Estimate of the centripetal force on a passenger

Problem

Estimate the centripetal force acting on a passenger sitting in a car taking a bend of radius $30$ m at $50$ km/h. Compare it with their weight.

Solution

Data and assumptions. Bend of radius $r \approx 30$ m, speed $v = 50$ km/h. For the passenger’s mass we assume a typical value $m \approx 70$ kg.

Speed conversion. $v = \frac{50}{3{,}6} \approx 13{,}9\ \mathrm{m/s}$

Centripetal acceleration. $a_c = \frac{v^2}{r} = \frac{(13{,}9)^2}{30} \approx \frac{193}{30} \approx 6{,}4\ \mathrm{m/s}^2$

Centripetal force. This is the horizontal resultant that the seat (and the seatbelt) must exert on the passenger for them to curve together with the car: $F_c = m\,a_c \approx 70 \cdot 6{,}4 \approx 4{,}5\cdot 10^2\ \mathrm{N}$

$\ev{F_c \sim 4{,}5\cdot 10^2\ \mathrm{N} \ (\text{of the order of a few hundred newtons})}$

Comparison with the weight. The passenger’s weight is $P = mg \approx 70 \cdot 9{,}81 \approx 6{,}9\cdot 10^2\ \mathrm{N} \approx 690$ N. The centripetal force is thus of the same order of magnitude as the weight (about two-thirds). Since this is an estimate, the message is clear: on a tight, fast bend the seat must push the body sideways with a force comparable to that with which the floor supports it vertically. That is why on a bend you feel “pushed sideways”.

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Topics: Dinamica Concepts: Forza centripeta · Accelerazione centripeta · Forza peso Skills: Stime di Fermi