The bobbin with thread around it lies on a horizont floor...

The bobbin with thread around it lies on a horizont floor and can roll along it without slipping. Both pulle $P_{1}$ and $P_{2}$ are light and frictionless. The pulley $P_{1}$ mov downwards with constant velocity $v_{0}$. The velocity centre of mass of bobbin is : (a) $\frac{v_{0} R}{R-r}$ in forward direction (b) $\frac{2 v_{0} R}{R-r}$ in backward direction (c) $\frac{v_{0} R}{R-r}$ in backward direction (d) $\frac{2 v_{0} R}{R-r}$ in forward direction

The bobbin with thread around it lies on a horizont floor and can roll along it without slipping. Both pulle $P_{1}$ and $P_{2}$ are light and frictionless. The pulley $P_{1}$ mov downwards with constant velocity $v_{0}$. The velocity centre of mass of bobbin is :
(a) $\frac{v_{0} R}{R-r}$ in forward direction
(b) $\frac{2 v_{0} R}{R-r}$ in backward direction
(c) $\frac{v_{0} R}{R-r}$ in backward direction
(d) $\frac{2 v_{0} R}{R-r}$ in forward direction

Key Concepts

Center of Mass Kinematics

The center of mass of an object is the weighted average of the positions of all its mass elements and follows a trajectory that reflects the overall motion of the body. In mechanics, determining the velocity of the center of mass is often essential for understanding the motion of a system, particularly when multiple constraints and forces are involved. In problems with connected objects, the kinematics of the center of mass can provide insights into the net effect of the various motions created by the constraints.

Dynamics of Frictionless Pulleys

Frictionless pulleys are idealized systems where the pulleys do not add any frictional resistance to the motion of a string running over them. In such systems, the tension in the string remains constant over the pulley and the pulleys primarily change the direction of the forces involved. This simplification is used to derive kinematic relations between different parts of the system without having to account for energy losses or additional forces.

Rolling Without Slipping

Rolling without slipping is a constraint where the point of contact between a rolling object and the surface is momentarily at rest relative to the surface. This condition links the translational velocity of the center of mass and the object’s rotational velocity, ensuring that the motion is purely rolling without any sliding. This relationship is crucial in problems where rolling motion and friction are involved, such as calculating the velocity of a rolling body.

Inextensible String Constraints

An inextensible string does not change its length, meaning that the motion of one part of the system is geometrically linked to the motion of another. This yields constraint equations connecting the velocities and accelerations of various elements in the system. In problems with pulleys, these constraints are key to relating the velocity of one moving part to another, which is essential for finding unknown velocities or accelerations.

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