Lec 11: Work, Energy, and Universal Gravitation | 8.01 Classical Mechanics, Fall 1999 (Walter Lewin)

Main ideas and lessons

1) Definition of work in 1D (along a single line)

• Work by a force as an object moves from point A to B (motion strictly 1D): [ W=\int_A^B F\,dx ]

• Units: Newton-meters = joule (J)

• Work is a scalar:
  • Can be positive, zero, or negative
  • Positive work: force and displacement are in the same direction
  • Negative work: force and displacement are in opposite directions
• If multiple forces act in the same 1D direction:
  • Add them (effectively) and compute work using the resulting net force.

2) Work–energy theorem (derivation and meaning)

• Using (F=ma) and kinematics:
  • (a=dv/dt)
  • (dx=v\,dt)
• Substituting into the integral converts it from position to velocity.

• Result: [ W=\Delta K=\frac12 m v_B^2-\frac12 m v_A^2 ]

• Interpretation:

  • Work done from A to B equals the change in kinetic energy
  • If (W>0): kinetic energy increases
  • If (W<0): kinetic energy decreases
  • If (W=0): kinetic energy unchanged

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3) Examples using work–energy

Example 1: Gravity moving an object from A to B

• Setup:
  • Object starts at A with speed (v_A)
  • It rises/falls by height (h)
  • Gravity force is (mg)
  • Object “comes to a halt” at B, so (v_B=0)

• Work by gravity:

  • Force and displacement oppose: [ W_g=-mgh ]

• Apply work–energy: [ -mgh = 0-\frac12 m v_A^2 ]

• Solve for height: [ h=\frac{v_A^2}{2g} ]

Example 2: Lifting an object against gravity (Walter Lewin lifting)

• Setup:
  • Object goes from A to B under an upward force
  • (v_A=0) and (v_B=0) (implied by “no speed here”)

• Work done by the lifter:

  • Same magnitude, opposite sign to gravity: [ W_{\text{lifter}}=+mgh ]

• Gravity does: [ W_g=-mgh ]

• Net work: [ W_{\text{net}}=0 \Rightarrow \Delta K=0 ]

• Key conceptual takeaway:

  • You can do positive and negative work over a day and still have zero net work (even if you still get tired).
  • “Getting tired” is not the same as doing work—work in physics is defined via energy transfer.

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4) Work in 3D

General definition

• In 3D, work is: [ W=\int_A^B \mathbf{F}\cdot d\mathbf{r} ]

• For an infinitesimal displacement: [ dW = F_x\,dx + F_y\,dy + F_z\,dz ]

• Decomposition idea:

  • Treat motion component-by-component; each component contributes like a 1D work–energy relation.
• Result: [ W = \Delta\left(\frac12 m v^2\right) ] where (v^2) is the speed squared (magnitude squared of velocity).

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5) Gravity as a conservative force and path independence

Path independence criterion

• If the work done from A to B depends only on the start/end points (not the path), the force is conservative.
• Gravity is stated to be conservative.

Work done by gravity in 3D (near-Earth constant (g) case)

• With gravity only: [ \mathbf{F} = -mg\,\hat{y} ] (no (x) or (z) components)

• Work from A to B depends only on the vertical height difference (h=y_B-y_A): [ W_g = -mgh ]

• This is independent of the path (even if the trajectory is “crazy”).

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6) Mechanical energy conservation and gravitational potential energy

Work–energy form for gravity

• For gravity only: [ -mgh = K_B - K_A ]

• Rearrangement: [ mg y_B + K_B = mg y_A + K_A ]

Define gravitational potential energy

• Define: [ U = mgy ]

• Therefore: [ (K+U)_B = (K+U)_A ]

• Mechanical energy (E_{\text{mech}}=K+U) is conserved when forces are conservative.

Important caution: conservative vs non-conservative forces

• Mechanical energy conservation holds only for conservative forces.
• Friction is explicitly non-conservative:
  • Work done by friction depends on the path (longer route → more negative work → more energy lost).

Choice of zero potential energy

• The reference level for (U=0) is arbitrary.
• Only differences (U_B-U_A) matter.
• Near Earth (constant (g)), using (U=mgh) differences is convenient.

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7) Roller-coaster / minimum height condition (conservation of mechanical energy + centripetal requirement)

Setup and logic

• Frictionless roller-coaster-like track (circle of radius (R)).
• Released from rest at height (h) above the chosen zero level.
• Use conservation of mechanical energy (constant from release to any point on the track).

• Velocity at height (y): [ v^2 = 2g\,(h-y) ]

• At the top of the circular section (point D), centripetal acceleration is required: [ a_c=\frac{v^2}{R} \ge g ]

• Apply at point D where (y=2R):

  • Substitute (v^2 = 2g(h-2R)): [ \frac{2g(h-2R)}{R}\ge g ]

  • Simplify: [ h \ge \frac{5}{2}R ]

Conclusion (classic result)

• The ball will not reach the top unless released from at least (2.5R) above the zero level.
• With friction, it won’t return to the same height it started from; it stops lower.

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8) Far-away gravity: Universal gravitation and general gravitational potential energy

Gravitational potential energy definition (general)

• Formal definition:
  • Gravitational potential energy at point (P) is the work required to bring a mass from infinity to point (P).
• Equivalent reasoning:
  • Because gravity is conservative, (U) can be related to (minus) work gravity does, depending on sign conventions.

• Newton’s Universal Law of Gravity: [ F = \frac{G\,Mm}{r^2} ]

• Gravity is always attractive (“gravity sucks”).

• Gravitational acceleration: [ g(r)=\frac{GM}{r^2} ]

• Near Earth:

  • Plugging Earth’s values yields (g \approx 9.8\,\text{m/s}^2).

Compute potential energy from infinity to radius (R)

• Take work along radial inward motion.

• Result: [ U(R) = -\frac{GMm}{R} ]

• Key consequences:

  • (U(\infty)=0) by definition (not arbitrary, unlike constant-(g) potential)
  • (U) is negative everywhere except at infinity
  • Physically relevant is how (U) changes with position

Near-Earth limit recovers (mgh)

• For small height changes (h \ll R_{\oplus}): [ \Delta U \approx mgh ] (via Taylor expansion / first-order approximation)

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9) Real-world energy example: pendulum / wrecking ball

• A heavy mass lifted by 1 meter gains potential energy (mgh).
• Releasing converts potential energy into kinetic energy.
• Example claim:
  • For 15 kg lifted 1 m: about (150) joules (enough to demolish when used as a wrecking ball).

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10) Demonstration of energy conservation with a pendulum-like bob

• Lewin releases a bob from a certain height expecting it:
  • Not to rise higher on the return swing (assuming no added initial speed and conservation of mechanical energy).
• The demonstration is performed as a safety-critical act:
  • He closes eyes and waits as the bob swings back.

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Speakers / sources featured

• Walter Lewin (primary lecturer and demonstrator)
• Isaac Newton (mentioned for “Universal Law of Gravity”)
• No other named individuals appear in the subtitles.