Lec 12: Resistive Forces | 8.01 Classical Mechanics, Fall 1999 (Walter Lewin)

Scientific concepts & nature phenomena presented

1) Drag / resistive forces in fluids

• An object moving through a medium (gas or liquid) experiences a drag (resistive) force.
• Drag depends on:
  • Shape and size of the object
  • Medium properties
  • Speed of the object
• Unlike kinetic friction between solids (where the coefficient is roughly constant and independent of speed), drag varies with speed.

2) Speed-dependent force model (opposite the motion)

• In general, the resistive force is modeled as the sum of terms linear and quadratic in speed:

  • [ F_\text{res} \approx k_1 v + k_2 v^2 ]

  • The force always opposes the velocity.

  • The constants (k_1) and (k_2) depend on object geometry/size and the medium.
  • For spheres, the magnitude is given as:

  • [ F_\text{res} = C_1\, r\, v + C_2\, r^2\, v^2 ]

  • (C_1) has units (\mathrm{kg/(m\cdot s)})

  • (C_2) has units (\mathrm{kg/m^3}) (dimensionally similar to density)

The two contributions

• Viscous (linear) term: (\;C_1 r v)
  • Related to the medium’s stickiness / viscosity
  • Strongly temperature-dependent (e.g., heating tar reduces viscosity → lowers (C_1))
• Pressure (quadratic) term: (\;C_2 r^2 v^2)
  • Involves (v^2) (with the underlying motivation deferred to a later “transfer of momentum” topic)
  • The (r^2) scaling arises from the sphere’s cross-sectional area (\sim r^2)

3) Terminal velocity and critical velocity

• Consider an object falling under gravity:
  • Gravity: (\;mg)
  • Drag: (\;C_1 r v + C_2 r^2 v^2)

• As speed increases, drag increases until terminal velocity is reached (acceleration becomes zero):

  • [ mg = C_1 r v + C_2 r^2 v^2 ]

• If (m), (r), (C_1), and (C_2) are known, terminal velocity can be found (it becomes a quadratic equation in (v), with one nonphysical solution discarded).

Critical velocity

• Define a critical velocity where viscous and pressure terms are equal:

  • [ C_1 r v_\text{crit} = C_2 r^2 v_\text{crit}^2 ]

• Solving gives:

  • [ v_\text{crit} = \frac{C_1}{C_2 r} ]

4) Two drag regimes (scaling laws with radius)

Let:

• Regime 1: linear term dominates
• Regime 2: quadratic term dominates

Regime 1 (viscous-dominated)

• Condition: (\;v \ll v_\text{crit})

• Terminal velocity:

  • [ v_t = \frac{mg}{C_1 r} ]

• Using (m \propto r^3) for objects of the same density:

  • [ v_t \propto r^2 ]

Regime 2 (pressure-dominated)

• Condition: (\;v \gg v_\text{crit})

• Terminal velocity:

  • [ v_t = \sqrt{\frac{mg}{C_2 r^2}} ]

• With (m \propto r^3):

  • [ v_t \propto \sqrt{r} ]

5) Key empirical/numerical demonstrations & examples

Ball bearings in syrup (regime 1)

• Demonstration: drop steel ball bearings of known radii into sticky Karo light corn syrup (a viscous medium).
• Used multiple bearing diameters (examples: (1/8) in, (5/32) in, (3/16) in, (1/4) in).
• Observational/log-plot confirmation:
  • At low speeds relative to the critical speed, behavior matches regime 1 scaling:
    • terminal speed proportional to (r^2)
  • On a plot of (100/d^2) vs time, the data align approximately linearly (consistent with the predicted dependence).
• Critical-speed meaning:
  • Even if the terminal speed is far below (v_\text{crit}) in syrup, (v_\text{crit}) marks the crossover scale between linear and quadratic drag.
  • For small balls, (v_\text{crit}) can be extremely high (hundreds of m/s), so syrup remains viscous-dominated in typical conditions.

Times to approach terminal velocity

• In viscous regime (regime 1), there is a differential equation for speed vs time; the speed approaches terminal velocity asymptotically.
• Example: for the quarter-inch ball bearing, the time to reach about 99% terminal speed is about 9 ms (after emerging from the syrup surface film).

