A heavy disc with radius Road is rolling down hanging on two non-streched string wound around the disc the disc very tightly. The free ends of the string are always tensed during the motion. At the dance instant the angular velocity of the disc is w, and the angle between the string is x. Find the velocity of the centre of mass of the disc at this moment

To find the velocity of the center of mass of the disc as it rolls down while being suspended by two strings, we need to analyze the relationship between the angular velocity of the disc and its linear velocity. This scenario involves both rotational and translational motion, so let's break it down step by step.

Understanding the Setup

We have a disc with a radius denoted as R that is rolling down while being held by two strings. The strings are wound tightly around the disc, and the angle between the strings is x. The disc has an angular velocity w at a given moment.

Key Relationships

In this situation, the linear velocity v of the center of mass of the disc is related to its angular velocity w through the following equation:

• v = R * w

This equation holds true for rolling motion without slipping, where the point of contact between the disc and the surface is momentarily at rest.

Considering the Geometry

The angle x between the two strings affects how the forces are distributed but does not directly change the relationship between the angular and linear velocities. However, it can influence the tension in the strings and the dynamics of the system. For our calculation of the velocity of the center of mass, we will focus on the linear relationship derived from the angular velocity.

Calculating the Velocity

Given that the disc is rolling down, we can express the velocity of the center of mass as:

• v = R * w

Here, R is the radius of the disc, and w is the angular velocity at that instant. This equation shows that the linear velocity of the center of mass is directly proportional to both the radius of the disc and its angular velocity.

Final Thoughts

In summary, the velocity of the center of mass of the disc at the moment when its angular velocity is w and the angle between the strings is x can be expressed simply as: