Q.If, in Exercise 4.21, the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly describes the trajectory of the stone after the string breaks:

Concept: Newton's First Law and Centripetal Force

When the stone moves in a circle, the tension in the string provides the centripetal force that continuously changes the direction of the velocity vector. The velocity at any instant is tangent to the circular path.

The moment the string breaks, the centripetal force vanishes. By Newton's first law, a body continues in its state of uniform motion in a straight line when no net force acts on it. Since the velocity was tangential at the breaking instant, the stone continues moving in that tangential direction. …

Why does the stone fly off tangentially?

While the string is intact, it pulls the stone radially inward, providing the centripetal force that keeps the stone moving in a circle. The stone's velocity at every instant is tangent to the circular path — velocity is always in the direction of motion, and circular motion is motion along the circumference.

The moment the string breaks, the centripetal force disappears. No force acts on the stone (ignoring gravity for horizontal circular motion, or considering the instant before gravity becomes significant). Newton's first law tells us that an object with no net force continues in a straight line with constant velocity. The velocity the stone "inherits" at the breaking instant is precisely the tangential velocity it had while moving in the circle.

There is no radial component of velocity. The stone was never moving radially outward or inward — it was always moving tangent to the circle. The centripetal force changed the direction of that tangential velocity continuously, bending the path into a circle. Remove the force, and the bending stops; the stone proceeds straight ahead along the tangent.

Step-by-step reasoning

1. Identify the velocity at the instant of breaking.

   At any point on the circular path, the stone's velocity $\vec{v}$ is tangent to the circle. Its magnitude is $v$ (the speed), and its direction is perpendicular to the radius at that point.

2. Recognize what the centripetal force does.

   The tension $T$ in the string provides centripetal acceleration $a_c = \frac{v^2}{r}$ directed radially inward. This acceleration changes the direction of $\vec{v}$ continuously, keeping the stone on the circular path. Crucially, centripetal force does not change the speed (it is perpendicular to velocity), only the direction.

3. Apply Newton's first law when the string breaks.

   The instant the string snaps, $T = 0$, so the net force becomes zero (in the idealized horizontal case). With no force to change its velocity, the stone continues with the velocity it had at that instant: magnitude $v$, direction tangent to the circle.

4. Evaluate the options. …

Concept: Newton's First Law Applied at the Instant a Centripetal Force Vanishes

Step 1: Identify the stone's velocity direction during circular motion

At every instant, the stone's velocity is tangent to the circle — the string's tension

only ever changes the direction of this velocity (providing centripetal

acceleration), never adds a radial component.

Step 2: Apply Newton's first law the instant the string breaks

Once $T=0$, no net horizontal force acts on the stone. By the first law it continues

with the velocity it had at that instant — tangential, unchanged in magnitude or

direction, no radial component whatsoever.

Step 3: Eliminate the wrong options …