Lec 13: Equation of Motion for Simple Harmonic Oscillators | 8.01 Classical Mechanics (Walter Lewin)

Main ideas, concepts, and lessons

1. Potential energy vs. force (and the role of sign)

   • For near Earth (1D vertical coordinate (y)):

     • Gravitational force points downward while (y) increases upward, so [ F=-mg ] (the minus sign is crucial because force opposes increasing (y)).

     • Gravitational potential energy depends on your chosen zero:

       • Choose (U=0) at (y=0) (\Rightarrow) [ U(y)=+mgy ]

     • Work-energy relationship (qualitative sign logic):

       • If you move from A to B where (U) increases:
         • you do positive work
         • gravity does negative work
       • If you move from A to B′ where (U) decreases:
         • you do negative work
         • gravity does positive work
     • The choice of zero potential energy is arbitrary; shifting the zero does not change physical work differences.

2. Gravitational potential energy at large distances (Newtonian gravity)

   • For a mass (m) at distance (r) from a massive body (M):

     • [ F=-\frac{GMm}{r^2} ] (1D simplification: direction encoded by the minus sign)

     • Gravitational potential energy: [ U(r)=-\frac{GMm}{r} ]

     • Qualitative behavior:

       • (U(r)) is negative and varies like (1/r).
       • Near Earth, where gravity is approximately constant, the relation becomes effectively linear, matching the near-Earth behavior (U=mgy).

3. General rule: force opposes increasing potential energy

   • If an object is released from rest, it moves toward lower potential energy, because the force always points opposite the direction of increasing (U).

4. Spring potential energy and how it’s derived

   • Setup:
     • Spring relaxed at (x=0)
     • Extended to (x)
     • Spring force: (F=-kx)
     • In 1D, the direction sign is handled by the minus sign; arrows are omitted.

   • Work required to move from (x=0) to (x):

     • [ W=\int F\,dx ]

     • Since the force during the move is opposite the direction of increasing (U), the chosen work sign convention yields: [ U_{\text{spring}}(x)=\frac12 kx^2 ]

     • Potential energy zero is set at (x=0) by convention.

       • Shape:
     • (U(x)) is a parabola.
       • Restoring behavior:
     • Force always opposes increasing potential energy, so releasing from rest moves toward equilibrium.

5. Recovering force from potential energy (the key differential relationship)

   • 1D case (spring):

     • (U(x)=\dfrac12 kx^2)

     • Differentiate: [ \frac{dU}{dx}=kx ]

     • Relation to force: [ \frac{dU}{dx}=-F ] So force points opposite increasing (U).

   • 3D case:

     • If (U=U(x,y,z)), then: [ \frac{\partial U}{\partial x}=-F_x,\quad \frac{\partial U}{\partial y}=-F_y,\quad \frac{\partial U}{\partial z}=-F_z ]

     • Interpretation of partial derivatives:

       • To get the (x)-component, vary only (x) while keeping (y,z) fixed, etc.

6. Equilibrium, stability, and the role of curvature of (U(x))

   • Consider a curved gravitational potential energy profile (U(x)=mg\,y(x)).

   • Equilibrium points occur where: [ \frac{dU}{dx}=0 ] so the corresponding force component (e.g., (F_x)) is zero.

   • Stability criterion using second derivative:

     • Stable equilibrium: (\dfrac{d^2U}{dx^2} > 0) (minimum of (U))
       • If displaced, force pushes it back.
     • Unstable equilibrium: (\dfrac{d^2U}{dx^2} < 0) (maximum of (U))
       • If displaced slightly, it rolls away to lower (U).
   • Demonstrations:
     • Marble on a surface with a stable vs unstable equilibrium shows “stay put” vs “roll away.”

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Methodology / instruction-like steps explicitly presented

A) Derive gravitational potential energy (near Earth)

• Choose coordinate (y) (increasing upward).
• Write gravitational force: (F=-mg).
• Choose reference level where (U(y=0)=0).

• Then potential energy: [ U(y)=mgy ]

• Use sign/work logic:

  • Moving to higher (U) (\Rightarrow) you do positive work (gravity does negative work).

B) Derive gravitational potential energy (general spherical gravity)

• Use Newton’s law: [ F=-\frac{GMm}{r^2} ]

• Integrate/relate to potential energy to get: [ U(r)=-\frac{GMm}{r} ]

• Choose (U(\infty)=0), implying (U) is negative for finite (r).

C) Compute spring potential energy from work

• Setup:
  • Relaxed spring at (x=0), stretched to (x).
• Use spring force: (F=-kx).

