Kvant Physics Problem 186
A passenger is modeled as a point mass $m$ moving along a curved trajectory of radius $R$ at speed $v$ in a horizontal plane for a car or bus, and along a banked circular path for an airplane.
Verified: no
Verdicts: SKIP + SKIP
Solve time: 1m26s
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, a turn of an airplane is not felt by its passengers. Explain this difference.
I. Sh. Slobodetsky
Setup and Assumptions
A passenger is modeled as a point mass $m$ moving along a curved trajectory of radius $R$ at speed $v$ in a horizontal plane for a car or bus, and along a banked circular path for an airplane. The analysis is performed in an inertial reference frame attached to the Earth, neglecting air drag variations, suspension compliance in the car, and compressibility effects of air. The Earth’s rotation is neglected. The unknown to be explained is the presence or absence of a lateral force sensation on passengers during a turn.
Physical Principles
Newton’s second law in an inertial frame is written as $m\mathbf{a} = \sum \mathbf{F}$.
Uniform circular motion requires a centripetal acceleration of magnitude $a_c = \dfrac{v^2}{R}$ directed toward the center of curvature.
A passenger feels the normal reaction force from the seat, and the sensation of being pushed sideways corresponds to a horizontal component of the contact force acting on the body in the passenger’s frame.
In an airplane executing a coordinated turn, the lift force $\mathbf{L}$ is tilted by a bank angle $\theta$, with components $L\cos\theta$ vertically and $L\sin\theta$ horizontally.
Derivation
For a car or bus turning on a flat road, the required centripetal acceleration $a_c = \dfrac{v^2}{R}$ must be provided by static friction between the tires and the road. In the inertial frame, the passenger must share this acceleration, so the net horizontal force on the passenger satisfies $m \dfrac{v^2}{R}$ directed toward the center of the turn.
The seat exerts a normal force on the passenger, but in a non-inertial frame attached to the vehicle the passenger experiences an effective inertial force $m \dfrac{v^2}{R}$ directed outward. This force is not aligned with the seat normal, producing a lateral sensation pressing the passenger toward the outer side of the turn.
For an airplane in a coordinated turn, the lift force is tilted by an angle $\theta$ such that its horizontal component supplies the centripetal acceleration,
$$L \sin\theta = m \frac{v^2}{R}.$$
The vertical component balances weight,
$$L \cos\theta = mg.$$
Dividing the two equations gives
$$\tan\theta = \frac{v^2}{Rg}.$$
The resultant of lift and weight is aligned with the aircraft’s floor, so the contact force between seat and passenger has no horizontal component in the cabin frame. The passenger experiences a change in effective weight magnitude but no sideways imbalance, eliminating the sensation of being pushed laterally.
Result
For a car or bus on a flat turn, the effective lateral force in the passenger frame is
$$F_{\text{side}} = m \frac{v^2}{R}.$$
For an airplane in a coordinated banked turn, the lift components satisfy
$$L \sin\theta = m \frac{v^2}{R}, \quad L \cos\theta = mg, \quad \tan\theta = \frac{v^2}{Rg}.$$
In the airplane case the horizontal force in the cabin frame is zero, while the effective load factor is
$$n = \frac{L}{mg} = \frac{1}{\cos\theta}.$$
No numerical substitution is required because no specific values of $v$, $R$, or $\theta$ are provided.
Sanity Checks
The expression $m \dfrac{v^2}{R}$ has units $,\text{kg}\cdot \text{m}/\text{s}^2$, matching force units of newtons, confirming dimensional consistency. In the limit $R \to \infty$, both the car and airplane require zero centripetal acceleration, eliminating any lateral effect. In the limit of large speed at fixed radius, $v^2/R$ grows, increasing the required friction force in a car and increasing the bank angle in an airplane, consistent with stronger apparent effects in ground vehicles and steeper banking in aircraft.
A sign error would most easily occur in assigning the direction of the effective inertial force in the car frame, which must point outward from the center of curvature, opposite to the required centripetal acceleration in the inertial frame.