Since the cylinder moves without sliding, the centre of the...

Since the cylinder moves without sliding, the centre of the cylinder rotates about the point $O$, while passing through the common edge of the planes. In other words, the point $O$ becomes the foot of the instantaneous axis of rotation of the cylinder. It at any instant during this motion the velocity of the C.M. is $v_{1}$ when the angle (shown in the figure) is $\beta$, we have $$ \frac{m v_{1}^{2}}{R}=m g \cos \beta-N $$

Key Concepts

Force Analysis in Rotational Motion

This concept involves understanding and resolving the forces acting on a rotating body. Here, it includes accounting for gravitational forces, normal reaction, and the resultant net force that provides the centripetal acceleration. A detailed force analysis is essential to set up the proper equations that relate these forces to the motion parameters like velocity and angle, as seen in the problem equation.

Rolling Motion with No-Slip Condition

This concept describes the motion of an object where its rotational motion is synchronized with its translational motion such that there is no sliding at the point of contact with the surface. In the problem, the no-slip condition ensures that the velocity of the cylinder’s center is directly related to its angular velocity, leading to relationships crucial for determining forces like the centripetal force.

Centripetal Force

Centripetal force is the force required to keep an object moving in a circular path and is directed towards the center of the circle. In this case, the expression m*v?²/R represents the necessary centripetal force acting on the cylinder's center of mass as it follows a curved path defined by its instantaneous rotation about point O.

Rigid Body Dynamics

This concept involves the study of the motion of bodies assumed to be non-deformable. In the context of the cylinder, it refers to analyzing how a solid object rotates and translates as a whole without any internal deformations, which is fundamental in solving problems involving complex motions that include both rotational and translational components.

Instantaneous Center of Rotation

The instantaneous center of rotation is the point about which a body appears to rotate at a given instant. For the cylinder, even though the center of mass is moving, there exists a point (denoted as O) that acts as the pivot for the rotational motion at that moment. This concept simplifies the analysis of rotational movement by allowing the use of pure rotation principles at the instant considered.

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