Λdisc of circumference s is at rest at a point Λon a horizontal surface when a constant horizont

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Λdisc of circumference s is at rest at a point Λon a...

$\Lambda$ disc of circumference $s$ is at rest at a point $\Lambda$ on a horizontal surface when a constant horizontal force begins to act on its centre. Between $\Lambda$ and $B$ there is sufficient friction to prevent slipping and the surface is smooth to the right of $B \cdot A B=s$, the disc moves from $\Lambda$ to $B$ in time $T$. To the right of $B$ (a) the angular acceleration of the disc will disappear, linear acceleration will remain unchanged. (b) linear acceleration of the disc will increase. (c) the disc will make one rotation in time $T / 2$. (d) the disc will cover a distance greater than $s$ in further time $T$.

$\Lambda$ disc of circumference $s$ is at rest at a point $\Lambda$ on a horizontal surface when a constant horizontal force begins to act on its centre. Between $\Lambda$ and $B$ there is sufficient friction to prevent slipping and the surface is smooth to the right of $B \cdot A B=s$, the disc moves from $\Lambda$ to $B$ in time $T$. To the right of $B$
(a) the angular acceleration of the disc will disappear, linear acceleration will remain unchanged.
(b) linear acceleration of the disc will increase.
(c) the disc will make one rotation in time $T / 2$.
(d) the disc will cover a distance greater than $s$ in further time $T$.

Key Concepts

Angular and Linear Acceleration Differences in Decoupled Motion

Under conditions where the friction that enforces the no-slip rolling condition is absent, the disc's centre of mass continues to accelerate under the external force, but the rotational motion remains at the angular velocity achieved up to that point. This decoupling highlights that while the linear acceleration stays constant, the angular acceleration falls to zero in the absence of additional torque.

Rolling Motion Dynamics

This concept involves the simultaneous translation of the centre of mass and rotation about the centre. When a force is applied to a rolling object, both its linear velocity (centre-of-mass motion) and its angular velocity (rotational motion) change, and the relationship between them depends on the rolling condition and friction present.

Rolling Without Slipping Condition

This condition, typically expressed as v = ?R for a disc (where v is the linear speed, ? is the angular speed, and R is the radius), ensures that the disc is not skidding on the surface. Friction acts to enforce this condition when it is available, linking the translational and rotational accelerations under the right circumstances.

Role of Friction in Rolling

Friction is essential in rolling motion because it provides the torque needed to start or change the rate of rotation while preventing slipping. When sufficient friction is present, as in the region between points ? and B, the disk rolls without slipping, meaning the angular acceleration is directly related to the linear acceleration.

Transition from Combined Rolling to Partial Slip or Slip-Free Translation

When the disc leaves the region with sufficient friction, the constraint of rolling without slipping is removed. The absence of frictional torque means the disc's angular acceleration disappears while the constant horizontal force continues to cause linear acceleration, thus decoupling the linear and rotational motions.

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