Kvant Physics Problem 186

A passenger of mass $m$ moves along a curved path of radius $R$ with speed $v$ during a turn.

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Verified: yes
Verdicts: PASS + PASS
Solve time: 2m22s
Source on kvant.digital

Problem

When a bus or car turns, passengers are thrown to the side opposite the direction of the turn. At the same time, an airplane’s turn is not felt at all by its passengers. Explain this difference.

I. Sh. Slobodetsky

Setup and Assumptions

A passenger of mass $m$ moves along a curved path of radius $R$ with speed $v$ during a turn. The motion is considered in an inertial reference frame fixed to the Earth.

For a road vehicle, the car is assumed to turn on a flat horizontal road, so the centripetal acceleration must be provided entirely by lateral contact forces transmitted through the seat and friction between tires and road.

For an airplane, the turn is modeled as a coordinated banked turn in steady flight, with constant speed $v$ and constant turn radius $R$. The lift force generated by the wings is assumed to act perpendicular to the wing plane and is tilted by a bank angle $\theta$ relative to the vertical. The passenger is assumed to be rigidly attached to the aircraft cabin so that only forces transmitted through the seat and cabin structure act on them. Air resistance and transient maneuvers are neglected.

The unknown is the physical reason for the different sensation of lateral force in the two cases, which corresponds to the presence or absence of a horizontal component of the contact force acting on the passenger.

Physical Principles

The motion in a circular path requires a centripetal acceleration

$a_c = \frac{v^2}{R}.$

Newton’s second law in an inertial frame gives

$\sum \mathbf{F} = m\mathbf{a}.$

For the passenger, the apparent sideways force corresponds to the horizontal component of the contact force from the seat or cabin, which must supply the required centripetal acceleration when no other force provides it.

For an airplane in a banked turn, the lift force $L$ is tilted, so its components satisfy

$L\cos\theta = mg,$

$L\sin\theta = \frac{mv^2}{R}.$

Derivation

For a car turning on a flat road, the only significant horizontal force acting on the passenger is the lateral force $F_{\text{lat}}$ transmitted through the seat. Applying Newton’s second law in the radial direction toward the center of curvature gives

$F_{\text{lat}} = m\frac{v^2}{R}.$

This force acts perpendicular to the velocity of the car in the horizontal plane. In the car’s accelerating frame this force is felt as an outward push due to inertia, producing the sensation of being thrown outward from the turn.

For an airplane in a steady banked turn, the passenger experiences the gravitational force $mg$ downward and the normal force from the seat, which is aligned with the resultant of gravity and lift. The lift force has components in the vertical and horizontal directions. Resolving the lift,

$L\cos\theta = mg,$

ensures no vertical acceleration, and

$L\sin\theta = m\frac{v^2}{R},$

ensures that the horizontal component of lift provides exactly the required centripetal acceleration.

The resultant of gravity and inertial effects is aligned with the cabin floor when the aircraft is correctly banked, so the contact force from the seat has no horizontal component in the passenger’s frame. The passenger therefore experiences only a normal force perpendicular to the seat surface, with no sideward imbalance requiring correction by the body.

Result

For a car on a flat turn,

$F_{\text{side}} = m\frac{v^2}{R}.$

For a correctly banked airplane turn,

$L\sin\theta = m\frac{v^2}{R}, \quad L\cos\theta = mg,$

which implies

$\tan\theta = \frac{v^2}{Rg}.$

In this case the net contact force on the passenger is aligned with the cabin floor, so the sideways force component vanishes in the passenger’s frame:

$F_{\text{side, airplane}} = 0 \quad \text{(in steady coordinated turn)}.$

Sanity Checks

The expression $m v^2 / R$ has units $[\mathrm{kg} \cdot \mathrm{m}^2 \mathrm{s}^{-2} / \mathrm{m}] = \mathrm{N}$, consistent with a force.

In the limit $R \to \infty$, both systems approach straight-line motion and the lateral force tends to zero, matching $F_{\text{side}} \to 0$ for both car and airplane.

For increasing speed $v$, the required centripetal force grows as $v^2$, so a car produces rapidly increasing sideways load on passengers when turning sharply, while an airplane compensates by increasing the bank angle $\theta$ so that the lift vector tilts and maintains alignment of the resultant force with the cabin floor.

A sign error would most likely occur in assigning the horizontal component of lift or in mixing inertial and non-inertial frames, since both describe the same physics but distribute the apparent forces differently between real forces and fictitious ones.