Lec 10: Hooke's Law, Simple Harmonic Oscillator | 8.01 Classical Mechanics, Fall 1999 (Walter Lewin)

Main ideas, concepts, and lessons

1) Exam results and motivation (intro)

• Walter Lewin congratulates the class for scoring unusually well.
• He frames two possibilities for the high average:
  1. The class is exceptionally smart.
  2. The exam was too easy.
• He mentions the pass/fail threshold and compares it to typical failure rates.
• He previews today’s topic: springs, pendulums, and simple harmonic oscillators (SHO).

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2) Hooke’s Law for ideal springs

Definition of the setup

• Let the spring’s relaxed length correspond to (x=0).
• If you extend the spring by a displacement (x), a restoring force acts to drive it back toward equilibrium.

Hooke’s Law (experimental fact)

• For many ideal springs, the restoring force is proportional to displacement:
  • (F \propto x)
  • In 1D: (F = -kx)
• The minus sign encodes direction:
  • If (x>0), the force is negative (opposite direction).
  • If (x<0), the force is positive.

Spring constant

• (k) is the spring constant, with units newtons per meter (N/m).

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3) Measuring the spring constant (k) (two approaches)

A) Static method (force vs. displacement using gravity)

Procedure (conceptual steps)

1. Hang a mass (m) from the spring.
2. Wait until the system reaches a new equilibrium (no acceleration).
3. At equilibrium, the forces balance:
   • (F_{\text{spring}} = mg)
4. Vary the hanging mass to change the equilibrium displacement (x).
5. Record data pairs ((x, F)).

Data analysis

• The points form an approximately straight line (for ideal springs).
• Using increments:
  • (k = \Delta F / \Delta x)

B) Dynamic method (oscillation period)

Setup

• Place a mass (m) on a frictionless surface attached to a spring.
• Ignore gravity for the horizontal dynamics.

Key result

• Measure the period (T) of oscillation:
  • (T = 2\pi\sqrt{m/k})

Inference

• If you know (m), you can calculate (k) from (T).
• If you know (k), you can calculate (m) even “in the absence of gravity.”

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4) Limits of Hooke’s Law (real-world behavior)

• Hooke’s Law works only within limits.
• If you stretch too far:
  • The force no longer increases linearly with (x).
  • The spring can suffer permanent deformation.
• Lewin illustrates that behavior may start out linear, then transitions to non-Hookean stretching, leaving a net extension after release.

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5) Deriving Simple Harmonic Oscillator (SHO) behavior for springs

Governing differential equation

• Newton’s 2nd law in the horizontal direction:
  • (m\ddot{x} = -kx)
• Rearranging:
  • (\ddot{x} + (k/m)x = 0)

Trial solution and angular frequency

• Use a sinusoidal form:
  • (x(t)=A\cos(\omega t+\phi))
• Substitution implies:
  • (\omega^2 = k/m)
  • (\omega = \sqrt{k/m})
  • and therefore:
    • (T = 2\pi\sqrt{m/k})

Amplitude independence (nonintuitive result)

• The period is independent of amplitude (within ideal assumptions).
• It is also independent of phase (\phi).

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6) Role of initial conditions: determining (A) and (\phi)

Example given

• Initial conditions:
  • At (t=0): (x=0)
  • Velocity at (t=0): (\dot{x}=-3\ \text{m/s})
• Parameters:
  • (k=10\ \text{N/m}), (m=0.1\ \text{kg})

Computations

• Angular frequency:

  • [ \omega=\sqrt{k/m}=\sqrt{10/0.1}=10\ \text{rad/s} ]

• Using (x(0)=0):

  • (0=A\cos(\phi)) ⇒ (\cos(\phi)=0) ⇒ (\phi=\pi/2) (or (3\pi/2))
• Using (\dot{x}(0)=-3):
  • gives (A=0.3\ \text{m})

Final form

• (x(t)=0.3\cos(10t+\pi/2))
• Equivalent phase choices produce the same physical motion.

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7) Demonstration: period independence from amplitude (air track)

• Lewin measures 10 periods at different amplitudes to reduce timing uncertainty.
• Reported times:
  • ~20 cm amplitude: 15.16 s
  • ~40 cm amplitude: 15.13 s
• Conclusion: period is essentially constant within measurement uncertainty, supporting the ideal Hooke’s law / SHO prediction.

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8) Demonstration: period dependence on mass ((T \propto \sqrt{m}))

• He doubles the mass (adds two objects):
  • Prediction: (T_2 = T_1 \sqrt{2})
• Comparison:
  • measured (10T) for doubled mass ≈ 21.36 s
  • predicted (based on (\sqrt{2})) ≈ 21.42 s
• Conclusion: “spectacular” agreement within uncertainty supports the ideal-spring approximation.

