Lec 08: Frictional Forces | 8.01 Classical Mechanics, Fall 1999 (Walter Lewin)

Main ideas / lessons

1) Friction basics: why it “adjusts itself” and has a maximum

When an object rests on a horizontal surface and there is no vertical acceleration:

• The normal force balances gravity: [ N = mg ]

When a horizontal force is applied:

• At first, the object does not accelerate because static friction increases to exactly oppose the applied tendency to move.
• As the applied force increases:
  • static friction grows, but only up to a maximum value.
• Once static friction can no longer match the applied force, the object begins to accelerate.

The maximum static friction is: [ F_{f,\max} = \mu_s N ]

Key friction coefficients:

• Static friction: (\mu_s)
• Kinetic friction: (\mu_k)

Typically: [ \mu_s > \mu_k ] So it’s harder to start moving than to keep moving.

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2) Measuring (\mu_s) using an incline (“critical angle” method)

Procedure concept:

1. Place an object on an incline at angle (\alpha).
2. Increase (\alpha) slowly until the object is just about to slide.

Force decomposition:

• Along incline (“x”): [ mg\sin\alpha ]

• Perpendicular (“y”): no vertical acceleration, so [ N = mg\cos\alpha ]

At the threshold of sliding:

Along the incline: [ mg\sin\alpha - F_{f,\max} = 0 ] and since (F_{f,\max} = \mu_s N = \mu_s mg\cos\alpha): [ mg\sin\alpha = \mu_s mg\cos\alpha ]

Solving gives: [ \mu_s = \tan\alpha ]

Key takeaway: measure the critical angle (\alpha) experimentally to find (\mu_s).

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3) “Nonintuitive” experimental conclusions about what friction does NOT depend on

Experiments emphasize:

• Mass independence (approximately):
  • The critical angle (\alpha) (and thus (\mu_s)) changes little with object mass.
• Contact area independence (approximately):
  • Changing contact area (e.g., different footprint size) produces little to no change in the critical angle.

Therefore, for the same materials:

• The “start sliding” angle and inferred (\mu) are mainly material-dependent, not geometry- or mass-dependent.

Real-world implication example:

• Rubber on concrete: (\mu \approx 1)
• Then (\tan\alpha \approx 1 \Rightarrow \alpha \approx 45^\circ)
• This suggests similar slide tendency for different vehicles as long as road/rubber materials are similar.

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4) Friction with a pulley system: three possible regimes

Consider:

• Two masses ((m_1) on an incline, and (m_2) hanging)
• Connected by a string over a frictionless, massless pulley
• Friction direction on (m_1) is not known ahead of time; it depends on whether (m_1) tends to go up or down the incline.

Always true for (m_1):

No acceleration perpendicular to the plane: [ N = m_1 g\cos\alpha ]

Static friction limit: [ F_{f,\max} = \mu_s N = \mu_s m_1 g\cos\alpha ]

Ideal pulley/string means the same tension throughout. At rest: [ T = m_2 g ]

Regimes at the point motion is about to occur

Assuming the system starts at rest:

1) Condition 1: onset of uphill acceleration ((m_1) going uphill) Static friction acts downhill to oppose uphill tendency. [ m_2 g = m_1 g\sin\alpha + F_{f,\max} ] [ m_2 g = m_1 g\sin\alpha + \mu_s m_1 g\cos\alpha ]

2) Condition 2: onset of downhill acceleration ((m_1) going downhill) Static friction acts uphill to oppose downhill tendency. [ m_2 g = m_1 g\sin\alpha - F_{f,\max} ] [ m_2 g = m_1 g\sin\alpha - \mu_s m_1 g\cos\alpha ]

3) Otherwise: no acceleration - (\,a = 0) - Static friction adjusts (within its maximum) to prevent acceleration.

Worked example (numerical)

Given:

• (m_1 = 1\,\text{kg}), (m_2 = 2\,\text{kg})
• (\alpha = 30^\circ)
• (\mu_s = 0.5), (\mu_k = 0.4)
• (g = 10\,\text{m/s}^2)

Compute:

• (m_2 g = 20)
• (m_1 g\sin\alpha = 1\cdot 10 \cdot 0.5 = 5)
• [ F_{f,\max}=\mu_s m_1 g\cos\alpha = 0.5\cdot 1\cdot 10 \cdot (\cos 30^\circ \approx 0.866)\approx 4.33 ]

Test condition 1 (uphill):

• (20 > 5 + 4.33) ⇒ true So the system accelerates uphill for (m_1).

After it begins moving:

• Friction becomes kinetic: [ F_f = \mu_k m_1 g\cos\alpha ]

• Tension is no longer (m_2 g) because the system accelerates.