Air examples (mostly regime 2)

• Approximate air parameters at 1 atm and room temperature:

  • [ C_1 \approx 3.1 \times 10^{-4} ]

  • [ C_2 \approx 0.85 ]

• Therefore:

  • (v_\text{crit}) in air scales roughly like (\;v_\text{crit} \propto 1/r)
  • (v_\text{crit}) is much lower than in syrup

• Conclusion: nearly all macroscopic falling spheres in air operate in regime 2, giving:

  • [ v_t \propto \sqrt{r} ]

• Terminal speed examples:

  • Pebble ((\sim)1 cm radius) dropped from high up: limited to ~75 mph
  • Human skydiver (approx. sphere radius (\sim 40) cm) without parachute: ~150 mph
  • Skydivers opening parachute late: ~120 mph (consistent with the estimate)
• Very small drops/crossover:
  • For sufficiently tiny droplets, motion can shift toward regime 1
  • The assignment suggests using oil drops to find the radius where the crossover occurs.

Large “balloon/ball” in air: regime 2 but transient buildup matters

• Estimated object: (\sim 34) g with radius (\sim 35) cm (about 70 cm across).
• Estimated terminal speed: ~1.8 m/s
• Time to reach terminal speed is not instantaneous; it requires numerical computation when both (v) and (v^2) terms matter.
• A graduate-student simulation is referenced.
• Estimated experimental drop from 3 m produced touchdown around ~2.0 s, matching transient dynamics.

Empire State Building pebble drop

• Pebble radius: (\sim 1) cm
• Drop height: 475 m
• Predicted terminal speed in air: ~75 mph
• Claimed no-drag speed would have been much higher (~225 mph).
• Emphasis: drag substantially limits speed even for relatively “small” objects dropped from great height.

Drag correction for an “apple drop” gravity measurement

• Apple parameters:
  • Mass (\sim 134) g, radius (\sim 3) cm (nearly spherical)
• Air drag effect estimated:
  • touchdown delayed by about ~2 ms from a 3 m drop
• Discussion: whether this delay lies within measurement uncertainty for determining (g).

6) Drag alters trajectories (not just vertical motion)

• With air drag, the trajectory becomes asymmetric, unlike ideal no-drag projectile motion.
• Horizontal and vertical components both reduce because drag opposes motion in the direction of the velocity vector.
• Demonstration with balls:
  • Tennis ball vs Styrofoam ball (similar radius → similar drag magnitude due to the (r^2) scaling)
  • Mass differs significantly:
    • larger mass → less deceleration for the same force ((F=ma))
• Result:
  • Tennis ball shows near-parabolic symmetry
  • Styrofoam shows noticeable asymmetry due to greater deceleration from drag

7) Final conceptual question posed

• Compare time intervals for a projectile between labeled points:
  • time from O to P vs time from P to S (with drag present)
• Key idea: drag generally makes ascent and descent times not equal, but the specific comparison is left as a confident-answer exercise.

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Methodologies / steps outlined (experimental & analysis approach)

• Model setup

  • Write drag as: [ F_\text{res} = C_1 r v + C_2 r^2 v^2 ] (opposite velocity)

  • At terminal speed, use force balance: [ mg = C_1 r v + C_2 r^2 v^2 ]

  • Define crossover speed: [ v_\text{crit} = \frac{C_1}{C_2 r} ]

• Regime selection

  • Use (v \ll v_\text{crit}) → regime 1 → (v_t \propto r^2)
  • Use (v \gg v_\text{crit}) → regime 2 → (v_t \propto \sqrt{r})
• Syrup experiment
  • Drop ball bearings into syrup with marked positions.
  • Start timing when the ball crosses one mark; stop when it crosses another.
  • Use travel time to infer terminal-velocity scaling with radius/diameter.
  • Check that the plotted data matches the predicted linear relation for regime 1.

• Transient-time calculation

  • Use Newton’s second law with drag: [ m\,\frac{dv}{dt} = mg - (C_1 r v + C_2 r^2 v^2) ]

  • Solve analytically in simplified limits (noted as an assignment hint) or numerically when both terms matter (done by a graduate student).

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Researchers or sources featured

• Walter Lewin (course lecturer; also performed/led the demonstrations)
• Dave Pooley (graduate student/instructor who performed numerical solutions and prepared plots)
• Referenced skydiving article (mentioned qualitatively; no author named)