• Work done by you while moving from (0) to (x):

  • [ W=\int_{0}^{x} F_{\text{(your applied/sign convention)}}\,dx ]

• Resulting spring potential energy: [ U(x)=\int_{0}^{x} kx\,dx=\frac12 kx^2 ]

• Convention:

  • Set (U(0)=0).

D) Recover force from potential energy

• 1D: [ \frac{dU}{dx}=-F ]

• 3D: [ \frac{\partial U}{\partial x}=-F_x,\quad \frac{\partial U}{\partial y}=-F_y,\quad \frac{\partial U}{\partial z}=-F_z ]

E) Use potential-energy curvature to identify stability

• Find equilibrium points where: [ \frac{dU}{dx}=0 ]

• Determine stability:

  • If (\dfrac{d^2U}{dx^2}>0): stable equilibrium (restoring).
  • If (\dfrac{d^2U}{dx^2}<0): unstable equilibrium (runaway).

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How simple harmonic motion is obtained from energy

1. Spring ⇒ SHM from energy conservation

   • Start with energy conservation (no friction; spring is conservative):

     • Total energy: [ E=\frac12 kx_{\max}^2 ]

   • At any position (x): [ E=\frac12 mv^2+\frac12 kx^2 ]

   • Substitute (v=\dot{x}) and rearrange: [ \dot{x}^2+\frac{k}{m}x^2=\frac{k}{m}x_{\max}^2 ]

   • Take time derivative and use (\dfrac{d}{dt}(\dot{x}^2)=2\dot{x}\ddot{x}):

     • leads to: [ \ddot{x}+\frac{k}{m}x=0 ]

   • Solution:

     • [ x(t)=x_{\max}\cos(\omega t+\phi) ]

     • [ \omega=\sqrt{k/m} ]

     • Period: [ T=\frac{2\pi}{\omega} ]

2. Circular track (pendulum-like) ⇒ SHM via small-angle approximation

   • For a mass on a circular arc of radius (R), gravitational potential:

     • [ U(\theta)=mgR(1-\cos\theta) ] (zero chosen at the lowest point)

   • Kinematics:

     • tangential speed (v=R\dot{\theta})

   • Energy conservation: [ \frac12 m(R\dot{\theta})^2+mgR(1-\cos\theta)=\text{constant} ]

   • Small-angle approximation (radians):

     • [ \cos\theta\approx 1-\frac{\theta^2}{2} ]

   • Substitute to make potential quadratic in (\theta), yielding: [ \ddot{\theta}+\frac{g}{R}\theta=0 ]

   • Therefore SHM:

     • [ \theta(t)=\theta_{\max}\cos(\omega t+\phi) ]

     • [ \omega=\sqrt{g/R} ]

     • Period: [ T=2\pi\sqrt{R/g} ]

   • Connection: it matches the pendulum form when the effective length is (l=R).

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Work done by constraints (tension/normal force)

• Even with a tension (pendulum) or normal force (circular track), energy conservation still works because:
  • those forces do no work (they are always perpendicular to the instantaneous direction of motion),
  • and work is (W=\vec{F}\cdot \vec{d}), so perpendicular force (\Rightarrow) dot product (=0).

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Demonstrations and experimental checks

1. Large-radius air-track experiment (radius (\sim 115) m)

   • Predict period:

     • [ T=2\pi\sqrt{R/g} ]

     • with (R=115\pm 5) m (\Rightarrow) predicted (T\approx 21.5\pm 0.5) s

       • Timing:
     • observes 3 oscillations total time: (3T=64.05) s
     • measured (T\approx 21.35) s
     • agrees with prediction within uncertainty
       • Notes:
     • small amplitude decay due to slight friction, but the key point is agreement with SHM/potential-energy reasoning.

2. Smaller-radius arc experiment (radius (\sim 85) cm)

   • Predicted period (small-angle justification):
     • estimated (T\approx 1.85) s (with stated uncertainty)
   • Actual measurement:
     • measured about 22.7 s for what was presumably intended as oscillations (instructor notes: “it should have been 18!”)
   • Suggested resolution (brief):
     • friction was mentioned, but the conclusion was that a different effect becomes important when the timing/conditions change.

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Speakers / sources featured

• Walter Lewin (lecturer, speaking throughout)
• Class / students (unidentified; occasional student answers and “class murmurs”)
• YouTube source: “Lec 13: Equation of Motion for Simple Harmonic Oscillators | 8.01 Classical Mechanics (Walter Lewin)”