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9) Pendulum derivation: small-angle simple harmonic motion

Setup

• Pendulum length (l), bob mass (m).
• Displaced by angle (\theta) from vertical.
• Forces:
  • gravity (mg)
  • tension (T) (with a note that symbol confusion is possible)

Coordinate directions

• (x): tangential restoring direction
• (y): radial/perpendicular direction

Coupled equations (before approximation)

• Decompose tension:
  • (T_y = T\cos\theta)
  • (T_x = T\sin\theta)
• Newton’s 2nd law yields two coupled differential equations in (x) and (y).

Small-angle approximations (key assumptions)

• Assume (\theta\ll 1) radians:
  • (\cos\theta \approx 1)
  • tangential motion dominates; radial acceleration is approximately 0
• Consequences:
  • radial equation gives (T \approx mg) (approximately independent of (\theta))
  • tangential equation becomes:
    • (\ddot{x} + (g/l)x = 0)

Pendulum SHO results

• Angular frequency:
  • (\omega = \sqrt{g/l})
• Period:
  • (T = 2\pi\sqrt{l/g})
• Restrictions:
  • small angles
  • bob mass concentrated in the bob (massless string)

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10) Physical interpretation: why formulas differ for springs vs pendulums

• Springs: (T = 2\pi\sqrt{m/k})
  • Restoring force depends on extension, not on the mass.
  • Doubling (m) increases inertia; with the same restoring force, acceleration halves ⇒ period increases.
• Pendulum: (T = 2\pi\sqrt{l/g}) (mass cancels)
  • Doubling bob mass increases:
    • gravitational force (and thus weight/inertia effects),
    • and the relevant tension components proportionally,
  • so acceleration and period remain unchanged.

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11) “Mother of all pendulums” experiment (Walter Lewin’s large pendulum)

• Very long pendulum:
  • length ~5.1 m (uncertainty ~5 cm)
  • bob mass ~15 kg (does not enter period formula)
• Predicted period:
  • (T = 2\pi\sqrt{l/g} \approx 4.57\ \text{s}) (length uncertainty gives ~0.02 s)
• Tests:
  • ~5° (about 10 periods timed): 45.70 s
  • ~10° (about 10 periods timed): 45.75 s
• Conclusion: changing amplitude within small-angle regime does not measurably affect period (given timing uncertainty ~0.1 s).

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12) Independence from bob mass (additional comparison)

• Since pendulum period is mass-independent, changing the effective “bob-plus-body geometry” should keep the period nearly the same.
• Lewin performs an awkward setup and measures 10T again:
  • 45.6 ± 0.1 s
• Conclusion: results match the predicted mass-independent pendulum period within experimental uncertainty.

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Methodology / “instructions” presented (bullet format)

Measuring spring constant (k) using static equilibrium

• Hang a mass (m) from the spring.
• Wait for equilibrium (no acceleration).
• Use gravitational force:
  • record equilibrium displacement (x) where spring force balances (mg).
• Vary (m) across multiple trials to get several ((x,F)) points.
• Fit/compute slope:
  • (k = \Delta F / \Delta x)
• Notes:
  • Ideal-spring behavior should look approximately linear.
  • Starting from a pre-tensioned spring is allowed; it’s not required to begin at relaxed length.

Measuring spring constant (k) dynamically with oscillation period

• Put mass (m) on a frictionless surface connected to the spring.
• Displace (ensure Hooke’s law still holds).
• Measure oscillation period (T) (preferably over multiple periods).
• Use:
  • (T = 2\pi\sqrt{m/k})
• Solve:
  • (k = m\left(\frac{2\pi}{T}\right)^2)

Verifying amplitude independence for SHO

• Use a spring-mass system (ideal behavior assumed).
• Measure the time for 10 periods at:
  • smaller amplitude
  • larger amplitude (e.g., double)
• Compare implied periods; expectation: same (T) within uncertainty.

Verifying mass dependence for springs

• Measure 10T for mass (m_1).
• Combine masses to create (m_2) (e.g., doubling).
• Predict:
  • (T \propto \sqrt{m})
  • so (T_2 = T_1\sqrt{m_2/m_1})
• Measure 10T for (m_2) and compare.

Deriving pendulum SHO using small-angle approximation

• Resolve forces into radial and tangential components (via tension decomposition).
• Write Newton’s 2nd law in both directions.
• Assume:
  • (\theta) small (radians), so (\cos\theta \approx 1)
  • radial acceleration is negligible while tangential motion dominates
• Use radial equation to approximate (T \approx mg).
• Substitute into tangential equation to get:
  • (\ddot{x} + (g/l)x = 0)
• Conclude:
  • (T = 2\pi\sqrt{l/g})

Pendulum experiment testing amplitude independence

• Set the pendulum to a maximum angle (e.g., 5°).
• Time about 10 oscillations; record total time.
• Repeat at a larger angle (e.g., 10°).
• Compare implied periods; expectation: no significant change within uncertainty.

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Speakers / sources featured

• Walter Lewin (lecturer; demonstrations and calculations)
• Students (audience volunteers responding during demonstrations)