Stated results:

• Acceleration: (a \approx +3.85\,\text{m/s}^2)
• Tension: (T \approx 12.3\,\text{N})

Reminder: the sign of (a) must match the direction implied by the threshold test.

Second numerical case (comparison)

Same setup, but:

• (m_2 = 0.4\,\text{kg}) ⇒ (m_2 g = 4)

Threshold tests:

• Not large enough for condition 1 (uphill)
• Not small enough for condition 2 (downhill)

Therefore:

• (a = 0)
• Static friction adjusts to maintain equilibrium (including choosing direction and magnitude appropriately).

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5) Additional demonstrations: extracting (\mu_s) and showing friction’s complexity

Lewin demonstrates measuring (\mu_s) by:

• Using an incline
• Increasing (\alpha) until the block slides

Example reported:

• Slide begins around (\alpha \approx 20^\circ)
• Hence: [ \mu_s \approx \tan(20^\circ) ]

He also suggests a pulley-based alternative:

• Fix (\alpha) (e.g., (20^\circ))
• Increase (m_2) until the block is just about to slide uphill
• Use that threshold to compute (\mu_s)

Sources of discrepancy:

• Grain/roughness directionality (anisotropy of wood)
• Small uncertainty in measuring angles
• Variability such as humidity

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6) Friction’s practical consequences and ways to reduce it

Friction leads to:

• Wear/tear
• Energy loss/costs (e.g., fuel in cars)

Methods to reduce friction:

• Lubrication: oil/water can reduce friction dramatically
  • Example: hydroplaning risk—water plus dust can reduce tire-road friction toward near-zero
• Air cushion / air-track idea:
  • Lift objects on a thin air layer to reduce contact friction
  • Turning off the air causes the object to stop quickly, showing the dependence on the cushion
• Hovering / slip-film demonstration (dry ice / CO(_2)):
  • A disc floats on a thin film of escaping CO(_2) gas
  • The reduced friction lets small pushes start motion easily and it glides far/long

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Methodology / instruction-style content (detailed bullet points)

A) Critical-angle method to determine static friction coefficient (\mu_s)

• Set up:
  • Place an object on an incline at angle (\alpha)
  • Increase (\alpha) gradually
• Identify the threshold:
  • Find the smallest angle where the object begins to slide

• Apply force balance at the threshold:

  • Perpendicular direction: [ N = mg\cos\alpha ]

  • Parallel direction: [ mg\sin\alpha = F_{f,\max} = \mu_s N ]

• Compute: [ \mu_s = \frac{mg\sin\alpha}{mg\cos\alpha} = \tan\alpha ]

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B) Pulley + incline method to determine/compare friction regimes

• Model assumptions:
  • String and pulley are ideal: massless, frictionless ⇒ same tension on both sides
• Choose a sign convention for motion along the incline axis (“x”)

• Compute normal force: [ N = m_1 g\cos\alpha ]

• Use the static friction limit before motion: [ F_{f,\max}=\mu_s N ]

• Consider three mutually exclusive cases:

  • Case 1 (uphill onset): [ m_2 g = m_1 g\sin\alpha + \mu_s m_1 g\cos\alpha ]

  • Case 2 (downhill onset): [ m_2 g = m_1 g\sin\alpha - \mu_s m_1 g\cos\alpha ]

  • Case 3 (no acceleration):

    • If neither threshold is met:
      • (a = 0)
      • Static friction adjusts within its maximum so net force along the incline is zero

• After motion begins:

  • Replace (\mu_s) with (\mu_k)
  • Tension is no longer (m_2 g)
  • Solve the coupled Newton’s 2nd law equations for acceleration (a) and tension (T)
  • Ensure the sign/direction of (a) matches the regime deduced from the static thresholds

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C) Qualitative experimental demonstrations (conceptual “procedure”)

• Compare materials:
  • Use different contacting surfaces (e.g., rubber vs smooth plastic)
  • Raise incline angle until each begins to slide
  • Observe which starts sliding first
• Compare masses:
  • Use similar objects with different masses
  • Measure the critical angle and see minimal change
• Compare contact area:
  • Use the same object but different footprint/flat contact area
  • Measure critical angle for each; observe little change

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D) CO(_2) “dry ice disc” friction reduction concept

• Fill a container with dry ice (solid CO(_2))
• CO(_2) evaporates; gas must escape through a bottom opening
• Gas forms a thin film between the disc and glass
• The disc effectively rides on the escaping gas layer
• Result: motion begins with a very small applied force and glides with little noticeable slowing

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Speakers / sources featured

• Walter Lewin (primary lecturer)
• Unidentified students (murmurs and occasional replies during Q&A prompts)
• Film narrator (voiceover during the CO(_2)/dry ice demonstration)