How Does Tension Affect Acceleration in a Falling Cylinder?

AI Thread Summary
The discussion focuses on analyzing the dynamics of a falling cylinder connected to a weightless rope. The tension in the string and the acceleration of the cylinder are derived from the equations of motion, considering both linear and rotational dynamics. The tension T acts tangentially at the radius R, creating a torque that influences the cylinder's angular acceleration. The relationship between linear acceleration and angular acceleration is crucial for solving the problem. Understanding the interplay of forces and torques is essential for determining the tension and acceleration accurately.
bluejay
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one end of a weightless rope is tied to the ceiling of a building and the other end is wrapped around a uniform solid cylinder that has a radius R and mass M. The cylinder is then released and falls toward the floor. The moment of inertia of a solid cylinder about an axis through its center of mass is: I=0.5MR^2
a. find the tension T in the string?
b. what is the acceleration of M?

Homework Statement


I=0.5MR^2

Homework Equations


T-mg=-ma

The Attempt at a Solution

 
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bluejay said:

Homework Statement


I=0.5MR^2


Homework Equations


T-mg=-ma

You have a start with this, in that you have an equation for the net force on the cylinder as it falls. But this by itself won't get us far because we also need to look at where on the cylinder the forces are acting: this is necessary because the cylinder is rotating as it falls, so there is an additional relationship between T and mg that we will be able to find that will let us solve for all the quantities.

Where do we treat the weight mg as acting on the body of the cylinder? How does the tension T in the string act on the cylinder, and where?
 
I have no idea.
 
You are going to need to review how torques are worked out, because that is the only way you'll be able to solve problems involving rotation.

The linear acceleration a applies to the center of the cylinder; the weight force mg effectively acts there. Since the string is wound around the cylinder, the tension acts along a tangent to the cylinder, so the force T is acting at a distance R (the radius of the cylinder) from the center, about which the cylinder will rotate. The tangent to the cylinder is perpendicular to the radius, so the tension T acts at right angle to that radius. So what torque is the tension applying about the central axis of the cylinder? How does this torque relate to the angular acceleration of the cylinder? (That is where we are going to connect up to the force equation you already wrote.)
 
Never mind i got it. Thanks
 

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