Sci-fi physics: zero gravity on a curve

Problem

A car takes a curve at $60\;\mathrm{km/h}$. What happens if gravity suddenly vanishes? Discuss the causal chain "$g \to N \to F_a \to a_c$".

Solution

On a flat curve, a curved trajectory is possible only if something provides the centripetal force $\vv F_c$ directed towards the centre. That “something” is the static friction between the tyres and the asphalt. Let’s follow the causal chain, link by link.

1) $g \to N$. Vertical equilibrium requires the normal force to balance the weight: $N = mg$. It is gravity that “presses” the car against the asphalt.

2) $N \to F_a$. The available static friction is limited by the normal force: $F_a \le \mu_s N = \mu_s m g$. The stronger the normal force, the more friction can be exploited.

3) $F_a \to a_c$. Friction is the only horizontal force, so it is what produces the centripetal acceleration: $F_a = m\, a_c = \frac{m v^2}{r}$

Now let’s make $g$ vanish. Then $N = mg \to 0$: without weight the car no longer presses on the asphalt. Consequently the maximum friction $\mu_s N \to 0$, so $F_a \to 0$. Without a horizontal force there is no longer a centripetal force: $a_c \to 0$.

By the first law of dynamics, in the absence of forces the car continues in uniform rectilinear motion, i.e. tangentially to the curve, leaving the track. Moreover, without weight, it would tend to “float”. All the ability to turn ultimately rests on gravity: friction on a curve rests on the normal force, which rests on the weight.

$\ev{\text{Without } g:\; N = 0 \Rightarrow F_a = 0 \Rightarrow a_c = 0 \;\Rightarrow\; \text{straight tangential trajectory}}$

Linked items

Topics: Dinamica · Attrito Concepts: Forza normale · Attrito statico · Forza centripeta · Forza peso Skills: Analisi di casi limite e fantafisica