Cut string

Problem

A stone rotates tied to a string. If you cut the string, in which direction does the stone fly off? Debunk the common feeling that it “flies outward”.

When the string is cut, the stone continues along the tangent (direction of $\vec v$), not along the radius.

Solution

What the string was doing. During the circular motion the string exerted the centripetal force: it continuously pulled the stone towards the centre $O$, forcing it to curve. Without that force the stone has no reason to curve.

Newton’s first law. At the instant you cut the string, no net force acts on the stone any more (neglecting gravity in the horizontal plane). By the principle of inertia, a free body continues in uniform rectilinear motion, i.e. in a straight line with the velocity it had at that instant.

Direction of escape. The velocity $\vec v$ in circular motion is always tangent to the trajectory. So the stone sets off along the tangent to the circle at the point where it was, not along the radius outward: $\vec v \perp \vec r \quad (\text{tangent to the circle})$

Why it seems to “go outward”. Watching from outside, the stone progressively moves away from the centre — but this is due to rectilinear motion along the tangent, not to a radial push. The “centrifugal force” that seems to fling it outward is only an apparent effect: in reality there is no longer any force, and the stone obeys inertia.

$\ev{\text{The stone flies off along the tangent to the orbit (direction of } \vec v\text{), by the first law.}}$

Linked items

Topics: Dinamica Concepts: Moto circolare uniforme · Forza centripeta Objects: Fune ideale