1.3.5 · D5Work, Energy & Power
Question bank — Potential energy — definition, gravitational (mgh and −GMm - r), elastic (½kx²)
True or false — justify
Gravitational PE near the surface can be negative.
True — is negative for any point below your chosen line, e.g. the bottom of a well. Sign depends entirely on where you place the reference. See Work done by a force.
The formula can ever be positive.
False — , , , are all positive and there is an explicit minus sign, so always. It only approaches zero (from below) as .
A single isolated object has potential energy.
False — PE lives in the configuration between two interacting bodies (book+Earth, ball+spring). "The book's PE" is shorthand for the book–Earth system's PE.
Compressing a spring stores less energy than stretching it by the same amount.
False — depends on , so . Push in 5 cm or pull out 5 cm: identical stored energy.
Friction has a potential energy function just like gravity.
False — friction's work is path-dependent (a loop around and back loses energy every cm), so no single-valued exists. Only conservative forces have a PE.
Doubling the height doubles the near-surface gravitational PE.
True — is linear in , so gives . (Contrast the spring, which is quadratic.)
Doubling the stretch of a spring doubles its stored energy.
False — is quadratic, so gives . Four times the energy, not two.
As , the gravitational PE goes to zero.
True — that's precisely the chosen reference: at infinite separation. The mass sits at the "rim" of the well.
The deepest (most negative) gravitational PE is at the smallest .
True — as shrinks, becomes a larger negative number, so the mass sits deeper in the well and needs more energy to climb out.
If PE increases, the conservative force must have done positive work.
False — the opposite. , so rising means ; the force resisted the motion while you did work against it.
Spot the error
"Spring PE is because the force is ."
The force must be integrated, not multiplied: . The missing is the average factor of a linearly-rising force. See Hooke's Law.
" and disagree, so one is wrong."
Neither is wrong — is the flat-Earth limit of for , giving using . They agree on differences, which is all that's physical.
"Energy to escape Earth is because that's the average PE."
": since is always positive, the force is always positive."
The sign of is irrelevant; the force comes from the slope. , which flips sign with — restoring on both sides. See Force from potential — F = -dU/dx.
"Since gravity pulls down, lifting a book does negative work on it, so is negative."
The gravitational force does negative work (), but . The minus sign in is exactly what makes storage come out positive.
"PE at the top of a shelf is 29.4 J, so the book contains 29.4 J."
That number is the difference from your chosen floor reference, not an absolute content. Move the reference to the shelf and the same book reads 0 J.
"A spring at its natural length has zero force and zero PE, so those two facts are the same statement."
They coincide only because both the slope and the height of vanish at . In general zero force () marks a flat point of , which can sit at any nonzero PE value.
Why questions
Why does carry a minus sign?
Positive work by the force spends stored energy so falls; work done against the force banks energy so rises. The minus flips "work by force" into "energy stored."
Why do only conservative forces get a potential energy?
Their work between two points is path-independent, so can be a single-valued function of position. A path-dependent force would assign a point many different values — a contradiction. See Conservative vs non-conservative forces.
Why is bound gravitational PE negative rather than positive?
The reference sits at infinity; as the mass falls inward gravity does positive work, driving below zero into a "well." Negative just means "below the infinitely-far reference."
Why is spring PE quadratic in but gravitational only linear in ?
The spring force grows with displacement, so integrating a rising line gives a quadratic. Near-surface gravity is constant, so integrating a flat force gives a straight line.
Why does force point toward lower PE?
means the force pushes down the steepest slope of the energy landscape — like a ball rolling into a valley. The steeper the drop, the stronger the push.
Why can we ignore the varying and use constant near the ground?
Over heights the distance changes by a negligible fraction, so barely moves. The uniform-field approximation errs only at the level of . See Gravitation — Newton's law.
Why is PE a property of the system and not one object?
PE measures the relative configuration of two interacting bodies; move one and the stored work changes. With only one body there is nothing to be conservative between. Tied to Conservation of mechanical energy.
Why does escaping to infinity require exactly and not more?
Because is defined at infinity, so the whole climb equals the well depth . Any extra energy would arrive as leftover kinetic energy, not as climbing cost.
Edge cases
What is the gravitational PE at ?
It diverges to — a mathematical signal that a true point mass is idealised; real bodies have finite radius, so you never reach outside them.
What is the spring PE when the spring is at natural length?
Zero by our chosen reference , and it's the minimum of , so the force also vanishes there — a stable equilibrium.
If you set the PE reference at the shelf instead of the floor, does the book's motion change?
No — only enters the physics, and differences are unchanged by shifting the reference. The predicted speeds and forces are identical.
What happens to for a point below the reference ()?
becomes negative, correctly saying the object sits in a valley relative to the reference. The formula handles both signs of without change.
At the exact top of a thrown ball's flight, is its gravitational PE maximal or its force zero?
PE is maximal (highest point), but the force is not zero — gravity still pulls down; that's what reverses the motion. Max PE ≠ zero force here because is constant, not slope-dependent.
For two masses infinitely far apart, what is the system's gravitational PE and force?
Both approach zero: and . Infinitely separated bodies barely feel one another — the natural "nothing stored, nothing pulling" state.
Can total mechanical energy be negative for a bound orbit?
Yes — if the (negative) gravitational PE outweighs the kinetic energy, , which is exactly the signature of a bound, non-escaping orbit. See Conservation of mechanical energy.
Recall One-line self-test before you leave
Cover these and answer with a reason. Sign of bound gravitational PE and why? ::: Negative, because the reference is at infinity and falling inward does positive work, dropping below zero. Why in spring PE? ::: It is the average factor from integrating the linearly-rising force . Whom does PE belong to? ::: The interacting system, never a single object.