherever is convenient for your problem! For everyday problems (ball dropping), put zero PE at ground level or on a table. For satellite problems, put zero at infinity (standard convention). It doesn't matter where you choose — only differences in PE have physical meaning. Choose different zeros and the absolute energies differ, but the total mechanical energy comes out the same.
Forces and potential energy
6. How do I understand \( \mathbf{F} = -dU/dx \)?
Force points in the direction of decreasing PE. If \( U \) increases (moving to the right), then \( dU/dx > 0 \), so \( F = -dU/dx < 0 \) (force is negative, pointing left). The force pushes the object back toward lower PE. This is the deep physics of conservative forces: systems naturally want to minimize their PE.
7. How can I tell if a force is conservative or not?
Three ways:
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Path independence: Work done is the same no matter what path you take between two points. Gravity is conservative (work depends only on height difference). Friction is not (work depends on path length).
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Potential energy exists: If you can write \( U(x) \) such that \( F = -dU/dx \), the force is conservative.
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Closed-loop work is zero: If an object travels in a closed loop and returns, the net work is zero (for conservative forces). Friction doesn't satisfy this — it always does negative work.
8. If a force is not constant, how do I find the potential energy?
Integrate the force: \( U(x) = -\int F(x) \, dx \). Example: if \( F(x) = -kx \) (ideal spring), integrating gives \( U = \frac{1}{2}kx^2 + \text{const} \). If \( F(x) = -\frac{Gm_1m_2}{x^2} \) (gravity), then \( U = -\frac{Gm_1m_2}{x} + \text{const} \). The constant is fixed by your choice of zero point.
Energy diagrams and motion
9. How do I read an energy diagram (graph of \( U \) vs position)?
Plot \( U(x) \) and draw a horizontal line at \( E_{\text{total}} \). The vertical gap between the line and the curve is the kinetic energy at that point: \( KE = E - U \). Where the line touches the curve, \( KE = 0 \) (turning point — particle stops). Where the line crosses the curve, the particle can move. If \( E \) is above the entire curve, the particle can escape. If \( E \) is trapped in a valley, the particle oscillates.
10. What is an equilibrium point, and how do I find it on an energy diagram?
Equilibrium is where \( dU/dx = 0 \) (force is zero). On the graph, these are the points where the slope of \( U \) is zero — the peaks and valleys. A minimum in \( U \) is stable equilibrium (small push → returns). A maximum in \( U \) is unstable equilibrium (small push → flies away).
11. What is a turning point?
A point where kinetic energy becomes zero: \( v = 0 \). On an energy diagram, these are where the horizontal line \( E_{\text{total}} \) touches the curve \( U \). The particle reverses direction here. If the particle is in a potential well, it bounces back. If it has enough energy, it escapes past the turning point.
12. Why can't an object enter regions where \( E < U \)?
Because kinetic energy cannot be negative. If \( E < U \) anywhere, then \( KE = E - U < 0 \), which is unphysical. The object cannot go there — it's blocked by the potential barrier. This forbidden region defines the boundaries of motion.
Energy conservation
13. What is mechanical energy?
The sum of kinetic and potential energy: \( E = KE + U = \frac{1}{2}mv^2 + U(x) \). If only conservative forces act, this value \( E \) remains constant. If non-conservative forces (friction, air resistance) act, \( E \) decreases.
14. If friction is present, why isn't energy conserved?
Friction converts mechanical energy to heat. Rather than energy disappearing, it changes form: mechanical → thermal. The total energy (including heat) is conserved, but the mechanical energy (KE + PE) is not. This is why you must be careful: if friction is present, use the work–energy theorem, not simple energy conservation.
15. What's the energy conservation law when friction acts?
\( E_i = E_f + Q \), where \( Q \) is the energy dissipated (to heat, sound, deformation). Equivalently: \( W_{\text{non-cons}} = \Delta E = E_f - E_i \). Example: if an object slides distance \( d \) on a friction surface, friction does work \( W_f = -fd \), so mechanical energy decreases by \( fd \).
16. Why doesn't the choice of zero point for PE matter?
Because only the difference in PE is physical. If you shift your zero point up by 10 meters, all PE values increase by \( 10mg \). But the change in PE remains the same, and the total mechanical energy remains the same. This is the freedom to choose the reference point.
17. Can total mechanical energy be negative?
Yes. If you set zero PE at infinity (for gravity), bound orbits have negative total energy: \( E = -\frac{GMm}{2r} < 0 \). Negative energy means the object is trapped — it cannot escape completely. To escape, you'd need to add energy until \( E \geq 0 \).
Escape velocity and orbits
18. What is escape velocity?
The minimum speed needed to completely escape a gravitational field. If an object leaves a planet's surface with escape velocity, its total energy is exactly zero (at infinity). Formula: \( v_{\text{esc}} = \sqrt{2GM/r} \). For Earth: \( v_{\text{esc}} \approx 11.2 \) km/s. For a black hole, \( v_{\text{esc}} = c \) — nothing escapes!
19. What is orbital velocity for a circular orbit, and how does it come from energy?
For circular orbit, gravitational force = centripetal force:
\[ \frac{Gm_1m_2}{r^2} = \frac{m_2v^2}{r} \]
Solving for \( v \): \( v = \sqrt{GM/r} \). Energy-wise: total energy \( E = KE + U = \frac{1}{2}m_2v^2 - \frac{Gm_1m_2}{r} \). Substituting \( v = \sqrt{GM/r} \) gives \( E = -\frac{Gm_1m_2}{2r} \) (negative, bound). This negative energy shows the satellite is trapped in orbit.
20. How do I know if an orbit is circular, elliptical, or escaping?
Check the total energy:
- \( E < 0 \): Closed orbit (circular or elliptical)
- \( E = 0 \): Boundary case (parabolic escape)
- \( E > 0 \): Open orbit (hyperbolic, escaping)
The shape of the orbit is determined by angular momentum (Chapter 10), but whether the orbit is bound or not is determined solely by energy.
Oscillations and energy
21. In simple harmonic motion (SHM), how does energy behave?
Energy oscillates between kinetic and potential, with the sum constant. When the mass passes through equilibrium, all energy is kinetic and speed is maximum. When it reaches maximum displacement, all energy is potential and speed is zero. Total energy: \( E = \frac{1}{2}kA^2 \) where \( A \) is the amplitude.
22. Why can't an oscillating object go beyond the amplitude \( A \)?
Because the points \( x = \pm A \) are turning points. At these locations, \( KE = 0 \) and all energy is potential. The force restores the object. If you increase the total energy, the amplitude grows, but the object never escapes the potential well (unless friction or external force adds energy).
Advanced concepts
23. What role does angular momentum play in planetary orbits?
Angular momentum determines the shape of the orbit (circle, ellipse, or hyperbola), but total energy determines whether the orbit is bound or open. Both are conserved if no external torque acts. This chapter focuses on energy; angular momentum appears more centrally in Chapter 10.
24. If two objects interact, how do I use their interaction potential energy?
The interaction PE depends only on the separation: \( U_{12}(r_{12}) = -\frac{Gm_1m_2}{r_{12}} \). Total system energy: \( E = KE_1 + KE_2 + U_{12} \). This is conserved if the system is isolated (no external forces). For deeper understanding, energy diagrams in the center-of-mass frame are very useful.
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