The angular velocity of the rolling cylinder is expressed as .
Further Explanation:
Since the cylinder rolls along with the string which is wrap around it. There is a tension developed in the string due to weight of the cylinder and its acceleration.
The tension developed in the string is given as:
This tension developed in the string produces a torque about the center which makes the cylinder to roll. The torque developed about center is expressed as:
The torque developed in a rotating body is also expressed as:
Here, is the moment of inertia of the cylinder and is the angular acceleration of the cylinder.
The moment of inertia of a cylinder is and the angular acceleration for a acceleration body can also be expressed as .
Thus comparing the expressions of torque:
The linear velocity of the cylinder after time will be:
The angular velocity of a rotating body is expressed as:
Thus, the angular velocity of the rolling cylinder is expressed as .
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Answer Details:
Grade: High School
Subject: Physics
Chapter: Rotational Motion
Keywords: String constrains, rotational motion, translational motion, angular velocity, falling cylinder, torque, center of mass of cylinder, linear speed of center of mass.