The Concept of Potential Energy (NCERT 5.7)
1. Energy by Position or Configuration
A brick sitting on a windowsill does nothing — yet it can fall and smash a table. A stretched spring holds still — yet it can launch a ball. Both are carrying potential energy: energy stored by position (the brick's height in Earth's gravity) or by configuration (the spring's stretched arrangement of coils), ready to be cashed into motion. Potential energy is the savings account of mechanics: the work-energy theorem handles the current account (kinetic energy filled by net work), and U handles the deposits — work done now that is recoverable later.
Two ownership notes from NCERT that MCQs quietly test. First, U belongs to a system, not a single body: "the potential energy of the block" really means the block–Earth system's shared store (raise either, and the store grows). Second, U is defined only through changes — which is why the next section's signs and reference levels are the actual exam content.
2. Complete Theory: Only ΔU Matters, and the Conservative-Force Test
The defining relation. For a conservative force, the change in potential energy is the negative of the work that force does: ΔU = −Wcons. A falling brick: gravity does +mgh, so U drops by mgh. A rising spring-compressor: the spring's force does negative work, so spring U rises by exactly that amount. The minus sign is the physics — the force's work drains the account it belongs to.
The two-path test. A force is conservative when its work between two points is path-independent — equivalently, when its work over any closed round trip is zero. Gravity passes: lift a book 2 m by ladder or by a 6 m zig-zag ramp and gravity's work is the same −mg(2 m) either way, returning untouched on the way down. Kinetic friction fails: a 2 m slide costs 2f, a 6 m slide costs 6f, and a round trip never cancels — the energy it extracts leaves as heat, irrecoverable. Only conservative forces get a potential energy; this is why tables of U contain gravity and springs but never friction.
The reference level. Since only ΔU is measurable, where you put U = 0 is a free choice: floor, tabletop, ceiling — anything. Declare it once and compute every height from it. The gravitational store near Earth's surface then reads U = mgh, with h measured from your chosen zero and g treated as constant — a local formula, valid while heights stay small against Earth's radius (the −GMm/r generalisation is Gravitation's story). Its dimensions M L² T⁻² match work and kinetic energy exactly, as the dimension tables of Chapter 1 confirm — one ledger, one unit, the joule.
3. Visualising: Two Paths, One Ledger Entry
Figure 5.4 — ΔU cares about the endpoints, not the route
Left: straight lift through h = 3 m. Right: frictionless 3-4-5 incline, path length 5 m but the same rise of 3 m. Compare the two energy deposits.
Exam read-out: the incline is 5 m long versus 3 m of ladder, yet the energy deposit is identical 30 J — that is the conservative-force test in pictures. What does change with the route: the time taken and (if friction existed) the heat billed. NTA's "different ramp, same height" questions are this figure with the numbers shuffled.
4. Solved Examples
Example 1 — Lifting, and reading all three work entries
A 2 kg block is lifted slowly through 4 m (g = 10 m s⁻²). Find the change in gravitational potential energy, the work done by gravity, and the work done by the lifting force.
Solution (step by step): ΔU = mgh = 2 × 10 × 4 = +80 J. Gravity opposes the rise: Wgravity = −mgh = −80 J — note ΔU = −Wgravity exactly. "Slowly" means negligible speed change, so by the work-energy theorem ΔK = 0 and the lifting force's work must close the books: Wlift = +80 J. Three entries, one audit — and the reference level (floor, say) never had to be justified because only the difference 4 m entered.
Example 2 — Path independence with numbers
A 1 kg block is dragged slowly up a frictionless 3-4-5 incline (height 3 m, length 5 m) and then returned to the bottom. Find the work done by gravity on the way up, on the way down, and over the round trip.
Solution: Up: gravity acts opposite the motion, W = −mgh = −1 × 10 × 3 = −30 J — only the height matters; the 5 m length never enters. Down: W = +30 J. Round trip: −30 + 30 = 0 — the closed-loop signature of a conservative force, which is exactly why U = mgh exists for gravity. (The puller's work: +30 J up, 0 down if simply released slowly, and +30 J over the trip — non-conservative bookkeeping, unlike gravity's.)
5. Practice Questions
Q1. A 5 kg sack of cement is raised 3 m by a crane at steady speed. Find the change in its potential energy and the work done by the cable.
ANSWER: ΔU = mgh = 5 × 10 × 3 = +150 J. Steady speed ⇒ ΔK = 0, so the cable's work equals the full deposit: W = +150 J, while gravity bills −150 J. If the crane took 2 s, its average power would be 150/2 = 75 W — the power page's opening move.
Q2 (MCQ). Which of the following forces is non-conservative? (a) gravitational force (b) ideal spring force (c) kinetic friction (d) the normal force on a level floor
ANSWER: (c) — kinetic friction's work depends on the path length and never cancels on a round trip, so no potential energy can be attached to it. Gravity and the spring force pass the round-trip test cleanly; the normal on a level floor does zero work trivially (perpendicular), costing nothing on any route.
Q3. A satellite moves in a circular orbit at constant speed. State the work done by Earth's gravity over one full orbit, and what this implies about the satellite's potential energy.
ANSWER: Gravity points radially inward, the velocity (and hence the small displacement each instant) is tangential: θ = 90° everywhere, so W = 0 over any arc, including one full orbit — the UCM zero-work result. Over a closed orbit the conservative gravity has done nothing, so U returns to its start: potential energy is constant in a circular orbit, as is kinetic energy.
6. Key Formulas & Takeaways
| Relation | Condition / remark |
|---|---|
| ΔU = −Wcons | Definition of PE change for a conservative force; mind the minus |
| U = mgh | Gravity near Earth's surface, g constant, h from your chosen zero |
| Round-trip W = 0 | The conservative-force test; gravity and springs pass, friction fails |
| Us = ½kx² | Spring configuration energy (full treatment next-next page) |
| Only ΔU is physical | Zero of U is a free convention; stay consistent within a problem |
| [U] = M L² T⁻² | Same joule ledger as work and KE |
Potential energy is the invention that lets forces "deposit" work for later withdrawal. The next page opens the account properly: if only conservative forces act, K + U never changes — the single most productive equation in school mechanics.
7. Frequently Asked Questions
What is potential energy?
Potential energy is the energy a system stores by virtue of the position or configuration of its parts — a raised block (position in Earth's gravity), a stretched spring (configuration of its coils), a drawn bow. It is energy waiting to be cashed into kinetic energy. Two fine print items matter for exams: potential energy belongs to the system (block plus Earth, spring plus block), not to one body alone; and only changes in U are measurable, because the zero level is a free choice.
What is a conservative force?
A force is conservative when the work it does moving a particle between two points is independent of the path taken — equivalently, its work over any closed round trip is exactly zero. Gravity and the ideal spring force are conservative; kinetic friction is not, since a longer slide bills more negative work. For a conservative force the change in potential energy is defined by ΔU = −W (the minus sign says the force's work drains the stored energy), and this is what lets a bookkeeping term U replace the force's work in energy equations.
Why is there no potential energy for friction?
Because friction's work depends on the path: sliding a box 2 m across a floor costs 2f joules, sliding it around a 6 m loop costs 6f, and a round trip costs a full 2f each way instead of cancelling. The energy friction takes is converted to heat and dispersed — it never sits in a recoverable store tied to position. Since no path-independent U can be defined, friction's work must always be carried as an explicit (usually negative) term in energy equations, never absorbed into a potential.
Does the choice of zero for potential energy matter?
No — U = 0 is a reference convention, not physics. Choose the floor, the tabletop or the ceiling; every formula that matters (K_i + U_i = K_f + U_f, speeds from energy conservation) involves only differences ΔU, and adding a constant to every U cancels out. Exam procedure: declare your zero level once, compute heights from it consistently, and never switch mid-problem. An answer quoted as an absolute U without a stated reference is physically incomplete — NTA tests this only indirectly, through consistency.
Saved on this device only — no account, no sign-in.